Image Reconstruction and Discrimination at Low Light Levels by Petros Zerom

Image Reconstruction and Discrimination

at Low Light Levels

by

Petros Zerom

Submitted in Partial Fulfillment of

the Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Robert W. Boyd

The Institute of OpticsArts, Sciences and Engineering

Edmund A. Hajim School of Engineering and Applied Sciences

University of RochesterRochester, New York

2013

ii

Dedicated To My Parents

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Biographical Sketch

The author was born in Asmara, Eritrea. In 1997 he obtained a Bachelors of

Science in Physics from the University of Asmara, graduating with great distinction.

In the fall of 2000, he joined the Masters program in Physics at Washington State

University in Pullman, Washington and obtained his M.Sc. degree in 2002. He con-

tinued his graduate studies at the Institute of Optics at the University of Rochester.

His doctoral research in nonlinear and quantum optics was supervised by Professor

Robert W. Boyd.

Publications

1. P. Zerom, Z. Shi, M. N. O’Sullivan, K. W. C. Chan, M. Krogstad, J. H.Shapiro, and R. W. Boyd, “Thermal ghost imaging with averaged specklepatterns,” Phys. Rev. A 86, 063817 (2012).

2. P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photoncompressive ghost imaging,” Phys. Rev. A 84, 061804(R) (2011).

3. C. J. Broadbent, P. Zerom, H. Shin, J. C. Howell, and R. W. Boyd,“Discriminating orthogonal single-photon images,” Phys. Rev. A 79, 033802(2009). [March 2009 issue of Virtual Journal of Quantum Information]

4. M. Malik, H. Shin, M. O’Sullivan, P. Zerom, and R. W. Boyd, “Quantumghost image identification with correlated photon pairs,” Phys. Rev. Lett.104, 163602 (2010). [May 2010 issue of Virtual Journal of QuantumInformation]

5. P. Zerom and R. W. Boyd, “Self-focusing, conical emission, and otherself-action effects in atomic vapors,” (book chapter) in Self-focusing: Past andPresent - Fundamental and Prospects, Eds. R. W. Boyd, S. G. Lukishova andY. Shen, Topics In Applied Physics, Volume 114, 231-251 (2009)

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6. R. W. Boyd, N. N. Lepeshkin, and P. Zerom, “Slow light in a collection ofcollisionally broadened two-level atoms,” Laser Physics, 15, 1389 (2005)

7. G. S. He, C. G. Lu, Q. D. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, andM. Samoc, “Stimulated Rayleigh-Bragg scattering in two-photon absorbingmedia,” Phys. Rev. A. 71, 063810 (2005)

8. M. S. Bigelow, P. Zerom, and R. W. Boyd, “Breakup of ring beams carryingorbital angular momentum in sodium vapor,” Phys. Rev. Lett., 92, 083902(2004)

Conference Papers

1. J. Howell, G. Howland, R. Boyd, P. Zerom, and J. Schneeloch, “Entropy,information and compressive sensing in the quantum domain,” in Research inOptical Sciences (Optical Society of America, 2012), p. QT4B.5.

2. G. Howland, P. Zerom, R. W. Boyd, and J. C. Howell, “Compressive sensingLIDAR for 3D imaging,” CLEO - Laser Applications to Photonic Applicationsp. CMG3 (2011).

3. P. Zerom, K. W. C. Chan, J. C. Howell, and R. Boyd, “Compressive quantumghost imaging,” in International Conference on Quantum Information(Optical Society of America, 2011), p. QTuF3.

4. P. Zerom, G. Piredda, R. Boyd, and J. Shapiro, “Optical coherencetomography based on intensity correlations of quasi-thermal light,”, inConference on Lasers and Electro-Optics/International Quantum ElectronicsConference (Optical Society of America, 2009), p. JWA48.

5. R. W. Boyd, K. W. C. Chan, A. Jha, M. Malik, C. O’Sullivan, H. Shin, and P.Zerom, “Quantum imaging: enhanced image formation using quantum statesof light,” in Proc. of SPIE Vol. 7342, 73420B (2009).

6. R. W. Boyd, G. M. Gehring, G. Piredda, A. Schweinsberg, Z. Shi, H. Shin, J.Vornehm, and P. Zerom, “Slow, fast, and backwards light propagation inerbium-doped optical fibers,” in Nonlinear Optics: Materials, Fundamentalsand Applications (Optical Society of America, 2007), p. WB1.

7. Y. Chen, Z. Shi, P. Zerom, and R. W. Boyd, “Slow light with gain induced bythree photon effect in strongly driven two-level atoms,” in Slow and Fast Light(Optical Society of America, 2006), p. ME1.

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8. Y. Chen, P. Zerom, Z. Shi, and R. W. Boyd, “Slow light using thethree-photon effect in a dressed two-level atomic system,” in Frontiers inOptics (Optical Society of America, 2006), p. JWD29.

9. R. W. Boyd, N. Lepeshkin, A. Schweinsberg, P. Zerom, G. Gehring, G.Piredda, Z. Shi, H. Shin and Q.-H. Park, “What are the limits to the timedelay achievable using ”slow-light” methods?,” in Proceedings of SPIE Vol.5924, 592402 (2005).

10. R. W. Boyd, M. S. Bigelow, N. Lepeshkin, A. Schweinsberg, and P. Zerom,“Ultraslow and superluminal light propagation in room temperature solids,”in Nonlinear Optics: Materials, Fundamentals and Applications (OpticalSociety of America, 2004), p. FA5.

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Acknowledgments

This thesis would not have come to fruition without the assistance of many people

whose contributions I would like to gratefully acknowledge here.

First of all, I would like to express my sincere appreciation of the help, both

scientific and otherwise, I have received through the years from my thesis supervisor

Prof. Robert W. Boyd, without whom this thesis would not have been possible.

I am thankful to Professors Nicholas Bigelow and Carlos R. Stroud for agreeing

to serve on my Ph. D. committee. I appreciate their feedback and comments.

I would like to thank current and former members of Prof. Boyd’s and Prof.

Stroud’s research groups for their constant support and collaborations. I would like

to specifically thank Dr. Matt Bigelow (for work on spatial solitons), Dr. Giovanni

Piredda, Colin O’Sullivan, Dr. Kam Wai Clifford Chan, Dr. Anand Jha (for many

discussions on quantum related subjects and for introducing me to the games squash

and badminton), Dr. George Gehring (for the coherence propagation work), Dr. Luke

Bissell (for single photon sources using NV centers), Dr. Zhimin Shi and Dr. Heedeuk

Shin for the wonderful collaborations and great discussions. I would like to thank

Colin O’Sullivan for constantly answering all my questions and his collaborations

on the ghost imaging work. I would like to thank Dr. Kam Wai Clifford Chan for

collaborations on the compressive sensing and speckle averaging projects. I would like

to thank Dr. Zhimin Shi for all his efforts and collaborations on the ghost imaging

and slow light related projects. I would be remiss if I don’t thank Dr. Heedeuk

vii

Shin for his collaborations on the image discrimination projects. I specially extend

my gratitude to Dr. Giovanni Piredda for collaborations on coherence tomography

work, for being a wonderful roommate for two years and for educating me about the

Italian culture. I would also like to thank Dr. Svetlana Lukishova for her help and

collaborations on the single photon sources projects.

I would like to thank all the staff of the Institute for their invaluable support and

for being available whenever I needed their help: Maria Schnitzler, Joan Christian,

Lissa Cotter, Noelene Votens, Betsy Benedict, Gina Kern, Lori Russell, Marie Banach,

and Per Adamson.

I would like to thank my sisters Harnet (and her family), Lidia and Senait for

their love and moral support. A special thanks goes to Tsega K. for all her love and

support. Finally, I would like to dedicate this thesis to my parents Hiwet Mosazghi

and Zerom Tesfayesus. I would be eternally grateful for your constant love, support

and encouragement.

viii

Abstract

Quantum imaging is a recent and promising branch of quantum optics that exploits

the quantum nature of light. Improving the limitations imposed by classical sources

of light in optical imaging techniques or overcoming the classical boundaries of image

formation is one of the key motivations in quantum imaging. In this thesis, I describe

certain aspects of both quantum and thermal ghost imaging and I also study image

discrimination with high fidelity at low light levels.

First of all, I present a theoretical and experimental study of entangled-photon

compressive ghost imaging. In quantum ghost imaging using entangled photon pairs,

the brightness of readily available sources is rather weak. The usual technique of

image acquisition in this imaging modality is to raster scan a single-pixel single-photon

sensitive detector in one arm of a ghost imaging setup. In most imaging modalities,

the number of measurements required to fully resolve an object is dependent on

the measurement’s Nyquist limit. In the first part of the thesis, I propose a ghost

imaging (GI) configuration that uses bucket detectors (as opposed to a raster scanning

detector) in both arms of the GI setup. High resolution image reconstruction using

only 27% of the measurement’s Nyquist limit using compressed sensing algorithms

are presented.

The second part of my thesis deals with thermal ghost imaging. Unlike in quantum

GI, bright and spatially correlated classical sources of radiation are used in thermal

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GI. Usually high-contrast speckle patterns are used as sources of the correlated beams

of radiation. I study the effect of the field statistics of the illuminating source on the

quality of ghost images. I show theoretically and experimentally that a thermal GI

setup can produce high quality images even when low-contrast (intensity-averaged)

speckle patterns are used as an illuminating source, as long as the collected signal is

mainly caused by the random fluctuation of the incident speckle field, as opposed to

other noise sources.

In addition, I describe transverse image discrimination and recognition using holo-

graphic matched filtering techniques using heralded single photons from a spontaneous

parametric downconversion source. Heralded single photons are used for encoding and

discriminating images from our predefined orthogonal basis set. Our basis set consti-

tutes two locally spatially orthogonal objects. We show that if the object is a member

of a predefined set, we can discriminate the objects in the set with high confidence

levels.

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Contributors and Funding Sources

This thesis is a result of a collaboration with many colleagues. All work is done

under the supervision of my thesis advisor Prof. R. W. Boyd. If no affiliation of a

collaborator is mentioned below, the University of Rochester is assumed.

The research in chapter 2 was performed in collaboration with Prof. R. W. Boyd,

Dr. K. W. C. Chan of Rochester Optical Manufacturing Company and Prof. John

C. Howell. Equation 2.12 in section 2.3.1 was derived by Dr. K. W. C. Chan. I

carried out all the experiments. I created all the figures, except Figures 2.3 and 2.4

(by Dr. K. W. C. Chan). I wrote the paper with help from Dr. Chan, Prof. Boyd and

Prof. Howell. Most of this work was published in Physical Review A 84, 061804(R)

(2011). This work was supported through a quantum imaging MURI grant and the

DARPA/ARO InPho grant.


Dr. Z. Shi, M. N. O’Sullivan, Dr. K. W. C. Chan of Rochester Optical Manufacturing

Company, M. Krogstad of the University of Colorado at Boulder and Prof. J. H.

Shapiro of Massachusetts Institute of Technology. I carried out all the experiments

with help from Dr. Z. Shi and M. N. O’Sullivan. I carried out all the analysis and

created all figures except for Figs. 3.1 and 3.2 (by Z. Shi). Equation 3.18 in section

3.2 was derived by M. N. O’Sullivan. I wrote the first draft of the paper. Z. Shi

took over after that, with input from the rest of the coauthors. Most of this work

was published in Physical Review A 86, 063817 (2012). This work was supported

xi

by the DARPA/DSO InPho Program and by the Canada Excellence Research Chairs

Program.


Dr. C. J. Broadbent, Dr. H. Shin and Prof. John C. Howell. Prof. R. W. Boyd

designed the project. I conducted the initial experiment using highly attenuated

coherent light in our lab, together with H. Shin. The experiment described in this

thesis was conducted using spontaneous parametric downconversion source in Prof.

Howell’s lab. The experiment was mainly carried out by Dr. C. J. Broadbent, with

help from both myself and H. Shin. I was also involved in developing the multiplexed

holograms used in the experiment. All analysis was carried out together with Dr.

C. J. Broadbent and Dr. H. Shin. I created all the figures with the help of Dr. H.

Shin and Prof. R. W. Boyd, except for Figs. 4.2 and 4.3 (created by Dr. C. J.

Broadbent). Prof. R. W. Boyd wrote the paper with help from the rest of coauthors.

C. J. Broadbent wrote the experimental part of the paper. Most of this work was

published in Physical Review A 79, 033802 (2009). Note that I am the second author

of the published paper. This work was supported by the U.S. Army Research Office

through a MURI grant.

Contents

Acknowledgments vii

Abstract viii

Contributors and Funding Sources x

List of Tables xv

List of Figures xx

1 Background 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Ghost Imaging–Introduction . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Compressive Sensing–Introduction . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Sparse and Compressible Signals . . . . . . . . . . . . . . . . 14

1.3.2 Incoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.3 Restricted Isometry Property . . . . . . . . . . . . . . . . . . 18

1.3.4 Signal Reconstruction Algorithms . . . . . . . . . . . . . . . . 20

xii

CONTENTS xiii

1.3.5 Sparse Signal Reconstruction – A Numerical Example . . . . . 21

1.4 Image recognition – Introduction . . . . . . . . . . . . . . . . . . . . 23

1.5 Summary and Outline of Thesis . . . . . . . . . . . . . . . . . . . . . 24

2 Compressive Quantum Ghost Imaging 26

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Single-Pixel Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Compressive Quantum Imaging: Theory . . . . . . . . . . . . . . . . 34

2.3.1 Entangled-Photon Compressive Ghost Imaging . . . . . . . . . 35

2.3.2 Single-Photon Single-Pixel Compressive Imaging . . . . . . . . 38

2.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Image Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6 Photon Efficiency Comparison . . . . . . . . . . . . . . . . . . . . . . 50

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Speckle Averaging Effects in Thermal Ghost Imaging 52

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.1 Fast Detection Speed . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.2 Slow Detection Speed . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

CONTENTS xiv

4 Discriminating Orthogonal Single-Photon Images 73

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4 Hologram Characterization . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5 Single-Photon Image Discrimination . . . . . . . . . . . . . . . . . . . 85

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 Conclusions and Discussion 92

Bibliography 97

A Algorithms for Compressed Sensing 125

List of Tables

4.1 Image-discrimination results showing the total number of raw (C) and

accidental (A) coincidences and the C/A ratio. We also show singles

rate for the hearlding and image-discrimation channels . . . . . . . . 88

xv

List of Figures

1.1 Schematic of quantum and thermal ghost imaging setups . . . . . . . 8

1.2 Cameraman image and transform coding (signal compression) using

the Discrete Cosine Transform (DCT) basis . . . . . . . . . . . . . . 16

1.3 Compressive sensing at work: The original signal (spikes) is represented

by the red dots. A random sensing matrix was used in the reconstruc-

tion of the sparse signal using (a) ℓ1 and (b) ℓ2 minimizations. The

recovered signal is represented by blue circles. Exact reconstruction of

the sparse signal was achieved when the number of measurements was

set at four times the sparsity level of the signal as can be seen in (a).

The method of least squares (ℓ2 minimization) fails to approximate the

original sparse signal as can be seen in (b). . . . . . . . . . . . . . . . 22

1.4 Matched filtering technique . . . . . . . . . . . . . . . . . . . . . . . 23

xvi

LIST OF FIGURES xvii

2.1 Schematics for (a) the single-pixel camera [78] and (b) a computa-

tional ghost imaging [82] setup. In both cases, a single-pixel (bucket)

detector collects the signal. In (a), random patterns are impressed on

the amplitude-only spatial light modulator (SLM) working in reflective

mode. In (b), the field distribution at the object plane is computation-

ally determined for each controllably impressed random phase patterns

on the SLM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Experimental demonstration of single-pixel imaging. In (a), we show a

direct reconstruction of the ghost image of the logo of the University of

Rochester (UR), using computational ghost imaging techniques. In (b),

we use a transform basis (discrete cosine transform) and compressive

sensing techniques to reconstruct the object using fewer realizations

(measurements) than in (a). . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Schematics for compressive quantum ghost imaging. The object and

spatial light modulator (SLM) planes are conjugate to each other. In-

set: the corresponding unfolded Klyshko picture. . . . . . . . . . . . . 36

2.4 Schematics for compressive single-photon single-pixel imaging. The

object is imaged onto the plane of the spatial light modulator (SLM).

Inset: the corresponding unfolded Klyskho picture. . . . . . . . . . . 39

LIST OF FIGURES xviii

2.5 Setup for entangled photon compressive ghost imaging. PBS, polariz-

ing beam splitter; SLM, spatial light modulator; L, imaging lens; HWP,

half-wave plate; BBO, β-Barium Borate crystal. A and B represent

bucket detectors used for coincidence measurement. Inset: example of

a two-dimensional random binary pattern impressed onto the SLM. . 43

2.6 Experimental image reconstruction using compressive sensing algo-

rithms. Reconstructed ghost image of (a) the Greek letter Ψ and (b)

the University of Rochester (UR) logo. The insets show the masks

used in the test arm of the ghost imaging setup. (c, d) The absolute

value of the calculated two-dimensional discrete cosine transforms of

the insets in (a) and (b), respectively. . . . . . . . . . . . . . . . . . . 45

2.7 The calculated mean-squared error of the reconstructed ghost images

of the logo of the University of Rochester (UR) (●) and the Greek

letter Ψ (�) as functions of the number of measurements M . . . . . 46

2.8 The calculated signal-to-noise ratio of the reconstructed ghost images

of the University of Rochester (UR) logo (●) and the Greek letter Ψ

(�) as functions of the number of measurements M . . . . . . . . . . 47

LIST OF FIGURES xix

3.1 Normalized second- and fourth-order moments about the mean (a) and

kurtosis (γIM/σIM )4 (b) as functions of the speckle averaging factorM .

Here the lines are the theory [cf. Eqns. 3.25 and 3.26], and symbols

are the calculated results from one typical numerical simulation real-

ization [83, 118]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Schematics of our thermal ghost imaging setup. The spatial light mod-

ulator (SLM) is used to impress a sequence of random phase distribu-

tion on the laser field. BS: beam splitter; CCD: charge coupled device 68

3.3 (a) Representative speckle pattern of the sort used in our experiments

and (b) the intensity average of 25 patterns of the sort shown in (a).

The statistics of the two patterns are very different, as described in

the text. Nonetheless, ghost images obtained under the two conditions

are essentially identical. (c) A ghost image of a double slit mask (1.2

mm long, 100 µm wide, and with 40 µm gap in between) taken using

individual speckles and (d-f) a ghost image taken using the intensity

average of M = 5, 15 and 25 individual speckle patterns, respectively.

In each case, N = 10 000 measurements were used to obtain the ghost

image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

LIST OF FIGURES xx

3.4 CNR as a function of the number of measurements N for the ghost

imaging system that responds to different numbers M of averaged

speckle patterns for each measurement. Here the symbols are experi-

mental results, and the lines are simulation realizations. . . . . . . . 70

4.1 Concept for the single photon image discrimination experiment . . . . 79

4.2 Laboratory setup for writing the multiplexed hologram. Biplex holo-

grams are exposed using a HeNe laser and a pair of object-reference

beam combinations sequentially. A shutter is used to electronically

control the exposure time. For each exposure, the reference-object

pair is selected using a rotation stage and a translation stage. NPBS,

nonpolarizing beamsplitter . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3 Laboratory setup for the single-photon image readout. Heralded sin-

gle photons are sent through either object A or B, during the image-

discrimination phase of the experiment, and are then detected at either

detector A or B. TCSPC, time-correlated single-photon counter; BiBO,

Bismuth Borate crystal; NPBS, nonpolarizing beamsplitter . . . . . . 83

4.4 Object reconstruction using a plane wave read beam. . . . . . . . . . 85

4.5 Hologram readout with an image carrying read beam. . . . . . . . . . 86

4.6 Single-photon image-discrimination results. Total number of raw co-

incidences (C), accidental coincidences (A), and C/A ratio for each

object-detector combination. . . . . . . . . . . . . . . . . . . . . . . . 89

Chapter 1

Background

1.1 Introduction

Quantum imaging is a recent and promising branch of quantum optics that exploits

the quantum nature of light. Unlike optical imaging techniques that use classical

light sources, quantum imaging uses spatially multimode non-classical states of light.

The key motivation in this field is to improve limitations imposed by classical sources

of light in optical imaging techniques or overcome the classical boundaries of im-

age formation. Some of the possible applications that fall under quantum imaging

range from detection of small displacements with a precision beyond the Rayleigh

limit [1–8], subwavelength phase measurements [9, 10], noiseless amplification of op-

tical images [1, 11], quantum lithography [12–15], remote ghost spectroscopy [16],

entangled two-photon microscopy [17], high sensitivity imaging for the detection of

weak amplitude (or phase) objects beyond the standard quantum limit [18], quantum

1

1.1 Introduction 2

teleportation of optical images [19–21], quantum holography [22], to quantum imag-

ing using entangled-photon pairs, i.e., ghost (coincidence) imaging [6, 7, 23–29, and

references therein].

Non-classical states of light that display strong spatial (temporal) correlations can

be used as a source for most of the above applications. In this thesis, we are interested

in some aspects of ghost (coincidence) imaging. The source of radiation in ghost

imaging, in the quantum domain, is entangled photons created through the process of

spontaneous parametric down-conversion (SPDC) in a nonlinear optical medium [30–

32]. In SPDC, a strong pump photon of higher frequency ωp interacts with a non-

centrosymmetric crystal and results in the annihilation of a pump photon and the

creation of two down-converted photons of lower frequencies (usually called the signal,

of frequency ωs and idler, of frequency ωi). The nonlinear mixing results in signal

and idler photons that are entangled in frequency and momentum. Depending on the

phase-matching condition, the down-converted photons could be collinear (with the

direction of the pump) or noncollinear; same polarization (for type-I phase matching

condition) or orthogonal polarization (for type-II phase matching).

The first ghost imaging experiment used entangled photon pairs generated via

SPDC [23]. The inherent difficulties associated with quantum ghost imaging are two

fold: source brightness and detection inefficiencies. First, SPDC in bulk crystals pro-

duces a rather weak light source, as it is an inefficient process (depending on the

crystal length and nonlinearity, the generation efficiency is typical in the range of

1.1 Introduction 3

10−12 to 10−8). Second, the usual technique of raster scanning a single-element de-

tector requires long integration times to acquire an image with high resolution and in

addition to this, transverse single photon sensitive detector arrays are an expensive

and cumbersome resource. In the first part of the thesis, we discuss methods that ad-

dress and solve the detector inefficiency problem using single-pixel (bucket) detectors

(as opposed to a raster scanning detector) and improved (shorter) integration times

using compressive sensing methods.

Ghost (coincidence) imaging has also been demonstrated using bright, classically

correlated beams of light. In the literature (and for reasons that will be clear later),

this imaging modality is known as thermal ghost imaging. In thermal GI, unlike

ghost imaging using quantum entangled photon pairs that requires coincidence mea-

surement using single photon sensitive detectors, intensity correlation measurements

are carried out to form the ghost image. Though there are similarities and differences

between quantum and thermal GI, the photon-limited aspects of the source (bright-

ness problem) in quantum GI can be addressed and easily be overcome using brighter,

spatially correlated classical sources of radiation. In thermal GI, two copies of speckle

patterns (splitted thermal-like beams or spatially correlated beams) are used in each

arm of the optical path. Intensity correlation measurements of the photocurrents of

the two detectors (one spatially resolving and the other bucket) results in ghost image

formation. In the second part of the thesis, we study how the quality of ghost image

formation is affected when using intensity averaged speckle patterns with arbitrary

1.2 Ghost Imaging–Introduction 4

statistical properties.

The third part of the thesis deals with transverse image discrimination and recog-

nition in photon-limited situations. Ghost imaging using quantum entangled photons

have been shown to provide superior performance (better signal-to-noise) as com-

pared to thermal ghost imaging in photon-limited (low-light-level) situations [33].

Low-light-level imaging arises in many applications, such as radar, astronomy, medi-

cal imaging, to mention a few. Optical image recognition with photon-limited images

is also one such application [34–37]. It has been shown that random quantum fluc-

tuations of light at such low-light levels set the ultimate performance limit for such

optical recognition [5, 37]. We study transverse image discrimination using holo-

graphic matched filtering techniques using heralded single photons produced by an

SPDC source.

1.2 Ghost Imaging–Introduction

Ghost imaging (GI), also known as two-photon coincidence imaging, has attracted

tremendous attention in the quantum optics community since its first experimental

demonstration by Pittman et al. [23] using entangled photons generated via sponta-

neous parametric down-conversion [24–26,38–48]. The theoretical foundation of ghost

imaging was first laid out by Klyshko [49, 50]. Ghost imaging is a novel transverse

imaging technique that uses two spatially correlated beams that travel through two

separate optical paths, usually called the test and reference arms. In the test arm,


one of the two correlated beams impinges on the object, whose image we want to de-

termine, and the transmitted or reflected light is collected by a single pixel (bucket)

detector that does not provide any spatial information about the object. In the ref-

erence arm, we have a spatially resolving detector. The beam in this arm does not

interact with the object. No information about the object can be inferred from one

of the correlated beams alone. A cross correlation between the signals of the two

spatially separated detectors results in the image of the object.

The first experiment on ghost imaging used orthogonally polarized entangled pho-

tons generated from a type-II phase-matched spontaneous parametric down-conversion

source [23]. Because of the nature of the source used, it was claimed that ghost imag-

ing was a quantum effect and shortly thereafter it was argued theoretically that

entanglement is a prerequisite for achieving distributed quantum imaging [24,47,48].

However, Bennink et. al [25, 38] experimentally demonstrated ghost imaging using

classically correlated beams of light. They used a pair of collimated laser beams that

produced angularly correlated pulses (analogous to momentum-correlated photons

produced by an entangled source) and showed that a ghost image can be formed

using this classical source of light and argued that entanglement is not required for

ghost (coincidence) imaging. Since then other classical sources of radiation (thermal

or pseudothermal) have been used in ghost imaging experiments [26, 51–53].

The image acquisition of a generic ghost imaging system is described next (see

Fig. 1.1). In Fig. 1.1(a) a nonlinear crystal (NLC) is used as a source of entangled


photon pairs (commonly referred to as signal and idler photons). One of the photon

pairs (say, the signal) propagates through the test arm (say, arm 1) and the idler

photon traverses through the reference arm (arm 2). The unknown object is located

in the test arm. The impulse responses (optical transfer functions) of the test and

reference arms are given as h1(x1,x′1) and h2(x2,x

′2), respectively. The impulse re-

sponses describe the field propagation from the transverse plane x′i to xi, for i = 1, 2,

where x′i is the plane of the output face of the nonlinear crystal and xi is the plane

of the detectors. In the test arm, we have a fixed, spatially non-resolving (“bucket”)

detector (labeled D1 in Fig. 1.1). It collects the total intensity falling on plane x1 and

therefore no spatial information about the object can be retrieved using the test arm

alone. However, a spatially resolving detector is used in the reference arm to collect

light. In the case of ghost image formation using entangled photon pairs, usually

referred to as quantum ghost imaging, a scanning single-pixel detector (labeled D2

in Fig. 1.1) is used to scan the location of the idler photon in the transverse plane

x2. The image of the object is retrieved by measuring coincident events between the

signal and idler photon pairs as detector 2 is scanned in plane x2.

When the source of radiation is not quantum entangled photons, but rather ther-

mal or pseudothermal (usually produced by passing laser light through a rotating

ground glass), we refer to it as thermal ghost imaging [43, 54–58]. In Fig. 1.1(b) we

show a schematic for a thermal GI setup. Here two copies of spatially correlated

beams are created by splitting the incoming chaotic light with thermal statistics us-


ing a beam splitter. Similar to the quantum GI, the detection in the test arm uses

a spatially non-resolving detector. In the reference arm, however, spatially resolving

array of detectors (e.g. CCD) are usually employed. The intensity distribution of

the object (ghost image) is retrieved by measuring the correlation function of the

intensity fluctuations in both arms as a function of the pixel position of the array

detector in arm 2.

Before we describe some of the similarities and differences between quantum and

thermal ghost imaging techniques, we present a description of the theory of ghost

imaging when the source of radiation is entangled photons (quantum GI) and when

using spatially correlated thermal beams (thermal GI). In both cases, information

about the object is retrieved by measuring the fourth-order spatial correlation func-

tion of the intensities detected by the bucket (D1) and spatially resolving (D2) detec-

tors.

Let us first consider quantum ghost imaging. We will take the source of our entan-

gled photon pairs to be from a type-II phase matched χ(2) nonlinear crystal through

the process of spontaneous parametric down-conversion. In spontaneous parametric

down-conversion, the photons of a high-intensity pump field (of central frequency ω0)

are split into pairs of lower energy photons, usually called signal and idler (of central

frequency ω1 and ω2) through a nonlinear interaction with the medium. For a type-II

phase matching condition, the signal and idler photons are orthogonally polarized.

We briefly describe a parametric down conversion (PDC) process using a three-


(a)

(b)

x2

x1

x2

x1

D2

D1

D2

D1

source of

entangled

photons

source of

thermal

radiation

coincidences

,

h2 (x

2 , x2 )

,

h2 (x

2 , x2 )

,

h 1(x 1

, x 1)

,

h 1(x 1

, x 1)

,

h 1(x 1

, x 1)

<<

I2(x

2)I

1(x

1)

Figure 1.1: (a) Quantum ghost imaging setup. Entangled photons from a sponta-neous parametric down-conversion (SPDC) are used as a source of radiation. Thesignal and idler photons traverse the test and reference arms. Due to the low-fluxnature of such SPDC sources, photon-coincidence detection is carried out for ghostimage reconstruction. (b) Classical ghost imaging setup: Here a beam splitter isused to create two classical (thermal, pseudothermal) beams with a strong spatialcorrelation. Usually a laser beam passing through a rotating ground glass diffuser isused as a source of a pseudothermal radiation. Intensity cross correlation between thephotocurrents of the two detectors is carried out to reconstruct the object’s intensitytransmission function.


wave (pump, signal and idler) interactions inside a nonlinear crystal. Let the field

envelope operators for the signal and idler photons at the output face of a χ(2)

nonlinear crystal of length lc be b1(x) and b2(x), where x is the position in the

transverse plane. The Fourier transform of the field envelopes bi(x) are given by

bi(q) =∫(dx/2π) e−iq·x bi(x). For PDC initiated by vacuum fluctuations, no signal

(a1(x)) and idler (a2(x)) fields are present at the crystal input face. In the plane wave

pump approximation limit, the field envelope operators at the output of the crystal

are related to the input operators through [5, 59–61]:

b1(q) = U1(q)a1(q) + V1(q)a†2(−q) (1.1)

b2(q) = U2(q)a2(q) + V2(q)a†1(−q) (1.2)

where Ui(Vi) (i = 1, 2) are gain functions and exact expressions are given in [5,59,60].

The fields at the detection planes (x1 and x2) are related to the fields at the exit face

of the crystal (source planes) through ci(xi) =∫dx′

i hi(xi,x′i) bi(x

′i), (i = 1, 2) for a

lossless system and hi(xi,x′i) are the impulse response functions of the system. The

correlation function for the quantum GI is thus given by [7]

G(x1,x2) =

∣∣∣∣∫ ∫

dx′1dx

′2 h1(x1,x

′1) h2(x2,x

′2)〈b1(x′

1)b2(x′2)〉∣∣∣∣2

. (1.3)

For the thermal GI, we use a thermal field a(x) characterized by a Gaussian field

statistics as the source of our radiation and a beam splitter to create two spatially


correlated beams bi(x) (i = 1, 2). Let the complex transmission and reflection coef-

ficients of the beam splitter are t and r, respectively. The fields bi(x) (i = 1, 2) are

given interms of the input field a(x) using the standard beam splitter input-output re-

lations. Following similar formulation as in the quantum GI, the correlation function

is given by

G(x1,x2) = |tr|2∣∣∣∣∫ ∫

dx′1dx

′2 h

∗1(x1,x

′1) h2(x2,x

′2)〈a†(x′

1)a(x′2)〉∣∣∣∣2

. (1.4)

Here 〈a†(x′1)a(x

′2)〉 is the thermal second-order correlation in the source plane. We

see the analogy between the correlation functions for the quantum and thermal ghost

imaging techniques from Eqns. 1.3 and 1.4, where the signal-idler correlation function

〈b1(x′1)b2(x

′2)〉 takes the place of the thermal second-order correlation function. In

both cases, the transverse coherence length is determined by the correlation functions

(〈a†(x′1)a(x

′2)〉 and 〈b1(x′

1)b2(x′2)〉). In both configurations (thermal and quantum

GI), a bucket detector (with no spatial resolution) is used in the test arm (arm 1

in Fig. 1.1). All the object information is thus contained in the measured quantity

∫dx1G(x1,x2).

O’Sullivan et al. and Erkmen et al. have performed detailed comparison of dif-

ferent aspects, such as resolution, noise characteristics, image acquisition times, of

the two imaging modalities [33, 62]. For photon-limited applications, quantum GI

performs better (higher signal-to-noise ratio) than thermal GI, although thermal GI

needs only a slightly higher average illumination intensity to outperform quantum

1.3 Compressive Sensing–Introduction 11

GI. Image acquisition times for thermal GI are much shorter than those for quantum

GI since the readily available source of entangled photons are rather weak.

Some of the concepts described in this section will be used in chapters 2 and 3.

In chapter 2, we present a detailed theory and experiment on a quantum ghost image

reconstruction technique using compressive sensing methods. In chapter 3, the effects

of using intensity averaged speckle patterns on the quality of thermal ghost image

is studied, both theoretically and experimentally. In the next section, we present an

introduction to compressive sensing.

1.3 Compressive Sensing–Introduction

Compressive sensing (compressive sampling or sparse recovery) is a new and novel

sampling and signal reconstruction method that requires far less samples (measure-

ments) than that would be deemed necessary by the Nyquist-Shannon criterion [63–

66]. It exploits the fact that many natural signals and images are sparse or compress-

ible in certain basis, such as Fourier, wavelets and discrete cosine transforms (DCT),

to mention a few. That is, if a signal or image is sparse in some transform basis,

it can be expressed or approximated by a linear combination of a small set of basis

vectors. Compressive sensing provides a way of reconstructing the signal or image

from a small (highly incomplete) measurements of the signal.

The conventional approach for full signal recovery is such that the sampling rate

must be at least twice the bandwidth (the maximum frequency) of the signal (i.e., the


Nyquist rate). This approach inherently requires that we make as many observations

(measurements) as there are unknowns. This sampling technique does not exploit the

sparsity or compressibility of signals. For example, the traditional approach to many

lossy signal compression techniques, such as in music (MP3 standard), video (MPEG)

and images (JPEG, JPEG-2000 standards), is first to acquire the full information on

the signal. The transform coefficients of the acquired signal are then calculated in a

suitable basis where the signal is sparse or compressible. Compression is possible since

the largest coefficients and their locations are stored and the rest are set to zero. This

resource inefficient approach throws away most of the acquired information content.

In the compressive sensing paradigm, however, we sample not at the Nyquist rate,

but at the “information rate” and directly acquire the “compressed” data.

Consider the linear set of equations,

yi = 〈x, Ai〉 = ATi x, i = 1 . . .M (1.5)

which is the inner product between x and Ai and (·)T denotes transposition. . Here

x ∈ RN is the signal that we want to reconstruct (an N × 1 column vector), yi are

the measurements, and {Ai}Mi=1 are a collection of measurement (sensing) vectors.

Rearranging the sensing vectors ATi as columns in an M ×N matrix A, Eqn. 1.5 can

be re-written, in a matrix form, as

y = Ax, (1.6)


where y ∈ RM is an M × 1 column vector and A is a known M × N measurement

matrix. In order to ensure reconstruction of the signal x from the measurement y,

the fundamental theorem of linear algebra dictates that M ≥ N . The linear set of

equations in Eqn. 1.6 is underdetermined for M < N and no unique solution exists.

In compressive sensing, we are generally interested in the cases where M ≪ N .

The question that is addressed in the compressive sensing field is: if we know

that a signal or image x (of length N) is sparse or nearly sparse (compressible) in

a certain basis, that is, if x depends on a small number of unknown parameters,

can we recover x exactly (or with high probability) using M ≪ N measurements?

Rephrasing the question: can we reconstruct a signal x (of length N) with no or little

information loss by acquiring a condensed representation (of length M) through a

dimensionality reduction? It has been shown in the compressive sensing (sampling)

field that a signal that is sparse in one basis can be reconstructed using a small

set of projections onto a second measurement basis that is incoherent with the first.

The number of measurements required for full signal recovery is dependent on two

factors: (1) the sparsity level of the signal in the sparsifying transform basis and (2)

the degree of coherence between the sparsifying and the measurement bases. Below

a brief description of the two fundamental tenets of compressive sensing, namely,

sparsity and incoherence is presented.


1.3.1 Sparse and Compressible Signals

Any signal x ∈ RN can be expressed in terms of a basis of N × 1 vectors {φi}Ni=1 as

x =N∑

i=1

θiφi, (1.7)

or, in matrix notation, as x = ΦΘ. Here the weighting coefficients θi = 〈x, φi〉. A

signal x is k-sparse1 in the basis Φ, i.e., it has k non-zero components, if it can be

approximated by a linear combination of k vectors from the sparsifying transform

basis Φ. A signal x is compressible2 if it can be represented by a few large coefficients

θi in the representation given by Eqn. 1.7. The fact that k such non-zero or large

coefficients (and their locations) can be used to approximate (with a minimum error)

a vector of size N , where k ≪ N , is the basis for many efficient fundamental signal

processing methods such as transform coding (data compression). However such a

process requires that we make N measurements of the signal x as the locations of

the most significant coefficients θi are not known in advance. Consider the example

given in Fig. 1.2. The discrete cosine transform (DCT) of the original image given in

Fig. 1.2(a) was first calculated. The locations and coefficients of the transform were

stored. We then discard 50% of the smallest coefficients and inverse transform back

1A signal x ∈ RN is called k-sparse if

‖x‖0 = #{1 ≤ i ≤ N : xi 6= 0} ≤ k.

2A signal x ∈ Rn is called compressible if the sorted magnitude of the coefficients {|xi|}ni=1

decayrapidly.


into the canonical basis where the image was first acquired. Comparing Fig. 1.2(b)

with the original, we almost see no discernible differences. Generally speaking, the

disadvantage in such transform coding (data compression) techniques is that we have

wasted valuable resources in acquiring all N coefficients.

If a signal is k-sparse, i.e., it has k degrees of freedom, in the canonical basis (in

which case Φ is the identity matrix I) or is sparse in a different basis, using the signal

sparsifying basis, Eqn. 1.6 can be rewritten as

y = AΦΘ. (1.8)

Since we want to reconstruct a sparse signal from the measurements, a natural ap-

proach would be to solve the following ℓ0 optimization problem3:

Θ = minΘ

‖Θ‖0 subject to AΦΘ = y (1.9)

The ℓ0 norm counts the number of non-zero entries of the signal we want to recon-

struct. Solving Eqn. 1.9, however, is a hard combinatorial problem, is numerically

unstable and is computationally intractable as there are(Nk

)possible combinations

for the locations of the entries of the sparse signal Θ. For image acquisition, N can

easily be on the order of 106 or bigger. Due to the computational intractability of

3The ℓp norm of v is defined as ‖v‖p :=

(N∑

i=1

|vi|p)1/p

for 0 < p < ∞. The ℓ0 norm counts the

number of non-zero entries, the ℓ1 norm of v gives the sum of the absolute values of the elements of v,

i.e., ‖v‖1 =∑N

i=1|vi|, and the ℓ2 norm of v gives the Euclidean norm, i.e., ‖v‖2 =

(∑Ni=1

|vi|2)1/2

.


the ℓ0 optimization problem, Eqn. 1.9 is usually recast interms of the ℓ1 norm. The

equivalence of the above ℓ0 optimization problem with the computationally tractable

ℓ1 minimization problem will be shown below using a numerical example.

The number of measurements required to acquire a signal nonadaptively depends

on how sparse (or compressible) the signal is. The relationship between the number

of measurements and the sparsity level will be given after we review the concept of

incoherence in compressive sensing.

(a) (b)

Figure 1.2: (a) Image of a Cameraman. The discrete cosine transform (DCT) of theimage in (a) was calculated. Although not shown, most of the coefficients in thetransform basis are small. After taking the DCT of the image in (a), we set about50% of the coefficients to zero and transform it back into the measurement basis. Theresulting compressed image is shown in (b). It shows the effect of the sparse nature ofthe image in the chosen basis. We see that there is no discernible difference betweenthe original and the compressed image.


1.3.2 Incoherence

The mutual coherence between two orthonormal bases A and Φ is defined as [67]

µ(A,Φ) =√N max

i,j| 〈Ai, φj〉 | (1.10)

where Ai, φj ∈ RN denote the column vector of A and Φ, respectively, and N is the

length of the column vector. The mutual coherence measures the maximum correla-

tion between elements of the measurement (sensing) basis A and the representation

basis Φ. For highly incoherent basis pairs, there is no sparse representation of the

elements of one basis in terms of the other.

The mutual coherence takes values between 1 and√N , where the lower (upper)

bound is for completely incoherent (coherent) bases. For reasons that will be clear

later, highly incoherent basis pairs are preferable in compressed sensing applications.

One such example of maximally incoherent basis pairs is between canonical or spike

basis Ai(t) = δ(t − i) and the Fourier basis φj(t) = N−1/2ei2πjt/N . The maximum

of the inner product of the two basis pairs is N−1/2. Another example is random

orthonormal basis which are largely incoherent with any fixed basis Φ, as such they

are a universal measurement basis [67].

The main reason for requiring maximally incoherent basis pairs is that if our mea-

surement basis (A) is maximally coherent with the sparsifying basis (Φ), sampling

the transform coefficients will result in values very close to zero (for a compressible

signal) or zero (for a sparse signal) most of the time, since Θ is already sparse (com-


pressible) in the Φ basis. However, if the measurement basis is highly incoherent with

the sparsifying basis, every measurement returns a little bit of information about the

sparse signal.

The mutual coherence determines the number of measurements required for signal

recovery. Suppose we take m measurements uniformly at random from the measure-

ment basis A. If x is k-sparse in the sparsity basis Φ, then

m ≥ C · µ2(A,Φ) · k · logN (1.11)

measurements (for some positive constant C) guarantees exact recovery of the signal

x with high probability [67, 68]. Note that for highly (or maximally) incoherent

basis pairs, the number of samples is on the order of k logN instead of N . This is

crucial since we can exactly recover a k-sparse signal (of length N) using m ≪ N

measurements.

1.3.3 Restricted Isometry Property

The above discussion about the mutual coherence between the measurement and the

sparsifying bases does not address how we design a stable measurement matrix A.

The notion on constructing such a matrix, called restricted isometry property (RIP),

was put forward by Candes and Tao [69]. Let A be anM×N matrix (where as above

M < N) and K be a positive number. Then, the isometry constant δK is defined as


the smallest number such that, for each integer K = 1, 2, . . .

(1− δK)‖x‖22 ≤ ‖Ax‖22 ≤ (1 + δK)‖x‖22 (1.12)

for all K-sparse vectors x. The restricted isometry constant δK quantifies how close

to isometrically the measurement matrix A acts on the K-sparse vectors x. Note

that we want to reconstruct a length-N signal (x) from M < N measurements. The

above condition states that for any K-sparse signal x, if the K locations are known,

then the problem is not ill-conditioned provided that M ≥ K.

Designing a measurement matrix that satisfies Eqn. 1.12 requires checking all

(NK

)combinations for a K-sparse signal x of dimensions N . As mentioned above,

random matrices (for example, with elements Ai,j that are independent and identically

distributed (iid) random variables) are generally incoherent with any fixed basis and

are usually chosen in different applications of compressed sensing as measurement

matrices. For example, if the sparse signal representation basis is the spike basis, an

iid Gaussian matrix is shown to have the RIP with high probability provided that we

make M & O[K log(N/K)] measurements [65, 69, 70].

Given the above definition for the restricted isometry constant, Candes and Tao [69]

have shown that if the measurement matrix A satisfies:

δ3K + 3δ4K < 2, (1.13)


then the K-sparse vector x is the unique minimizer of the following ℓ1 optimization

problem:

x = minx

‖x‖1 subject to Ax = y (1.14)

1.3.4 Signal Reconstruction Algorithms

The traditional approach to determine the signal x from such a linear set of equations

(Eqn. 1.6) is to use the least-squared method, i.e., ℓ2 minimization. That is minimizing

the sparse signal x through

x = minx

‖x‖2 subject to y = Ax. (1.15)

Unlike the ℓ0 minimization problem (Eqn. 1.9) which is computationally intractable,

the method of least squares (Eqn. 1.15) is easy to solve. In fact, there exists a close-

form solution x = AT (AAT )−1y, where T denotes transposition. However, for a

sparse signal x, this minimizer does not necessarily guarantee a sparse solution. We

elaborate this statement by giving a sample numerical example below.

The most common algorithm used in finding sparse or compressible solution in

compressed sensing is the ℓ1 norm. Minimizing the ℓ1 norm encourages small com-

ponents of a sparse signal to become exactly zero, thus promoting sparse solutions.

In fact, Eqn. 1.14 can be recast as a linear program, for which efficient algorithms

exist. For example, when the measurement matrix A is taken to be a random matrix

(such as random matrices whose entries are independent and identically distributed


Gaussian), then a K-sparse signal can be recovered exactly using the ℓ1 optimization

if M & O[K log(N/K)] measurements are taken [70].

1.3.5 Sparse Signal Reconstruction – A Numerical Example

We next give a numerical example of sparse signal reconstruction from undersampled

measurements (see Appendix for code used to generate Fig. 1.3). Consider a signal x

(represented as red dots in Fig. 1.3 of length N = 512 that is sparse in the canonical

basis (spikes), with a sparsity level of K ≡ ‖x‖0 = 30. The measurement matrix

A is formed by sampling i.i.d entries from the normal distribution, with zero mean

and 1/M variance. Here the number of measurements M is taken to be 4 times

the sparsity level of the signal, i.e., M = 120. We can see from Fig. 1.3(a) that we

have exact recovery of the sparse signal when using ℓ1 minimization (blue circles in

the Figure). The results of reconstruction of a sparse signal using methods of least

squares (ℓ2 minimization) is also given for comparison purposes in Fig. 1.3(b), where

we see that the minimization fails to reasonably approximate the original signal (red

dots).

In chapter 2, we describe a theoretical and experimental work on entangled-photon

compressive ghost imaging. Concepts introduced in sections 1.2 and 1.3 will be

used. We will describe high-resolution compressive ghost imaging at the single photon

level using entangled photons produced by a spontaneous parametric down-conversion

(SPDC) source and using single-pixel (bucket) detectors.


100 200 300 400 500

−2

−1

0

1

2

(a) Sparse signal reconstruction using ℓ1 minimization

100 200 300 400 500

−2

−1

0

1

2

(b) Sparse signal reconstruction using ℓ2 minimization

Figure 1.3: Compressive sensing at work: The original signal (spikes) is representedby the red dots. A random sensing matrix was used in the reconstruction of the sparsesignal using (a) ℓ1 and (b) ℓ2 minimizations. The recovered signal is represented byblue circles. Exact reconstruction of the sparse signal was achieved when the numberof measurements was set at four times the sparsity level of the signal as can be seen in(a). The method of least squares (ℓ2 minimization) fails to approximate the originalsparse signal as can be seen in (b).

1.4 Image recognition – Introduction 23

1.4 Image recognition – Introduction

In chapter 4 of the thesis, we describe transverse image discrimination and recog-

nition using heralded single photons produced by a spontaneous parametric down-

conversion. We have used holographic matched filtering methods to achieve image

discrimination with high confidence level. Below we present a brief description of

matched filtering [71–73].

ζξ η

input

plane

output

plane

frequency

planeL

1L

2

Figure 1.4: Matched filtering technique.

Consider the Vander Lugt optical correlator schematically shown in Fig. 1.4. Let

the input signal, in plane ξ, be given by f(x, y) and let L1 and L2 be the Fourier

transforming lenses, i.e., plane η (ζ) is the Fourier plane of ξ (η), respectively. The

Fourier transform of the input signal, f(x, y), is given by [73]

F (fX , fY ) =

∫ ∫dx dy f(x, y) exp[−i2π(fXx+ fY y)]. (1.16)

1.5 Summary and Outline of Thesis 24

We say that the filter (in plane η) is matched to the input signal f(x, y) if its transfer

function is

H(fX , fY ) = F ∗(fX , fY ) (1.17)

as this maximizes the cross correlation function of h(x, y) and the input signal f(x, y).

For a signal centered in the input plane ξ, when the matched filtering condition is

satisfied, the signal after the mask in the frequency plane η is transformed into a

bright spot at the origin of the output plane ζ by the second Fourier transforming

lens L2. For any other input signal, the transmitted light will not be brought into a

bright spot by L2.

1.5 Summary and Outline of Thesis

In this chapter, I have reviewed ghost imaging (both quantum and thermal), compres-

sive sensing, and image discrimination using matched filtering technique. In chapter

2, I will describe a theoretical and experimental work on high-resolution compres-

sive ghost imaging at the single-photon level using entangled photons produced by a

spontaneous parametric down-conversion source and using single-pixel detectors. In

section 1.2, I have described both quantum and thermal ghost imaging. The raster-

scanning of single-photon sensitive detectors in one arm of the quantum GI setup in

order to retrieve information about the object was described. In our experiment, we

only use single-pixel detectors in both arms and compressive sensing algorithms re-

viewed in section 1.3 are used to reconstruct the ghost image. In chapter 3, I present

1.5 Summary and Outline of Thesis 25

a theoretical description and experimental results on how intensity-averaged (that

is, “blurred”) speckle patterns affect the quality of ghost image formation. I show

theoretically how the contrast-to-noise ratio depends on the speckle-averaging factor.

I also describe an experimental study in support of the theoretical work. In chapter

4, I show how to discriminate images using holographic-matched filtering technique

(section 1.4) using heralded single photon sources with high confidence level. I close

by presenting general discussions and conclusions in chapter 5.

Chapter 2

Compressive Quantum Ghost

Imaging

In this chapter, we present the results of a theoretical work and an experimental

demonstration of quantum compressive ghost imaging protocol at the single-photon

level using biphotons generated by spontaneous parametric down-conversion (SPDC)

and using spatially non-resolving (single-pixel) detectors only [74]. We show that

compressive sensing (CS) can be usefully implemented at the level of few-photon

imaging. A single-pixel compressive sensing setup using heralded single photons from

a SPDC source is also described. We also show that for a given mean-squared er-

ror of the reconstructed two-dimensional, high-resolution quantum ghost image, the

number of measurements and the number of photons needed by the compressive sens-

ing algorithm is much smaller than quantum ghost imaging experiments employing a

26

2.1 Introduction 27

raster scan. This implies both an improvement in the acquisition time of the ghost

image and a more economical use of photons for low-light-level imaging. The success

of this demonstration suggests that compressive sensing methods are likely to prove

useful much more generally in applications involving quantum light fields.

2.1 Introduction

One of the key goals of many imaging protocols is to form an image using as small a

number of photons as possible. Such strategies are especially useful for applications

in quantum information, where the quantum nature of the light field is a key aspect

of the problem at hand, or in other applications where photons are “expensive,” such

as in image formation at unusual wavelengths [75].

Due to the inherently weak sources of entangled photons, quantum imaging with

entangled photons suffers from low photon flux and resource-inefficient transverse de-

tection. Owing to the need for gating to achieve high temporal resolution, transverse

arrays are expensive and require intensive electronics even for low- to moderate-

resolution images [76, 77]. The most commonly employed technique used is to raster

scan a single-pixel detector to acquire the image. However, to obtain images with high

resolution and high signal-to-noise ratio, this cheaper and simpler method requires

long integration times. In this chapter we show that compressive sensing (sampling)

can be usefully implemented to address these problems at the level of few-photon

imaging. We describe an implementation of CS algorithm in the context of a specific

2.1 Introduction 28

quantum imaging protocol, that of single-photon ghost imaging, where the primary

goal is to transfer or acquire an image using the absolute minimum number of trans-

mitted photons.

As we have described in Sec. 1.3, compressed sensing (CS) is a novel sampling

and signal reconstruction method that requires far less data than would be deemed

necessary by the Nyquist-Shannon criterion [63–65]. As a resource-efficient sensing

paradigm, CS has proven to be extremely useful in the context of classical image for-

mation, the first application of which is the single-pixel camera developed by Duarte

and co-workers [78]. The method has also been recently applied to quantum state

tomography [79] and quantum process tomography [80].

The configuration of the classical single-pixel camera [78,81] is conceptually equiv-

alent to that of the computational ghost imaging, first described by Shapiro [82] the-

oretically and later verified experimentally by Bromberg et al. [83]. Conventionally,

a ghost imaging (quantum or thermal) setup involves two beams of light, which are

termed object and reference beams [23, 25, 39, 42, 43, 84] (see Sec. 1.2). The object

beam illuminates the object and the transmitted or reflected light is monitored by

a spatially non-resolving (bucket) detector. The light in the reference arm is moni-

tored by a spatially resolving detector. The image of the object is then formed by

a coincidence measurement (in the quantum case) or intensity correlation (for the

thermal case) between the object and reference signals, as described in section 1.2.

In computational ghost imaging [82, 83] a single-beam and a single spatially non-

2.2 Single-Pixel Imaging 29

resolving (single-pixel) detector are used. In both cases, the intensity distribution

of the reference beam is determined computationally. This flexibility in determining

the field distribution computationally lends itself to applications in three-dimensional

(3D) imaging [83].

This chapter is organized as follows. In section 2.2, we review classical single-pixel

imaging protocols. First the single-pixel camera developed by Duarte et al. [78] and

computational ghost imaging (CGI) put forward by Shapiro [82] are described. We

also present experimental results based on CGI and its compressive counterpart [85].

A detailed theoretical description of entangled-photon compressive ghost imaging and

heralded single-photon compressive imaging is laid out in section 2.3. In section 2.4,

we present details of the experiment on entangled-photon compressive ghost imag-

ing. This is followed by analysis of image reconstruction using compressive sensing

algorithms in section 2.5 and photon efficiency comparison between entangled-photon

compressive ghost imaging and GI based on raster scanning in section 2.6. A summary

is presented in section 2.7.

2.2 Single-Pixel Imaging

The theory of compressive sensing [63–65], introduced in section 1.3, can potentially

has a number of practical applications in astronomy, medicine such as in MRI [86,87],

computational biology [88], radar analysis [89–94], metrology [95], robotics, seismol-

ogy, generally in signal processing [96–98] and imaging (e.g. single-pixel terahertz


imaging system by Chan et al. [99]) and many others [100, a website dedicated to

compressive sensing, both theory and applications].

The classical single-pixel camera developed by Duarte and co-workers [78, 81] is

the first application of the theory of compressive sensing. In the experiment of [78],

a schematic of which is shown in Fig. 2.1(a), a coherent laser beam was used as

a light source and the image of the object was formed on the plane of a reflective

spatial light modulator (SLM) using a lens (not shown in Fig. 2.1(a)). In [78], a

digital micromirror device (DMD) of 1024 × 768 pixels is used as a reflective SLM.

Bucket

detector

source

ζ

x1

object

SLM

Bucket

detector

object

SLML

ζ x1

Laser

(a)

(b)

Figure 2.1: Schematics for (a) the single-pixel camera [78] and (b) a computationalghost imaging [82] setup. In both cases, a single-pixel (bucket) detector collects thesignal. In (a), random patterns are impressed on the amplitude-only spatial lightmodulator (SLM) working in reflective mode. In (b), the field distribution at theobject plane is computationally determined for each controllably impressed randomphase patterns on the SLM.


Each mirror of the DMD can be positioned into one of two states (the ’on’ and ’off’

positions). No light is collected from a DMD pixel in the ’off’ position and is deflected

away from the bucket detector. The light falling on the DMD can thus be reflected in

two directions depending on the orientation of the mirror. A bucket detector is used

to collect light reflected from the DMD from one of the two directions. The image of

the object can be reconstructed following the discussion used in section 1.3.

Consider the schematics of a computational ghost imaging setup shown in Fig. 2.1(b).

A phase-only spatial light modulator is used to impress a controllable random phase

φ(ξ, η) to the input field E0 incident onto the SLM. That is, the field distribution

right after the SLM is given by E(ξ, η) = E0 eiφ(ξ,η). The field distribution E(x, y) in

the object-plane, a distance L from the SLM-plane, can be easily calculated as

E(x, y) =

∫ ∫dξ dη E(x− ξ, y − η) h(ξ, η) (2.1)

where the Huygens-Fresnel free-space propagation Green’s function

h(ξ, η) =eikL

iλLexp[i

k

2L(ξ2 + η2)], (2.2)

where λ is the wavelength of the light source and the wavenumber k = 2π/λ. A

bucket (single-pixel) detector collects the light transmitted through or reflected from

the object for a transmissive or reflective [56, 101] ghost imaging setup, respectively.

For each controllable random phase φ(n)(ξ, η), where n = 1, . . . , N , impressed onto


the SLM, the corresponding intensity distribution I(n)(x, y) = |E(n)(x, y)|2 at the

object plane is calculated, where N denotes the total number of measurements. The

ghost image G(x, y) is determined from the correlation function

G(x, y) = 〈(IB − 〈IB〉)I(x, y)〉 (2.3)

where 〈· · · 〉 denotes ensemble averaging and

I(n)B =

∫ ∫dx dy I(n)(x, y) T (x, y) (2.4)

is the total signal collected by the bucket detector for the nth measurement and

T (x, y) is the intensity transmission function of the object. Note that the number

of measurements N required to reconstruct the object is governed by the number of

pixels needed to resolve the object.

The concepts of compressive sensing can be used to reduce the number of mea-

surements required to resolve the object (see section 1.3). In a similar fashion as

the single-pixel camera, the image of the object can be reconstructed following the

discussion used in section 1.3. This has recently been shown experimentally by Katz

et al. [85] using a pseudothermal ghost imaging setup (see Fig. 2.1(b)).

Next we show results of an experimental work on computational [82,83] and com-

pressive [85] thermal ghost imaging, for completeness. The same data collected using

the schematic shown in Fig. 2.1(b) was used for both experiments. We have used a


HeNe laser (λ = 632.8 nm) and a phase-only spatial light modulator (from Boulder

Nonlinear) to create speckle patterns. In both cases, the intensity distribution of the

reference beam (at the object plane) is determined computationally. A transparency

of the logo of the University of Rochester (UR), located a distance L = 152 cm from

the SLM surface, was used as our object and a large-area bucket detector (New-

port powermeter) collects light transmitted through the object for each realization n.

This process was repeated N times. The reconstruction of the computed ghost image

(object), using Eqn. 2.3, is shown in Fig. 2.2(a).

For the compressive ghost imaging experiment, we have used the discrete cosine

transform (DCT) as our sparsifying basis (see section 1.3.1). The reconstructed image,

with a better signal-to-noise ratio, is shown in Fig. 2.2(b).

(a) (b)

Figure 2.2: Experimental demonstration of single-pixel imaging. In (a), we show adirect reconstruction of the ghost image of the logo of the University of Rochester(UR), using computational ghost imaging techniques. In (b), we use a transformbasis (discrete cosine transform) and compressive sensing techniques to reconstructthe object using fewer realizations (measurements) than in (a).

2.3 Compressive Quantum Imaging: Theory 34

2.3 Compressive Quantum Imaging: Theory

In this section, we present a theoretical description for two compressive quantum

imaging setups. The first configuration is called entangled-photon compressive ghost

imaging. This configuration is also the focus of our experimental work, where bipho-

tons generated by a spontaneous parametric downconversion process (SPDC) are used

as light sources. We also describe a similar configuration we call single-photon single-

pixel compressive imaging. Here heralded single photons from an SPDC process are

used as a light source.

The coincidence rate at the two detector positions x1 and x2 for both configu-

rations (shown in Figures 2.3 and 2.4) is given by the normally ordered correlation

function [102, 103]

C(x1,x2) = 〈ψ| E(−)(x1)E(−)(x2)E

(+)(x2)E(+)(x1) |ψ〉 ,

=∣∣∣〈0| E(+)(x2)E

(+)(x1) |ψ〉∣∣∣2

(2.5)

where E(+)(x) and E(−)(x) is the positive- and negative- frequency part of the electric-

field operator at position x and |ψ〉 is the biphoton state. In most previous ghost

imaging configurations [23,25,38,39,42,43], a spatially resolving detector is used only

in the reference arm and a bucket detector is used in the object arm. The ghost image

is thus contained in

C(x2) =

∫dx1 C(x1,x2) (2.6)


However, unlike ghost imaging setups that employ a spatially resolving detector such

as a charge coupled device (CCD) or raster scanning a single-pixel detector in the

reference arm, bucket detectors are used in both arms in our case. All the object

information is thus contained in the integrated coincidence signal

Cm =

∫dx2 dx1 C(x1,x2) (2.7)

where m denotes the mth measurement, as we will describe in more detail in Sec. 2.4.

2.3.1 Entangled-Photon Compressive Ghost Imaging

The schematic for entangled-photon compressive ghost imaging is given in Fig. 2.3.

The inset shows the unfolded Klyshko picture. Note that the object is imaged onto

the amplitude-only spatial light modulator (SLM). The two-photon amplitude for

setup is given by

〈0| E(+)(x2)E(+)(x1) |ψ〉

=

∫dxs dξ dxi dη dζ h(ζ,x2) Am(ζ) h(η, ζ) L(η)

×h(xi,η)ψ(xs,xi) h(ξ,xs) T (ξ) h(x1, ξ) (2.8)

where

L(x) = exp[−ik/(2f)x2] (2.9)


bucket detector

BBO

SLM

f

object

bucket

detector

ζ

d1’

d2”

xs

ξ

η

x2

x1

xi

d2

d2’

d1

d2

d2’ d

1d

1’

objectSLM

BBOf

1

d2’

1

d1+d

2

1

f+ =

d2”

Figure 2.3: Schematics for compressive quantum ghost imaging. The object andspatial light modulator (SLM) planes are conjugate to each other. Inset: the corre-sponding unfolded Klyshko picture.

is the transfer function of the lens and h(x,x′) is the Fresnel free-space propagation

kernel, which under the paraxial approximation is h(x,x′) ∝ exp[ik/(2d)(x − x′)2].

Here f is the focal length of the lens, k = 2π/λ is the wavenumber and d is the longitu-

dinal separation between the x- and x′- planes. The quantity we wish to determine,

the transmission function of the object, is given by T (x) and Am(x) is the two-

dimensional random pattern imprinted onto the amplitude SLM with m = 1, . . . ,M ,

where M is the total number of realizations (measurements). The random pattern

Am(x) used in conjunction with the bucket detector map the spatial information con-

tained in the object transmission function T (x) into a sequence of coincidence signals

Cm

encoded by the different realizations of Am.

In the spontaneous parametric downconversion process (SPDC), the biphoton


state can be approximated by ψ(xs,xi) ∝ δ(xs − xi) for a thin nonlinear crystal and

narrow bandpass filters before the detectors [24]. Under such conditions (which are

good approximations in our experiment and many experiments on quantum ghost

imaging) and after substituting the expressions for the Fresnel free-space propagation

kernel and the transfer function of the lens, the two-photon amplitude becomes

〈0| E(+)(x2)E(+)(x1) |ψ〉

=

∫dζ dξAm(ζ) T (ξ) e

ik

2d′′2(ζ−x2)2

eik2[ 1d′1(x1−ξ)2+ 1

d1ξ2]

×∫dη e

ik2[

d1d2(d1+d2)

η2+ 1d′2(ζ2−2η·ζ)]

∫dxs e

ik2[( 1

d1+ 1

d2)x2

s−2( ηd2

+ ξd1

)·xs]

(2.10)

After carrying out the integration over xs and when the thin lens equation 1/(d1 + d2)+

1/d′2 = 1/f is satisfied, the two-photon amplitude further simplifies to

〈0| E(+)(x2)E(+)(x1) |ψ〉

∝∫dξAm(−Mξ) T (ξ) e

ik2(M

2

d′′2+ 1

d1+ 1

d′1+M2

d′2)ξ2

eik

2d′′2(x2

2+2Mξ·x2)e

ik

2d′1(x2

1−2x1·ξ)

(2.11)

The integrated coincidence signal for the setup shown in Fig. 2.3, after integrating


over x1 and x2 and using Eqn. 2.11 for the two-photon amplitude becomes

Cm =

∫dx1dx2

∣∣∣⟨0∣∣∣E(+)(x1) E

(+)(x2)∣∣∣ψ⟩∣∣∣

2

∝∫dξ |Am(−Mξ)|2 |T (ξ)|2

∝∑

n

|Am(−Mξn)|2 |T (ξn)|2 (2.12)

where M = d′2/(d1 + d2) is the magnification of the system and the finite size of

the SLM pixel are used, with n = 1, . . . , N , where N is the number of pixels in the

SLM. Note that the object and SLM planes are conjugate to each other and bucket

detectors are used in both arms. The last expression for the integrated coincidence

signal (Eqn. 2.12) can be rewritten in matrix form and will be used in the context of

compressed sensing later.

2.3.2 Single-Photon Single-Pixel Compressive Imaging

The schematic for the heralded single-photon compressive imaging is shown in Fig. 2.4.

In this configuration, similar to the entangled-photon compressive GI setup, biphotons

generated by SPDC are used. However, in this configuration, one of the biphotons, in

arm 2 of Fig. 2.4, is used to herald the presence of a single photon in arm 1. Similar

to the derivation given above in section 2.3.1, the two-photon probability amplitude

is given by


〈0| E(+)(x2)E(+)(x1) |ψ〉

=

∫dxs dξ dη dζ dxi h(xi,x2) ψ(xs,xi) h(ξ,xs) T (ξ) h(η, ξ)

×L(η) h(ζ,η) Am(ζ) h(x1, ζ)

∝∫dζ dξAm(ζ) T (ξ) e

ik2( 1d2

+ 12f

)ξ2e

ik2( 12f

+ 1d′1

)ζ2

e− ik

d′1ζ·x2

1

×∫dxs e

ik2( 1d2

+ 1d1

)x2s e

− ikd2

xs·x2 e− ik

d1xs·ξ

∫dη e−

ik2f

(ξ+ζ)·η

(2.13)

bucket

detector

BBO

SLM

f2f

2fobject

bucket detector

ζ

d1

d2

d1’

xs

xi

ξ

η

x2

x1

d2

d1

2f 2f

object SLM

BBO fd

1’

Figure 2.4: Schematics for compressive single-photon single-pixel imaging. The objectis imaged onto the plane of the spatial light modulator (SLM). Inset: the correspond-ing unfolded Klyskho picture.


After evaluating the ‘η’, ‘ζ ’, ‘xs’ integrals, the two-photon amplitude simplifies to,

〈0| E(+)(x2)E(+)(x1) |ψ〉

∝∫dξAm(−ξ) T (ξ) e

ik2( 1d1

+ 1f+ 1

d′1)ξ2

eik

d′1ξ·x2

1

× e− ik

2( 1d2

+ 1d1

)−1( 1d2

x2+1d1

ξ)2. (2.14)

The integrated coincidence signal for the setup shown in Fig. 2.4, after evaluating

the integrals over x1 and x2 and using the two-photon amplitude (Eqn. 2.14), is given

by

Cm =

∫dx1dx2

∣∣∣⟨0∣∣∣E(+)(x1) E

(+)(x2)∣∣∣ψ⟩∣∣∣

2

∝∫dξ |Am(−ξ)|2 |T (ξ)|2

∝∑

n

|Am(−ξn)|2 |T (ξn)|2 . (2.15)

Note that the object is imaged onto the SLM with unity magnification and the in-

tegrated coincidence signal for the heralded single-photon compressive imaging setup

becomes proportional to Eqn. 2.12 with M = 1.

The expressions for the integrated coincidence signal for both setups (see Eqns. 2.12

and 2.15) can be re-written in matrix form as C = AT, with Amn ≡ |Am(−xn)|2 and

Tn ≡ |T (xn)|2. Most natural images are sparse when expressed in the proper basis

2.4 Experimental Setup 41

such as that of the discrete cosine transform 1 or the wavelet transform used in JPEG

compression. Suppose the object intensity transmission function T is K-sparse in the

basis Φ, i.e., only K of its coefficients are non-zero. When the measurement matrix A

is taken to be a random matrix (such as a matrix whose entries are independent and

identically Gaussian or Bernoulli distributed), it has been shown that the restricted

isometry property (RIP) is satisfied [65, 69, 70]. Then according to the theory, the

vector T gives the desired result by minimizing ‖ΦTT‖1 subject to the condition

C = AT, in which ‖v‖1 =∑

i |vi| is the ℓ1 norm of v. The error in determining T

is bounded from above if M & O[Klog(N/K)] measurements are used. This number

can be much smaller than that of the Nyquist-Shannon criterion.

2.4 Experimental Setup

We demonstrate entangled photon compressive ghost imaging. The experimental

setup is depicted in Fig. 2.5. A continuous-wave Ar-ion laser, operating at a wave-

length of 363.8 nm, was used to pump a BBO (β-Barium Borate) nonlinear crystal.

1The two-dimensional discrete cosine transform (DCT) of a function f(x, y) is defined as

C(u, v) = α(u)α(v)

N−1∑

x=0

N−1∑

y=0

f(x, y) cos

[π(2x+ 1)u

2N

]cos

[π(2y + 1)v

2N

],

for u, v = 0, 1, . . . , N − 1 and α(u) and α(v) are defined as α(u) =√

1

N (√

2

N ) for u = 0 (u 6= 0)

respectively. The inverse DCT is defined as

f(x, y) =N−1∑

u=0

N−1∑

v=0

α(u)α(v)C(u, v) cos

[π(2x+ 1)u

2N

]cos

[π(2y + 1)v

2N

],

for x, y = 0, 1, . . . , N − 1


The BBO was cut for a type-II phase matching angle to generate a pair of orthog-

onally polarized signal and idler photons (degenerate at a wavelength of 727.6 nm)

that propagate collinearly. The pump was then spatially separated from the down-

converted degenerate photons using a UV grade fused silica dispersion prism. A

polarizing beam splitter (PBS1 in Fig. 2.5) was used to send the orthogonally po-

larized photons into the object and reference arms. A phase-only reflective spatial

light modulator (from Boulder Nonlinear: 512× 512 pixels, pixel pitch 15 µm), sand-

wiched between orthogonal polarizers (the two ports of PBS2 in Fig. 2.5), was used to

mimic an amplitude-only SLM. A half-wave plate was used to rotate the polarization

of the photons before impinging on the SLM. The face of the nonlinear crystal is

imaged using a lens (L in Fig. 2.5) of focal length f = 25 cm onto the object and the

amplitude-only SLM with a magnification of 3.

We group the native pixels of the SLM into cells with a size of 4 × 4 pixels, so

that we effectively have an array with N = 128× 128 pixels. We then impress known

but random binary patterns onto the SLM. An example of such two-dimensional

random binary pattern is shown in the inset of Fig. 2.5. We use identically dis-

tributed Bernoulli random variables (with values of 0 or 1) with equal probability.

The photons transmitted through the optical system are coupled into a multimode

fiber and registered by single-photon counting module (SPCM-AQR-14, from Perkin-

Elmer) detectors in both arms. A 10-nm FWHM bandwidth spectral filter (centered

at 727.6 nm) is placed in front of each detector. Coincidence circuitry (with a time

2.5 Image Reconstruction 43

window of 12 ns) was used to measure coincidence events between the avalanche

photodiodes (APDs) in the reference and object arms.

A

BBBO

Ar+ Laser

PBS2

PBS1

objectSLM

HWP

L

Figure 2.5: Setup for entangled photon compressive ghost imaging. PBS, polarizingbeam splitter; SLM, spatial light modulator; L, imaging lens; HWP, half-wave plate;BBO, β-Barium Borate crystal. A and B represent bucket detectors used for coin-cidence measurement. Inset: example of a two-dimensional random binary patternimpressed onto the SLM.

2.5 Image Reconstruction

It is important to point out the data acquisition techniques commonly used in ther-

mal and quantum ghost imaging (GI) setups here before we discuss the process for

the entangled photon compressive GI. In thermal GI, a random speckle pattern in the

reference arm is measured using a spatially resolving array of detectors and is corre-

lated with the bucket signal collected in the object arm. This procedure is repeated

M times and the ghost image is reconstructed using Eqns. 2.3 and 2.4. In quantum


GI, two single-photon sensitive detectors (APDs, for example) are used in the signal

and idler arms due to the inherently weak nature of entangled photon sources. In the

reference arm, a spatially resolving APD is scanned in the transverse plane. A bucket

detector is used in the object arm. For each detector position in the transverse plane

of the reference arm, a coincidence measurement between the two detector signals

results in the ghost image of the object.

The image reconstruction for the entangled compressive ghost imaging is as fol-

lows. A two-dimensional random binary amplitude mask (Am) was sent to the spatial

light modulator (SLM) via a computer and coincidence measurements (Cm) between

the two bucket detectors were performed form = 1, . . . ,M whereM is the total num-

ber of measurements. For each random pattern impressed onto the SLM, coincidence

events were integrated for 9 seconds. In the experiment, we have used two objects

(the logo of the University of Rochester and the Greek letter Ψ) in the test arm of the

ghost imaging setup, as shown in the insets of Fig. 2.6(a) and (b). On average, the

singles counts in the object (reference) arms are: 19.5 k counts/s (25.6 k counts/s)

for the logo of the University of Rochester and 28.3 k counts/s (25.7 k counts/s) for

the Greek letter Ψ, respectively. The coincidence rate is about ∼ 1% of the singles

rate.

Compressive sensing works, i.e. we are able to fully recover a signal (image) from

undersampled measurements, because most signals (images) are sparse under certain

basis transformation (representation). To show the sparsity of the objects used in the


experiment, we have used the two-dimensional discrete cosine transform (2D-DCT)

as a representational basis. As can be seen in Fig. 2.6(c) and (d), the objects are

sparse in the chosen basis (Φ).

(a) (b)

(c) (d)

0

1

2

3

4

Figure 2.6: Experimental image reconstruction using compressive sensing algorithms.Reconstructed ghost image of (a) the Greek letter Ψ and (b) the University ofRochester (UR) logo. The insets show the masks used in the test arm of the ghostimaging setup. (c, d) The absolute value of the calculated two-dimensional discretecosine transforms of the insets in (a) and (b), respectively.

The reconstruction of the object intensity transmission function (T), was accom-

plished by minimizing ‖ΦT‖ℓ1 subject to C = AT using the gradient projection

algorithm [104, the algorithm is described in Appendix A]. Here the ℓ1 norm of v


gives the sum of the absolute value of the elements of v, i.e., ‖v‖1 =∑N

i=1 |vi|. The

results of the reconstruction, for the maximum number of measurements (M = 6300),

are shown in Fig. 2.6(a) and (b). Comparisons with the original masks of the objects

(insets in the same figure) show that we have a good reconstruction.

2000 3000 4000 5000 60000.02

0.04

0.06

0.08

0.1

M

MS

E

Figure 2.7: The calculated mean-squared error of the reconstructed ghost imagesof the logo of the University of Rochester (UR) (●) and the Greek letter Ψ (�) asfunctions of the number of measurements M .

To better quantitatively characterize the fidelity of the compressed sensing image

reconstruction algorithm, we have used the mean-squared error (MSE) as our metric.

The MSE is defined as

MSE =1

N||x− x||22 . (2.16)

Here x is the reconstructed image, x represents the original mask, ‖v‖2 is the Eu-


clidean (ℓ2) norm of v and N is the number of resolution cells, in our case N =

128 × 128. Fig. 2.7 shows the calculated MSE as a function of the number of mea-

surements. As can be seen from the figure, the MSE flattens out for M > 4500 (27%

of the Nyquist limit of 128× 128), with values of 0.06 for the University of Rochester

(UR) logo and 0.03 for the Greek letter Ψ.

We have also characterized our CS results in terms of the signal-to-noise ratio

(SNR). The signal and noise are calculated as the mean intensity of the bright pixels

and the standard deviation of the dark background pixels, respectively [85]. The

maximum SNR we obtained, for the maximum number of measurements M = 6300,

is SNR=8 (SNR=10) for the object mask UR (Ψ) as shown in Fig. 2.8, respectively.

2000 3000 4000 5000 6000

1

3

5

7

9

M

SN

R

Figure 2.8: The calculated signal-to-noise ratio of the reconstructed ghost images ofthe University of Rochester (UR) logo (●) and the Greek letter Ψ (�) as functionsof the number of measurements M .


Here we mention how our entangled-photon compressive ghost imaging recon-

struction results [74] compare with other compressed sensing based reconstructions.

It is important to note here that the conditions under which most of the experimen-

tal results [78, 81, 85–87, 105–111] described below are found are quite different. The

light source (coherent, pseudothermal, quantum), the image acquisition techniques,

the compressed sensing algorithms used, the dimensions (pixel size) of the object, the

number of measurements used for reconstruction, the sparsity level of the object, and

so on, could be different.

In [78, 81, 105], Takhar and co-workers describe the operation of the single-pixel

camera, the first application of the theory of compressed sensing. Duarte et al. [78]

have shown image reconstruction of the letter R (of size N = 256× 256 pixels) using

compressed sensing methods using only M = 1300 random measurements, i.e., using

only 2% of the measurements Nyquist limit (50× sub-Nyquist). They have calculated

values of MSE of 0.01 for image reconstruction of the letter R (of size N = 1282 pixels)

using M = N/10 measurements.

In [85], Katz et al. show image reconstruction using compressive sensing methods

for a pseudothermal ghost imaging setup (similar to Fig. 2.1(b)). For a double-slit

mask, the mean-square error (MSE) is calculated to be 0.05 (0.04) for the image

reconstructed using M = 256 (512) realizations, i.e., using 15% (30%) of the number

of measurements corresponding to the Nyquist limit, respectively. They have also

shown image reconstruction of a grayscale object with an MSE value of 0.005 using


60% of the Nyquist limit.

One other application of compressed sensing (CS) is in magnetic resosance imaging

(MRI) [86, 87, 106–111]. For example, in [108], Ma et al. describe full body MR

image reconstruction using compressed sensing algorithms. For the full body image

of dimensions 924×208 pixels, they show image reconstruction with SNR values of

24.12 for a sampling ratio (M/N) of 38.38%, where M and N are the number of

measurements and the size of the image (number of pixels), respectively. In [107],

Ni et al. show image reconstruction of a 256×256 angiography image and a knee image

with a 320×320 pixel resolution using only 25% of the measurement Nyquist limit

with reconstruction errors of about -45 dB using chirp sensing matrices. In [111],

Huang et al. show results of brain MR image reconstruction using 20% sampling

using different CS algorithms [86, 108, 110, 111]. The maximum calculated value of

the signal-to-noise ratio was 20.35 dB.

We note here that in some cases, the result of our experimental work [74] is

comparable to others [85], where the MSE is about the same. Duarte et al. calculated

values of MSE of 0.01 for image reconstruction of the letter R (of size N = 128× 128

pixels) when using M = N/10 measurements. In our case, the results shown in

Fig. 2.6 (c) and (d) are taken using 38% of the measurements Nyquist limit. We

have used entangled photons for our experiment, while [78, 81, 105] have all used a

coherent light source. On the other hand, image reconstruction with much better

SNR are shown in [107–109].

2.6 Photon Efficiency Comparison 50

2.6 Photon Efficiency Comparison

We compare the performance of our CS procedure with other approaches to image

formation. For our demonstrations, there are 128 × 128 = 1.6 × 104 pixels in the

object. We obtain a very good image using 6300 measurements (see Fig. 2.6) and

a highly acceptable image using only 2000 measurements (see Fig. 2.8). Thus, we

are able to obtain good images while performing far fewer measurements than there

are pixels in the field to be imaged. We did not make any systematic attempt to

minimize the total number of photons used to form the image. It is nonetheless

interesting to examine the photon-efficiency of our CS process. Using the numbers

reported above, we estimate that approximately 1.4 × 107 detected biphotons were

used to obtain either of the images of Fig. 2.6. This number is considerably smaller

than the number required by conventional quantum ghost imaging, in which a point

detector is raster scanned in the reference arm. In this case, assuming a shot noise

limited system, we would need to collect approximately 100 photons per pixel to

achieve a SNR of 10 (the maximum SNR for the University of Rochester logo, see

Fig. 2.8). There are 128× 128 pixels in the image, but for raster scanning we utilize

only 1 part in 128×128 of the emitted photons. Thus, the required number of photons

is 100 × 1284 = 2.6 × 1010, which is three orders of magnitude higher than our CS

approach. Thus the CS method could be preferable in low-light-level, i.e., photon

starved, applications.

2.7 Summary 51

2.7 Summary

We have presented an experimental demonstration of image reconstruction at low

light levels using entangled photons from an SPDC source and using compressive

sensing (CS) algorithms. We have shown that CS can lead to high-resolution images

with a dramatically improved SNR. A relatively higher photon efficiency has also

been inferred as compared to ghost imaging using raster scanning. For the objects

used in the experiment, high-fidelity ghost image reconstruction was achieved using

only 27% of the number of measurements corresponding to the Nyquist limit. In

addition, unlike most ghost imaging (quantum or thermal) experiments where spa-

tially resolving detectors are a requirement, we have used only single-pixel (bucket)

detectors in both the reference and test arms. This could have an important impact

in quantum imaging where photon counting arrays are an expensive and cumbersome

resource and may have applications in secure image transmission [112] and optical

encryption [113].

Chapter 3

Speckle Averaging Effects in

Thermal Ghost Imaging

In chapter 2, we have presented a theoretical and experimental work on the impli-

cations of compressed sensing (CS) to quantum ghost imaging (GI). We have shown

experimentally that, compared to GI based on raster scanning, CS-based quantum

GI has a relatively high photon-efficiency. We have also shown that the need for pho-

ton counting arrays that are expensive and that require intensive electronics could

be mitigated by using a configuration that combines a spatial light modulator and

single-pixel (bucket) detector. The emphasis in the previous chapter has mainly been

on the detection part of a quantum GI setup.

In the present chapter, we study the effect of the field statistics of the illuminating

source on the quality of ghost images. We show theoretically and experimentally that

52

3.1 Introduction 53

a thermal GI setup can produce high quality images even when low-contrast speckle

patterns are used as an illuminating source. We show that as long as the collected

signal is mainly caused by the random fluctuation of the incident speckle field, as

opposed to other noise sources, the quality of the ghost image formed is not degraded

even when the detectors used are so slow that they respond only to the intensity-

averaged speckle patterns [114].

3.1 Introduction

Laser speckle patterns have attracted the attention of the scientific community and

have been studied after the first operation of the cw HeNe laser since the early

1960s [115–118, and references therein]. The term “speckle pattern” conventionally

refers to the intensity distribution produced by the mutual interference of a set of

randomly generated wave fronts, such as those obtained when scattering a coherent

laser beam off a rough surface or from a spatially disordered sample. The statistical

properties of such speckle fields have been studied by many authors [117, 118].

A diverse range of applications, based on the use of speckle phenomena, have

been developed in recent years. Some of these applications include 3D mapping

and range finding [119,120], metrology for biomedical applications [121–123], random

lasers [124], imaging of strongly interacting quantum systems [125], etc. The suc-

cess of many speckle-based technologies and techniques inevitably relies on the facts

that a speckle field can have high spatial and temporal randomness and that the de-

3.1 Introduction 54

tecting devices have enough spatial and temporal resolution to monitor the dynamic

variations of individual speckles [114].

In many circumstances, the contrast ratio is used to quantify the amount of vari-

ation within a speckle pattern. The contrast ratio can be defined as [126]

K =σ(I)

〈I〉 , (3.1)

where σ(I) ≡√

〈I2〉 − 〈I〉2 is the standard deviation of the intensity variation of the

speckles and 〈· · · 〉 denotes either temporal or spatial ensemble average.

If the response time of the detector is slow compared to the variation of the speckle

patterns, i.e., when the detector used in such a system is slow in the sense that its

refresh rate cannot keep up with the temporal variations of the illuminating speckle

field, the effective illuminating field that the system measures essentially becomes the

intensity average (or sum) of multiple mutually uncorrelated speckle patterns. The

contrast ratio of such an intensity-averaged speckle pattern obeys an inverse log-log

relation with the speckle averaging factorM , i.e., the contrast ratio K of such speckle

pattern scales with the speckle averaging factor M as M−1/2 [126]. The quantity M

can be calculated as the ratio between the integration time τd of the detector and the

coherence time τc of the random speckle field, i.e, as M = τd/τc. In other words, M

indicates the number of independent speckle patterns that are averaged together in

each measurement. Due to the reduction of the contrast ratio as M increases, the

performance of many speckle-based metrology and imaging techniques would quickly

3.1 Introduction 55

deteriorate as the integration (response) time of the detector increases [123].

Ghost imaging has recently been performed using speckle fields [23, 25, 39, 43, 55,

57, 127–132]. As it has been described before, ghost imaging is an indirect imaging

method that acquires the image of an object through spatial intensity correlation mea-

surements (see section 1.2). Unlike conventional imaging techniques, ghost imaging

uses a nonspatially resolving bucket detector to collect the optical signal directly from

the object either through reflection or transmission, and therefore it can be advanta-

geous in scenarios where using a detector array is restricted or difficult. We have also

seen, in chapter 2, how using compressive sensing in quantum ghost imaging setup

solves the need for expensive photon counting array of detectors. Ghost imaging also

offers great potential for imaging objects located in optically harsh environments [133]

or for imaging through turbulent and scattering media [134–136].

The use of slow detectors in many speckle-based imaging methods degrades the

image quality thus produced. One might naturally expect that this extends to ghost

imaging systems as well. However, in this chapter we show both theoretically and ex-

perimentally that this is actually not the case and that the image quality of a thermal

ghost imaging system can remain high even though the refresh rate of the detectors

is much slower than the coherence time of the illuminating speckle field, as long as

the fluctuations in the detected signal are due predominantly to the randomness of

the speckle pattern itself and not due to noise in the detection system [114, 130].

This chapter is organized as follows. In section 3.2, I present a detailed theoretical

3.2 Theory 56

study of the effect of the field statistics of the illuminating source on the quality of

ghost image formation. The effect of the speed of the detector on the quality of ghost

images is also studied. The experimental details for the thermal ghost imaging are

given in section 3.3. Analysis of the ghost image formation and the contrast-to-noise

ratio for different speckle averaging factors is carried out in section 3.4. A summary

is presented in section 3.5.

3.2 Theory

As we have described before in the introductory part to ghost imaging (see section 1.2)

and section 2.2, we can form the ghost image by calculating the correlation function

of the background-subtracted object and reference signals. Here a light source with

strong transverse spatial correlation, for example a speckle pattern split using a beam

splitter, propagates in the two arms of the ghost imaging configuration. In the object

arm, the speckle pattern is projected on a transmissive or reflective object, and all the

light that is transmitted or reflected is collected by a bucket detector. In the reference

arm, the speckle pattern is directly collected by a spatially resolving detector, e.g., a

CCD camera. The beam in this arm does not interact with the object. The object

is illuminated with N known random intensity patterns, and for each illuminating

pattern, the total energy that is transmitted through (or reflected from) the object

is recorded.

We now make the assumption that the detected signal is primarily given by the

3.2 Theory 57

random intensity variation of the speckle fields and that other noise sources, e.g.,

detector dark noise, can be neglected [137]. The image can then be acquired using

the following background-subtracted correlation formula:

G(x) =1

N

N∑

n=1

(I(n)B − 1

N

N∑

n=1

I(n)B

)(I(n)(x)− 1

N

N∑

n=1

I(n)(x))

=1

N

N∑

n=1

I(n)B I(n)(x)− 1

N2

N∑

n=1

I(n)B

N∑

n=1

I(n)(x), (3.2)

where x = (x, y) represents the two-dimensional Cartesian coordinates, N denotes

the total number of measurements, and

I(n)B =

∑

x

I(n)(x)T (x), (3.3)

is the total signal collected by the bucket detector for the nth measurement, I(n)(x) is

the intensity of the illuminating field collected by the camera for the nth measurement

at location x, and T (x) is the intensity transmission function of the object. For

simplicity, we here assume that the object has binary transmission; i.e., the value of

T (x) is either zero or unity. Note that the second term in Eqn. 3.2 is the product

of the background from two uncorrelated signals, and by subtracting this term, we

ensure that 〈G(x)〉 = 0 for any pixels where T (x) = 0.

The quality of the ghost image is characterized by the contrast-to-noise ratio

(CNR), i.e., by the ratio of the background-subtracted signal to its noise. For an

3.2 Theory 58

object with binary transmission, the CNR is defined by the following expression [137,

138]:

CNR =〈gin〉 − 〈gout〉√σ2in + σ2

out

, (3.4)

where 〈gin〉 ≡ 〈G(xin)〉 and 〈gout〉 ≡ 〈G(xout)〉 are the ensemble averages for the

ghost image signal at a point xin (inside the transmitting regions of the object) where

T (xin) = 1 and xout (outside the transmitting regions the object) where T (xout) = 0,

respectively. Similarly σ2in and σ2

out are the variances of the signal at xin and xout,

respectively.

We next derive an expression for the CNR of such a ghost imaging system analyt-

ically in terms of the statistical properties of the illuminating speckle field with three

simplified but reasonable assumptions: (1) the intensity of the illuminating patterns

are statistically independent of each other, (2) the intensity at each pixel is indepen-

dent from that at each other pixel, and (3) the detected signal fluctuation is primarily

given by the random intensity variation of the speckle fields, and other noise sources,

e.g., detector dark noise, can be neglected. Note that the second assumption implies

that we can replace the ensemble averages in Eqn. 3.4 with spatial average, and in

fact we use spatial average in our simulation and experiment to calculate the CNR of

the obtained ghost image.

The intensity transmission function of the object we wish to image, represented

by T (x), is assumed to be binary for simplicity, i.e. T (x) = 0 or 1. The object is

also assumed to be pixelated and to transmit at a total of T pixels. The number

3.2 Theory 59

of transmitting pixels T is generally calculated as the ratio of the transparent area

of the object to the speckle size of the illuminating field. The object is illuminated

with N known random intensity patterns given by I(n)(x) for n = 1, . . . , N . For each

illuminating pattern, the total energy transmitted through the object is recorded,

denoted by I(n)B .

Using the first assumption, i.e., the statistical independence of the intensity of the

illuminating patterns from each other, we calculate the expected imaging signal as

〈G(x)〉 = 〈 1N

N∑

n=1

I(n)B I(n)(x)− 1

N2

N∑

n=1

N∑

m=1

I(n)B I(m)(x)〉

= 〈 1N

N∑

n=1

I(n)B I(n)(x)− 1

N2

N∑

n=1

I(n)B I(n)(x)− 1

N2

N∑

n=1

N∑

m=1m6=n

I(n)B I(m)(x)〉

= 〈IBI(x)〉 −1

N〈IBI(x)〉 −

N(N − 1)

N2〈IB〉〈I(x)〉

〈G(x)〉 =N − 1

N

[〈IIB〉 − 〈I〉〈IB〉

], (3.5)

where we have introduced the shorthand I ≡ I(x) in the last line. Similarly, the

variance is given, after a lengthy calculation, by

σ2(x) =1

N

(N − 1

N

)2

〈I2I2B〉+(N − 1)(N − 2)

N3

[〈I2〉〈IB〉2 + 〈I〉2〈I2B〉 − 〈IIB〉2

]

− 21

N

(N − 1

N

)2 [〈I2IB〉〈IB〉+ 〈II2B〉〈I〉

]+ 2

(N − 1)(3N − 4)

N3〈IIB〉〈I〉〈IB〉

+N − 1

N3〈I2〉〈I2B〉 − 2

(2N − 3)(N − 1)

N3〈I〉2〈IB〉2. (3.6)

We can further simplify the equations for the expected imaging signal (Eqn. 3.5)

3.2 Theory 60

and the variance (Eqn. 3.6) by using the second assumption that the intensity at each

pixel is independent from one another by substituting in the following relations for

the various expectation values [114, 137]

〈I〉 = µ1, (3.7)

〈IB〉 = Tµ1, (3.8)

〈I2〉 = µ2, (3.9)

〈I2B〉 = T (T − 1)µ21 + Tµ2, (3.10)

〈IIB〉 = (T − T (x))µ21 + T (x)µ2, (3.11)

〈I2IB〉 = (µ3 − µ2µ1)T (x) + Tµ2µ1, (3.12)

〈II2B〉 =[µ3 − µ2µ1 + 2µ1(T − 1)(µ2 − µ2

1)]T (x) + Tµ2µ1

+ T (T − 1)µ31, (3.13)

〈I2I2B〉 =[µ4 − µ2

2 + 2µ1(T − 1)(µ3 − µ2µ1)]T (x)

+ Tµ22 + T (T − 1)µ2µ

21, (3.14)

where µr ≡ 〈Ir〉 is the r-th moment of the intensity probability distribution of each

illuminating pattern.

We see that, using Eqns. 3.7, 3.8 and 3.11, the expected imaging signal (Eqn. 3.5)

simplifies to

〈G(x)〉 =(N − 1

N

)(µ2 − µ2

1) T (x). (3.15)

3.2 Theory 61

The background-subtracted signal in the numerator of the CNR (Eqn. 3.4), after

using Eqns. 3.7– 3.14, is given by,

〈gin〉 − 〈gout〉 =(N − 1

N

)σ2I , (3.16)

where we have defined σ2I ≡ µ2 − µ2

1 as the variance of the intensities in each illumi-

nating pattern. Similarly we find the noise-squared (the denominator of the CNR) is

given by

σ2in + σ2

out =

(N − 1

N2

)[2Tσ4

I + (5− 6/N)(2µ2µ

21 − µ4

1

)

− (2− 3/N)µ22 − (1− 1/N) (4µ3µ1 − µ4)

](3.17)

=

(N − 1

N2

)[(2T − 2 + 3/N)σ4

I + (1− 1/N)γ4I

], (3.18)

where γ4I ≡ 〈(I−〈I〉)4〉 = µ4−4µ3µ1+6µ2µ21−3µ4

1 is the fourth-order moment about

the mean of each illuminating pattern.

Substituting these relations in Eqn. 3.4, the CNR is given by the remarkably

simple formula

CNR =

[N − 1

(2T − 2 + 3/N) + (1− 1/N) (γI/σI)4

]1/2. (3.19)

Here the total number of transmitting pixels T is generally given as the ratio of the

total transmitting area of the object and the average speckle size (spatial coherence

3.2 Theory 62

area) of the illuminating speckle pattern [137, 138]. Note that the second and fourth

moments about the mean of the intensity fluctuation for each illuminating speckle

field are given by σ2I ≡ 〈I2〉 − 〈I〉2 and γ4I ≡ 〈(I − 〈I〉)4〉, respectively, and we use

here the shorthand I ≡ I(x).

From the expression for the contrast-to-noise ratio (Eqn. 3.19), we see that the

CNR is determined by the number of measurements N , number of transmitting pixels

T , and the quantity (γI/σI)4, also known as the fourth standardized moment, or the

kurtosis, of the intensity distribution of the illuminating speckle fields. Note that

Eqn. 3.19 is valid for any illuminating field with arbitrary statistical properties, as long

as the pixel intensities are statistically independent. This expression for CNR also

indicates that the image quality of a thermal ghost imaging system is affected by the

fourth standardized moment (γI/σI)4 of the intensity fluctuation of the illuminating

field, rather than by the contrast ratio K = σI/〈I〉 of a speckle pattern as in many

conventional speckle-based methods.

In the usual limit of large N , this reduces further to

CNRN→∞ ∼[

N

2(T − 1) + (γI/σI)4

]1/2. (3.20)

3.2.1 Fast Detection Speed

When the detectors are fast enough to record individual speckle patterns (M = 1),

which obey negative exponential intensity statistics [126], we can easily show that

3.2 Theory 63

γ4I = 9〈I〉4 and σI = 〈I〉2. Consequently, the CNR of the ghost image with background

subtraction is given by [137]

CNR =

[N − 1

2T + 7− 6/N

]1/2. (3.21)

In the usual limit of large N and T ≫ 1, the CNR ∼ [N/(2T )]1/2.

3.2.2 Slow Detection Speed

The derivation given above for the contrast-to-noise ratio (CNR) is for the conven-

tional thermal ghost imaging (M = 1) method, whereby correlations are done between

the intensity distribution in the reference arm and the bucket signal for each real-

ization. Now we consider the case where we do collective frame averaging (M 6= 1)

before the correlations are carried out to form the ghost image, thereby simulating

slow detection speed.

When the detectors are slow, the ghost imaging system responds only to the inten-

sity average of M independent speckle patterns for each measurement. In such cases,

the ghost image can still be expressed using Eqn. 3.2, but with the expressions for the

bucket detector signal and the spatially resolved camera signal modified to take into

account the intensity sum of M independent speckle patterns for each measurement,

specifically,

I(n)B,M =

M∑

m=1

I(n,m)B , (3.22)

3.2 Theory 64

and

I(n)M (x) =

M∑

m=1

I(n,m)(x). (3.23)

Following Goodman [139–142], the sum of M speckle patterns follows a gamma

probability density function (PDF), i.e.,

p(x) =1

Γ(M)µMxM−1e−x/µ (3.24)

where µ is the mean intensity of a speckle pattern and Γ denotes the Gamma function.

The nth moment for such a distribution is

〈xn〉 =

∫xnp(x) dx

=1

Γ(M)µM

∫xn+M−1 e−x/µ dx

=µn

Γ(M)Γ(n+M)

where we have used the definition for the Gamma function.1

Using straightforward mathematics, one can show that the fourth-order moment

about the mean γ4IM and the variance σ2IM

of the intensity of the effective illuminating

speckle field are given by

γ4IM = 3(M + 2)M〈I〉4 = 3(M + 2)

M3〈IM〉4, (3.25)

1The Gamma function is defined as Γ(z) =∫∞

0e−t tz−1 dz

3.2 Theory 65

and

σ2IM

=M〈I〉2 = 1

M〈IM〉2 (3.26)

where 〈I〉 and 〈IM〉 are the expected values of the intensity for each independent

speckle pattern and the intensity-averaged speckle field at any pixel, respectively.

The ratio of the fourth moment (γIM ) to the variance (σIM ) is thus given by

(γIM/σIM )4 = 3(1 + 2/M). (3.27)

Expressions for the fourth and second order moments about the mean are given

explicitly in Eqns. 3.25 and 3.26, respectively. The dependence of speckle contrast

ratio K and kurtosis as a function of the speckle averaging factor M is shown graph-

ically in Fig. 3.1. Substituting Eqn. 3.27 into Eqn. 3.19 gives the general expression

for the CNR as a function of the collective frames (M), the number of transmitting

pixels (T ) and the number of measurements (N).

Substituting these relations into Eqn. 3.19, one can obtain the following remark-

ably simple expression for the CNR of a ghost imaging system that only responds to

intensity-averaged speckle fields:

CNR =

[N − 1

2T + 1 + 6/M − 6/(MN)

]1/2. (3.28)

We see that the CNR is very weakly dependent on M . The above expression for

the CNR shows that, even though the contrast ratio of the effective speckle fields

3.2 Theory 66

3

6

9

100 101 102 103

collective frame number M

100

100

10-4

101 102 103

collective frame number M

(a) (b)

Figure 3.1: Normalized second- and fourth-order moments about the mean (a) andkurtosis (γIM/σIM )4 (b) as functions of the speckle averaging factorM . Here the linesare the theory [cf. Eqns. 3.25 and 3.26], and symbols are the calculated results fromone typical numerical simulation realization [83, 118].

“seen” by a ghost imaging system decreases rapidly as the detectors become slow,

the quality of the ghost image actually remains approximately the same as long as

the transmitting area of the object is much larger than the spatial coherence area

of the individual speckle fields. This surprising result comes from the fact that the

quantity that affects the image quality of a ghost imaging system is the kurtosis of

the intensity fluctuation of the illuminating speckle field, which actually converges to

a constant value of 3 as M becomes larger than 10 (see Eqn. 3.27 and Fig. 3.1(b)).

Furthermore, in most practical situations, the transmitting area of the object is much

larger than the coherence area of individual speckle field, i.e., T ≫ 1. In such cases,

the image quality becomes essentially independent of the speckle averaging factor

M [cf. Eqns. 3.19 and 3.28]. Note that Eqn. 3.19 is a generalized result for thermal

ghost imaging systems using illuminating fields having arbitrary statistical properties,


however Eqn. 3.28 is a special case of this result for an illuminating field in the form

of the intensity sum of multiple speckle patterns.

3.3 Experimental Setup

Our thermal ghost imaging system is illustrated in Figure 3.2. A collimated HeNe

laser beam is used to illuminate a phase-only spatial light modulator (SLM, from

Boulder Nonlinear), which is programmed to impose a uniformly distributed random-

phase distribution onto the incident beam [83]. The first-order diffracted beam forms

speckle patterns with negative exponential intensity distributions [118] at the focal

plane of a Fourier lens and is used as the illuminating source. The generated illumi-

nating field is then projected, using an imaging lens and a beam splitter, onto the

object and reference planes. In the object plane, the speckle pattern is projected

onto a transmissive object, and all the transmitted light is collected by a large-area

bucket detector (Newport powermeter) placed behind the object. In the reference

arm, the intensity distribution of the illuminating speckle field is directly collected

by a detector array, a camera in our case. By using many uncorrelated speckle pat-

terns and correlating the signals collected by the bucket detector and the camera, one

can obtain a ghost image of the object using the background-subtracted correlation

function given by Eqn. 3.2.

3.4 Experimental Results 68

Laser

SLMFourier Lens

Aperture

Imaging

Lens

BS

CCD

Bucket

DetectorObject

beam-

expander

Figure 3.2: Schematics of our thermal ghost imaging setup. The spatial light modu-lator (SLM) is used to impress a sequence of random phase distribution on the laserfield. BS: beam splitter; CCD: charge coupled device

3.4 Experimental Results

In our experiment, our object is a double slit mask, whose transmitting area is ap-

proximately 200 times the average speckle size of each independent speckle field, i.e.,

T ≈ 200. The CCD and bucket signals are averaged for M uncorrelated speckle

patterns before the two signals are correlated using Eqn. 3.2 to mimic the use of slow

detectors that respond to the average of M independent speckle patterns. Note that,

for M = 1, our system reduces to a conventional ghost imaging system in which the

detectors respond to each independent speckle pattern.

We make the measurements for speckle averaging factor M equal to 1, 5, 15,

and 25, respectively, to study quantitatively the effect of the response time of the

detection system on the quality of the “ghost image” that we obtain. For each value


(a) (b)

(c)

normalized intensity0 1

(f)(e)

500 µm

(d)

Figure 3.3: (a) Representative speckle pattern of the sort used in our experiments and(b) the intensity average of 25 patterns of the sort shown in (a). The statistics of thetwo patterns are very different, as described in the text. Nonetheless, ghost imagesobtained under the two conditions are essentially identical. (c) A ghost image of adouble slit mask (1.2 mm long, 100 µm wide, and with 40 µm gap in between) takenusing individual speckles and (d-f) a ghost image taken using the intensity average ofM = 5, 15 and 25 individual speckle patterns, respectively. In each case, N = 10 000measurements were used to obtain the ghost image.


of M , we take 10 000 effective measurements. Figures 3.3(a) and 3.3(b) show two

typical illumination patterns recorded by the CCD camera for M = 1 and M = 25,

respectively. The speckle contrast ratio K for the two cases is 1 and 0.2, respectively.

The ghost images after 10 000 effective measurements for M = 1 is as shown in

Fig. 3.3(c). Similar ghost images for M = 5, 15 and 25 are shown in Figs. 3.3(d-f),

respectively. We can see that there is no obvious difference in image quality, which

is consistent with our theoretical prediction.

1000 3000 5000 7000 9000

1.5

2.5

3.5

CN

R

Number of Measurements N

M = 1

M = 5

M = 15

M = 25

Experiment Simulation

Figure 3.4: CNR as a function of the number of measurements N for the ghostimaging system that responds to different numbers M of averaged speckle patternsfor each measurement. Here the symbols are experimental results, and the lines aresimulation realizations.

To better demonstrate our theoretical predictions, we plot in Fig. 3.4 (as symbols)

3.5 Summary 71

our measured CNR of the ghost image as a function of the number of measurement

N for four different values of the speckle averaging factorM . It can be seen that even

though the response time of the detectors in the four cases is very different, there

is no obvious difference in the resulting image quality. Also shown in Fig. 3.4 (as

lines) are the results of numerical simulation. The agreement between simulation and

laboratory measurement is very good. The slight disagreement may be due to other

noise sources (such as camera dark noise) that are not considered in the simulation.

Note that in our experiment, the coherence area of the speckle field is approximately

100 pixels, whereas in our model we assumed that each pixel experienced independent

intensity fluctuations. However, we have performed extensive numerical simulations,

such as those reported in Fig. 3.4, which show that the predictions of our model are

not influenced by the average speckle size with respect the pixel size of the camera.

3.5 Summary

In this chapter, we have presented a theoretical analysis with experimental demon-

stration which shows that the image quality of a thermal ghost imaging system is

essentially independent of the response time of the detectors as compared to the co-

herence time of the illuminating speckle fields. This surprising result arises from the

fact that the image quality of a ghost imaging system is actually only weakly de-

pendent on the kurtosis of the intensity fluctuation of the illuminating speckle field

that the detectors respond to. As the detecting system becomes slow and sees only

3.5 Summary 72

an average of multiple speckle patterns, the contrast ratio K of the effective speckle

field decreases monotonically, but the kurtosis actually converges to a value of 3 for

thermal light (for M & 10). Consequently, the quality of the ghost image is almost

not affected by the detector speed as long as all nonspeckle noise, such as the detector

dark current noise, is small compared to the fluctuation of the averaged speckle fields.

While most thermal ghost imaging systems demonstrated to date have used pseu-

dothermal light whose coherence time can be controlled to match the speed of the

detectors, the possibility of performing ghost imaging with true thermal light has

always been considered intriguing and highly desirable [55,143]. The work presented

here shows that there need not be any blurring of the final image even when the

detection system is much slower than the coherence time of the thermal light source,

as long as the illumination is strong enough that shot noise and detector noise can

be neglected [114]. This result opens up the possibility of using slow detectors for

thermal ghost imaging with quickly varying thermal speckle fields and may shed light

on other applications using speckle fields as well.

Chapter 4

Discriminating Orthogonal

Single-Photon Images

4.1 Introduction

Quantum state discrimination (QSD) deals with finding optimal measurement schemes

(procedures) that determine the state of a quantum system [144–146]. In quantum in-

formation theory and quantum computing, the information is encoded in the state of

the quantum system. If, for example, a quantum system constitutes two nonorthogo-

nal states, distinguishing the states with certainty is not possible [147]. Determining

the exact state of a quantum system plays important roles in many quantum protocols

and QSD plays an important task in cryptography, in communication applications

and possibly in ultra-low light level image recognition [148].

Quantum state discrimination deals with the following problem. Suppose a quan-

73

4.1 Introduction 74

tum system is prepared in one of many nonorthogonal states given by {ρi}, with a

priori probabilities {pi}. Since exact determination of the actual state of the sys-

tem is generally not possible (unless the states are mutually orthogonal), what is the

best measurement scheme that leads to the determination of the initial state that

the quantum system was prepared in? Depending on the figure of merit that one

chooses to optimize, there are three different strategies (approaches) used for optimal

discrimination of quantum states.

The first scheme is called discrimination with minimum error [149,150]. Here the

probability of making an error in identifying the quantum state is minimized and

each possible outcome indicates some corresponding state. This was experimentally

demonstrated for two non-orthogonal polarization states at the Helstrom bound by

Barnett and Riis [151] and more states by [152–154].

Unlike in the first scheme, if we allow for the possibility of obtaining inconclusive

results, error-free discrimination is possible. This constitutes the second scheme and is

called unambiguous discrimination and is possible only when the allowed states are all

linearly independent [155–157]. Experimental verification of error-free measurement

of polarization for two non-orthogonal states was carried out by Huttner et al. [158,

159] and more states by [160].

The third scheme is a generalization of the unambiguous discrimination strategy

and allows for the possibility of linear dependence between the states. Due to this

possibility, there will always be errors associated when determining some states. The

4.1 Introduction 75

maximum confidence measurement strategy was introduced by Croke et al. [161]

and the scheme maximizes the confidence with which one identifies a given state

ρi. Here the confidence is defined as the the ratio between the number of correct

detection events and the total number of detection events when the outcome i is

detected [145,161]. Experimental verification of maximum confidence quantum state

discrimination has been shown by Mosley et al. [162] and Croke et al. [163].

Quantum state discrimination between two (or more) known orthogonal quantum

states has been performed experimentally for nearly all types of quantum states except

for images. Single photon image state discrimination is particularly difficult because

most single photon detector arrays are poorly suited to single photon image detection.

In this chapter we present results of an experiment demonstrating the quantum state

discrimination between two known single photon images using holographic methods.

Many protocols for encoding information onto an optical beam limit the infor-

mation content of an individual photon to one bit of information in classical optical

communication systems, or to one qubit of information for quantum information pro-

cessing protocols [164]. However, recent work has emphasized the vast Hilbert space

and thus the vast potential information content of a single photon. The use of the

orbital angular momentum states, such as Laguerre-Gauss (LG) states, of the photon

is one such example [165–167]. Since these states form an infinite basis, in principle

there is no limit to the information content that can be carried by a single photon.

Spatially encoded qudits [168], hyperentangled photon pairs [169] and multiphoton

4.1 Introduction 76

entanglement [170, 171] are such examples of the large information content of quan-

tum light field. Entanglement of a large number of photons, of the order of 100,

has been demonstrated recently [172]. Other examples include entanglement between

two photons generated by the process of parametric down conversion that can exist

in a high-dimensional Hilbert space; pixel entanglement between two optically en-

tangled d = 3 and d = 6 qudits was demonstrated experimentally using transverse

position-momentum entangled biphotons [173] and large-alphabet quantum key dis-

tribution using energy-time entangled biphotons with dimensionality d = 1024 was

experimentally demonstrated [174].

In the previous two chapters, the emphasis was on (ghost) image (GI) formation.

We have studied the use of compressive sensing in quantum GI in chapter 2 and the

effects of the statistics of the illuminating field on thermal GI quality in chapter 3.

The study in this chapter is concerned not with image formation, but with image

discrimination at low light levels. Here we describe an experimental procedure that

we have used to impress transverse image information onto an individual photon. If

the image is a member of a predefined basis set, we can determine which image is car-

ried by the photon by performing a single measure using holographic matched filtering

techniques. We should note here that Mair et al. [165] have used simplex holograms to

measure single-photon orbital angular momentum states. In this chapter we demon-

strate that arbitrary image states from a known basis set may be distinguished by

performing a single measure using a multiplexed hologram. This procedure should

4.2 Theory 77

be contrasted with the earlier work of [175], in which an image was impressed upon

a single photon, but the image was read out in a statistical fashion, one pixel at a

time, and thus required an ensemble of events to reproduce the image.

In this chapter, we present results from a proof-of-principle experiment which dis-

tinguishes between a basis set of two orthogonal images [164]. Quantum GI discrim-

ination for upto four spatially nonoverlapping objects have been recently carried out

by Malik et al. [176]. However, much larger sets of images (orthogonal or nonorthogo-

nal) can be used. In the classical field of matched filtering, as many as 10 000 images

have been fixed onto a single large-scale holographic memory [177].

This chapter is organized as follows. In section 4.2 we present the basic idea

of our single-photon image discrimination. The experimental details are given in

section 4.3 and the results on hologram characterization and single-photon image

discrimination are presented in sections 4.4 and 4.5, respectively. A summary is

presented in section 4.6.

4.2 Theory

Single photon image state discrimination was carried out using holographic methods.

The basic concept of our approach is illustrated in Fig. 4.1. A multiple-exposure

hologram was formed (parts a-d) using N different transmission objects (Ai) with

reference beams applied from N different directions (θi’s in the figure). We then

pass a single photon through one of the N objects as shown in part (e) and allow it

4.2 Theory 78

to fall onto the multiplexed-hologram constructed by the procedure shown in parts

(a)-(d). This photon will then diffract into one of N output directions depending

upon which image was impressed onto the photon. This procedure is well known in

optical information processing [72, 178] and is related to the more general method of

matched filtering [179] (see Sec. 1.4). The general methods used in classical image

discrimination apply equally well to quantum-mechanical light fields.

Let us first describe image reconstruction for a simplex hologram, i.e., a holo-

graphic material exposed by a single pair of object and reference beams. We assume

that the holographic recording material is illuminated simultaneously by an object

wave of field strength T (x) and a reference wave of field strength R(x) so that the

total field at the hologram is E(x) = T (x) + R(x), where x = (x, y) represents the

two-dimensional Cartesian coordinates. We assume that after development, the trans-

mission t(x) of the hologram is proportional to the local optical intensity [180, 181]

so that t(x) ∝ |T (x) +R(x)|2 or that

t(x) ∝ |T (x)|2 + |R(x)|2 + T (x)R∗(x) + T ∗(x)R(x). (4.1)

During the reconstruction process in the conventional holography [180, 181], the

hologram is illuminated with a wave identical to the reference wave R(x) used in

recording the hologram. The field leaving the hologram is thus given by Eout(x) ∝

4.2 Theory 79

object Ai hologram

unknown

image

single

photon

output wave

(if image is )

object A1

holographic

plate

reference

(plane wave)

reference

(plane wave)

holographic

plate

(a) exposure 1 (b) exposure 2

(e) single-photon readout

holographic

plate

reference

(plane wave)

reference

(plane wave)

holographic

plate

(c) exposure N-1 (d) exposure N

.

.

.

.

.

.

θ1

object A2

θ2

object AN-1 object A

N

θN-1

θN

A2

output wave

(if image is ) A1

output wave (if image is ) AN

Figure 4.1: Concept of the single photon image discrimination experiment. Amultiple-exposure hologram was formed using N transmission objects (Ai) and refer-ence beams applied from different directions (θi), as shown in parts (a)–(d). (e) Afterthe hologram was developed, an unknown image was impressed onto a single photonby passing a heralded single photon through an object (Ai) from the predefined set.The form of the image is determined by diffraction from the multiplexed hologramformed in parts (a)–(d). Depending on the image carried by the single photon, thehologram diffracts the image into the corresponding unique direction.

R(x)t(x) or by

R(x)|T (x)|2 +R(x)|R(x)|2 + T (x)|R(x)|2 + T ∗(x)R2(x). (4.2)

The third term in this expression is the one leading to standard holographic re-

4.2 Theory 80

construction, and if R(x) is nearly uniform across the aperture of the hologram we

see that this term just reproduces the amplitude distribution T (x) of the original

object. If, however, the hologram is illuminated (read) by a replica of the structured

object beam T (x), as in the case of holographic matched filtering, the situation is

more complicated. We find that Eout(x) ∝ T (x)t(x) or by

T (x)|T (x)|2 +R(x)|R(x)|2 + T (x)2R∗(x) + |T (x)|2R(x). (4.3)

In this case, the fourth term is the one leading to the diffracted output beam, and

we see that, due to the structured nature of the read beam, the transverse structure

of the object beam will be imprinted onto the diffracted beam.

We next consider a multiplexed hologram created by exposing N object-reference

field pairs sequentially. Let the ith reference field (assumed to be plane waves) be

represented by Ri(x) = R exp[iki ·x] and the object field as Ti(x) = Ai(x) exp[iφi(x)].

Here Ai(x) and φi(x) are the transverse amplitude and phase functions of the ith

object field at the hologram plane, respectively. Without loss of generality, we assume

that R, Ai, and φi are all real. Similar to Eqn. 4.1 for the simplex hologram, the

transmission function of the multiplexed hologram is approximated by

t(x) ∝N∑

i=1

|Ti(x) +Ri(x)|2

∝N∑

i=1

RAi(x) {exp[i(φi(x) + ki · x)] + c.c}+ Ai(x)2 +R2 (4.4)

4.3 Experimental Details 81

After the hologram was exposed and developed, if the jth object beam is incident

onto the hologram, the amplitude of the image diffracted into the ki-direction is given

by

Ej(x) ∝ Tj(x) t(x)

∝N∑

i=1

RAj(x)Ai(x) exp[i(φj(x) + φi(x) + ki · x)]. (4.5)

If we now assume zero local spatial overlap between the illuminating and the ith object

beams, i.e., Aj(x)Ai(x) = δij A2i (x), then the amplitude of the image diffracted into

the desired k-direction becomes

Ej(x) ∝ A2j(x) exp[i(φj(x) + kj · x)], (4.6)

and the multiplexed hologram projects the incident light from each image Aj(x) into

the corresponding kj-direction and each image can be uniquely detected.

4.3 Experimental Details

The experimental setup used for writing the multiplexed (biplexed, to be more spe-

cific) hologram is shown schematically in Fig. 4.2. We have used two spatially non-

overlapping transmission masks as our objects, i.e, we use stencils of yin and yang

symbol as objects A1 and A2, respectively (represented as A and B in the inset of

Fig. 4.2). Objects A and B have no spatial overlap and as such constitute orthogonal


objects. The hologram is a thick angularly multiplexed phase transmission holo-

gram and is made using PFG-01, a fine-grained red-sensitive silver halide emulsion

on a glass plate substrate [182]. The emulsion has a peak light sensitivity of about

100 µJ /cm2 at 630 nm.

A B

HeNeLaser

Timer/Shutter

Hologram

Object

object

rotationstage

NPBS

Figure 4.2: Laboratory setup for writing the multiplexed hologram. Biplex hologramsare exposed using a HeNe laser and a pair of object-reference beam combinationssequentially. A shutter is used to electronically control the exposure time. For eachexposure, the reference-object pair is selected using a rotation stage and a translationstage. NPBS, nonpolarizing beamsplitter

Since the holographic material has a peak sensitivity in the red, we use a HeNe

laser (λ =632 nm) as the light source for recording the holograms. A nonpolarizing

beamsplitter (NPBS) is used to split the HeNe laser into object and reference beams.

Each beam has a power of ∼ 300µW after the NPBS. The object beam passes through

the object stencil (A or B), which are mounted on a translation stage and is imaged

onto the hologram recording medium with a 50 mm focal length lens, along with one

of the two reference beams. A precision translation stage allows us to reproducibly


place either object A or B in the object plane. A mirror mounted on a rotation stage,

in the other port of the NPBS, sets the direction of the reference beam A (B) for each

exposure. The biplexed hologram is then formed by illuminating the hologram by the

object-reference pairs (A-A and B-B) sequentially. The exposure time for each pair

is set at 350 ms by an electronically controlled shutter.

TCSPC

A

Hologram

B

HeCdLaser

BiBO

Trigger

Object

A B

NPBS

Figure 4.3: Laboratory setup for the single-photon image readout. Heralded sin-gle photons are sent through either object A or B, during the image-discriminationphase of the experiment, and are then detected at either detector A or B. TCSPC,time-correlated single-photon counter; BiBO, Bismuth Borate crystal; NPBS, nonpo-larizing beamsplitter

The setup used for the single-photon readout (image-discrimination) phase of the

experiment is shown schematically in Fig. 4.3. A HeCd laser operating at λ=325 nm

is used to pump a 10-mm-long nonlinear crystal (BiBO) cut for collinear type-I phase

matching. The nonlinear cyrstal is angle-tuned to produce, through the process of

spontaneous parametric downconversion (SPDC), degenerate biphotons of the same

4.4 Hologram Characterization 84

polarization at 650 nm. A UV grade fused silica dispersion prism is used to separate

the biphotons from the pump beam and are sent to a nonpolarizing beamsplitter

(NPBS). The object beam is created by heralding single photons emitted by the

SPDC process. That is, one output port of the NPBS is coupled directly through a

multimode optical fiber to a single-photon sensitive detector which serves as a trigger.

The presence of a photon in the other output port of the NPBS, hereafter called the

image photon, is heralded by the trigger. The image photon passes through the

object transmission mask (A or B) and the biplexed hologram along the same path

as when the hologram was exposed. Depending on the image carried by the incoming

photons, the image photons are diffracted from the hologram and are coupled through

multimode fibers to detectors A or B. All three detectors used in the experiment are

Perkin-Elmer single-photon counting module (SPCM) detectors. Detection events

are counted with a PicoQuant PicoHarp 300, a time-correlated single-photon counter

(TCSPC).

4.4 Hologram Characterization

We have characterized the performance of the biplex holograms using a HeNe beam

used to write the hologram. The quality of the reconstructed images when the holo-

gram is read out by a plane-wave reference beams, as in conventional holography, is

shown in parts (a) and (b) of Fig. 4.4. Here we see that the reconstructed images are

accurate replicas of the stencil objects (yin and yang symbols). For the single-photon

4.5 Single-Photon Image Discrimination 85

readout phase of the experiment, parts (a) and (b) of Fig. 4.5 show the diffracted

(a)

(b)hologram

read beam

(plane wave)

read beam

(plane wave)reconstructed

image

reconstructed

image

Figure 4.4: Object reconstruction using a plane wave read beam.

beams when the hologram is illuminated by one of the image-bearing beams.

As we have described in Sec. 4.2, when the hologram is read out using the image-

bearing (object) beams, the transverse structure of the object beam will be imprinted

onto the diffracted beam. This behavior is apparent in the data shown in parts (a)

and (b) of Fig. 4.5. Quantitatively, we have measured a peak diffraction efficiency of

about 24% (19%) for objects A (B), and we find that the cross talk between them is

negligible.

4.5 Single-Photon Image Discrimination

As we have described in Sec. 4.2, when stencil A (B) is used as the object, the image

discrimination photons are diffracted by the hologram into the direction of reference

beam A (B), where single-photon sensitive detectors A (B) are located (see Fig. 4.3).


(a)

(b)hologram

input

output

input output

Figure 4.5: Hologram readout with an image carrying read beam.

Coincidence events between the heralding and the image-discrimination photons are

measured for the four object-detector combinations: (1) A-A, (2) A-B, (3) B-A, and

(4) B-B. The total number of coincidences for each object-detector combination for

54 min of integration are reported in Table 4.1. The singles rate (for the heralding

and image-discrimination photons) and the accidental coincidences are also included

in Table 4.1. For better visualization, these results (raw and accidental coincidences

and their ratio) are also shown graphically in Fig. 4.6.

The singles rate, s, for the ith channel, i.e., the heralding and image-discrimination

channels, is given by,

si = ǫiR + bi, (4.7)

where ǫi is the collection and detection efficiency of the ith channel, R is the pair

generation rate of the SPDC process and bi is the background counts of the ith

channel, which includes detector dark counts and counts from stray light measured


with the pump beam blocked.

Raw coincidences are the number of coincident events generated within the 500 ps

coincidence window. The raw coincidence C is the sum of real and accidental co-

incident events. Coincident events arising from coincidences between the herald

and image-discrimination photon pairs are real coincidences and the rate is given

by ǫhǫidR, where ǫid(ǫh) are the collection and detection efficiency of the image-

discrimination (heralding) channels, respectively. Accidental (or spurious) coincident

events arise between (a) a background count and a heralding photon (ǫhbidR∆t),

where ∆t is the coincidence window, (b) a background count and an image-discrimination

photon (ǫidbhR∆t), (c) two background counts (bhbid∆t), and (d) two uncorrelated

photons from a multipair event (ǫhǫidR2∆t). The accidental rate A is thus given by

the sum of the above four coincident events as

A = shsid∆t

= ǫhbidR∆t + ǫidbhR∆t + bhbid∆t + ǫhǫidR2∆t. (4.8)

The number of accidental coincidences arising from events (a–d) given above can

be measured by counting the number of image-discrimination photons that arrive

20±0.25 ns after the heralding photons. The total number of raw coincidences C is

given by

C = ǫhǫidR + A. (4.9)


The single-event count rates are sh ∼500 k counts/s for the trigger (in the herald-

ing arm) and sid ∼450 counts/s (sid ∼250 counts/s) for detector A (B), in the image

discrimination arm (see Table 4.1). In practice, the high degree of loss in the image

discrimination arm (sid ≪ sh) implies that the accidental coincidences are dominated

by coincidences between background counts and heralding photons. The low collec-

tion efficiency in the image discrimination arm is due to a combination of factors:

transmission losses at the image mask and the biplex hologram, reflection losses at

lenses, mirrors and beamsplitters, coupling losses from coupling a highly multimode

image into a multimode optical fiber, and alignment issues caused by using differ-

ent laser wavelengths for the hologram exposure (633 nm), single-photon generation

(centered at 650 nm), and single-photon alignment (using 660 nm laser).

Table 4.1: Image-discrimination results showing the total number of raw coincidences(C), accidental coincidences (A), and C/A ratio for each object-detector combination.Also shown are the heralding photon and image-discrimination photon singles rates,sh and sid. Background rates are bh ≃ 1000 ± 32 Hz, bid,A = 420 ± 20 Hz, andbid,B = 249± 16 Hz. 1/R∆t represents the maximum possible C/A ratio as discussedin the text.

Object-detector sh (kHz) sid (Hz) C A C/A 1/R∆tA-A 437 473 5738± 75 337± 18 16.99± 0.95 143± 3A-B 522 252 185± 14 201± 14 0.93± 0.09 N/AB-A 444 414 289± 17 287± 17 1.01± 0.08 N/AB-B 511 273 4401± 66 229± 15 19.24± 1.30 210± 5

We see from Table 4.1 (also Fig. 4.6) that when the readout beam is object A, the

total number of raw coincidence for the A-A combination is much higher than the

A-B combination, with accidental coincidences being about the same for both cases.


ABA

B

0

2000

4000

6000R

awC

oin

c.

ABA

B

0

2000

4000

6000

Acc

iden

tal

Co

inc.

ABA

B

0

5

10

15

20C

/AR

atio

DetectorObject

DetectorObject

Detector Object

(b)(a)

(c)

Figure 4.6: Single-photon image-discrimination results. Total number of raw coin-cidences (C), accidental coincidences (A), and C/A ratio for each object-detectorcombination.

Similar conclusion can be inferred when object B is used. We use the ratio of true

coincidences NAA (NBB) to false coincidences NAB (NBA) as a metric to quantify the

fidelity of our system. The ratio

fi = Nii/Nij, i, j = A,B and i 6= j, (4.10)

is found to be 31.2 (15.2) for A (B), respectively. This means that we can distinguish

object A from B with a confidence level of ∼96.8%. For object B, the confidence

level is ∼93.4%. Here the confidence level is calculated as 1 − 1/fi, where fi is as

defined above. The C/A ratios for the A-B and B-A object-detector combinations are


approximately unity as can be seen from the data in Table 4.1, i.e., nearly all of the

false events, NAB and NBA, can be attributed to accidental coincidences. One way

to increase the system fidelity is thus by improving the C/A ratios for the A-A and

B-B object-detector combinations. The C/A ratios can be improved by increasing

the total collection efficiency in the image-discrimination arm (the collection and

detection efficiency in the image-discrimination arm of the setup is extremely low for

image A and B as can be seen in Table 4.1) or by using detectors with reduced dark

counts for the image-discrimination photon, such as Perkin-Elmer SPCM-AQR-16

detectors that have only 25 dark counts/s.

Using Eqns. 4.8 and 4.9, the C/A ratio can be reduced to the following expres-

sion [164]:

C/A = 1 +1

R∆t

[(1 +

bhǫhR

)(1 +

bidǫidR

)]−1

, (4.11)

where, as before, ǫi is the collection and detection efficiency of the ith detector, bi

is the background count rate for the ith detector, R is the biphoton generation rate

of the crystal and ∆t is the duration of the coincidence window. The C/A ratio is

limited by (R∆t)−1 and the discrimination confidence level is limited by 1− R∆t in

the ideal setting where bi ≪ ǫiR. In the present experiment (R∆t)−1 ∼ 143 so that

the discrimination confidence level is bounded by 99.30%. To date, the best C/A ratio

in entangled biphoton sources is ∼1000 from a Raman-scattering process in optical

fiber [183].

4.6 Summary 91

4.6 Summary

In summary, we have studied image discrimination at the single photon level and

have shown that it is possible to impress an image onto an optical field comprised

of a single photon and subsequently sort these photons into classes determined by

the image that the photon carries [164]. We have used basis sets containing only two

locally spatially orthogonal (spatially separated) images, for which very good discrim-

ination was obtained with a multiplexed image hologram. For many applications, a

much larger basis set, possibly including nonspatially separated, or more generally,

nonorthogonal images, would be desirable. Image discrimination of nonorthogonal

images using numerical correlation methods has been investigated by [34] where Mor-

ris showed that distinguishing images with a confidence level of 97% requires about

250 photons. For a basis set involving nonorthogonal images the principles of unam-

biguous state discrimination in large Hilbert spaces [184] may be applied to design a

hologram which optimally discriminates images in the basis set. Limits to the number

of images that can be discriminated in a hologram are set by issues such as cross talk,

which tends to increase with the number of stored images, and diffraction efficiency,

which tends to decrease with the number of stored images. However, it is reassuring

to note that as many as 10 000 images have been stored in a holographic memory

under appropriate conditions [177].

Chapter 5

Conclusions and Discussion

In this thesis, I have presented both theoretical and experimental studies on ghost

image formation using compressive sensing methods and transverse image discrimina-

tion at low light levels. I have also presented a study of how the statistics of the light

field used in thermal ghost imaging affects the quality of ghost images thus formed.

The first two chapters deal with different aspects of ghost imaging. Ghost imaging

(GI) is a novel transverse imaging modality that exploits the spatial correlation be-

tween the reference and test beams to retrieve information about an unknown object.

The two spatially correlated beams travel through two separate optical paths and can

have the same (degenerate) or different (nondegenerate) wavelength [185]. The test

beam, which interacts with the object, is not spatially resolved while the reference

beam does not interact with the object but is spatially resolved by a detector.

In quantum ghost imaging, a spatially resolving single-pixel detector is raster

scanned in the reference arm (Dr) of the GI setup. Due to the inherently weak

92


sources of entangled photons used for experiments in quantum GI, the signal-to-noise

(SNR) is improved by increasing the integration time (T ) for each single-pixel detector

position in the reference arm. If the number of pixels used to resolve the unknown

object N ≫ 1, as is usually the case, the total integration time (NT ) needed to

acquire the image of the unknown object could be prohibitively large for practical

applications.

In chapter 2, a new quantum GI setup was introduced. The configuration of

the test arm is the same as in other quantum GI setups (see, for example, [23]).

However a spatial light modulator (SLM), at the location of the spatially resolving

single-pixel detector Dr, is used instead in the reference arm. All the light that is

reflected from the SLM is collected by a bucket detector. Here the new and highly

evolving field of compressive sensing is used in reconstructing the ghost image. The

number of measurements required to resolve the object is not N , unlike in raster-

scanned quantum GI, but is dictated by the the sparsity level (K) of the object in

the transforming basis and is given by O[Klog(N/K)]. One clear benefit here is the

improvement in image acquisition time.

The ghost image reconstruction using the gradient projection method was carried

our for two object: the logo of the University of Rochester and the Greek letter Ψ.

High quality images were reconstructed using only 27% of the number of measure-

ments corresponding to the Nyquist limit. The fidelity of the compressed sensing

image reconstruction algorithm was also characterized by using the mean-squared


error (MSE) as a metric. We have found values of 0.06 and 0.03 for the logo of the

University of Rochester (UR) and the Greek letter Ψ, respectively, at 27% of the

measurement Nyquist limit.

We have also shown that the total number of photons needed for image recon-

struction using the compressed sensing based approach is much smaller than raster

scanned quantum GI. We note here that no systematic effort was made to mini-

mize the total number of photons needed for image reconstruction. We have used

the discrete cosine transform (DCT) as our image sparsifying basis. We should note

that since the image reconstruction was done offline, we could also search for other

transforming basis where the object could be represented more compactly than in

DCT.

In chapter 3, a systematic study on how the field statistics of the illuminating

source affects the quality of the ghost image formation was carried out, both theo-

retically and experimentally. In thermal ghost imaging experiments bright and high

contrast speckle patterns illuminate the object. One would expect that the quality of

the ghost image would be degraded as the contrast of the speckle patterns illuminating

the object decreases.

We have experimentally studied ghost image formation for varying contrast of the

illuminating speckle patterns. One of the metric used to quantify the quality of the

ghost image is the contrast-to-noise ratio (CNR). We have shown that the quality of

the ghost image is not affected even when low-contrast illuminating speckle patterns


are used as long as the fluctuations in the detected signal are due predominantly to the

randomness of the speckle pattern itself and not due to noise in the detection system.

We have shown, both theoretically and experimentally, that the CNR depends on

the kurtosis and not on the contrast of the illuminating field. We derived how the

CNR depends on the number of measurements (N), the total number of transmitting

pixels of the object (T ) and the fourth standardized moment or the kurtosis given

by (γI/σI)4, where σ2

I and γ4I are the second- and fourth- order moments about the

mean, respectively, of the intensity fluctuation for each illuminating speckle field.

We have carried out an experiment where as many as 25 speckle patterns are

averaged together for each measurement. We have found good agreement between

the experimental results and the theoretical predictions. These findings could have

important practical implications in ghost imaging experiments where thermal source

with short coherence time and slow detectors are employed.

In chapters 2 and 3, we considered different aspects of ghost image formation and

reconstruction. In chapter 4, we studied image discrimination, using holographic-

matched filtering techniques, at low light levels. Here we considered locally spatially

orthogonal set of two objects as our basis. Heralded single photons from a sponta-

neous parametric downconversion process are used for encoding and discriminating

images from our predefined orthogonal basis set. We show experimentally that we

can discrimination two objects with a confidence level of >93.4%. Using better single-

photon detectors (that have lower dark counts than used in our experiment), we have


shown that the confidence level can be as high as 99.34%. Despite the fact that our

predefined basis set consists of only two locally spatially orthogonal masks (objects),

holographic memory [177] could be used to extend the number of the predefined set.

An extension of these work using a quantum ghost imaging setup (see for exam-

ple, [23]) for upto four spatially nonoverlapping objects have been recently carried

out [176].

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Appendix A

Algorithms for Compressed

Sensing

In compressed sensing [63–66] we deal with finding unique solutions to an underdeter-

mined set of linear equations. Before we introduce the algorithm used to reconstruct

the ghost images in chapter 2 (see Fig. 2.6), consider the following set of linear equa-

tions:

Ax = y (A.1)

where A is an invertible N × N matrix, y is a vector in RN . One way of finding a

solution to the linear system of equations in Eqn. A.1 is to minimize the objective

125


(cost) function F (x), explicitly given by,

F (x) ≡ 1

2〈x,Ax〉 − 〈y,x〉

=1

2xTAx− yTx. (A.2)

subject to x ∈ RN . Here (·)T represents the transpose of a vector and 〈a,b〉 denotes

the dot product of the vectors a and b. We see the equivalence of Eqns. A.1 and A.2

by setting the gradient of F (x) to zero.

Although there are many methods, such as conjugate gradient, gradient projec-

tion [186–188], of solving the unconstrained minimization of the objective function

F (x) in Eqn. A.2, we use the steepest-descent method to show the general approach.

In this method, we start with an arbitrary initial guess for x, i.e. x(0). The objective

function decreases most quickly if we choose our search direction dk ∈ RN as the

negative gradient of F at xk, i.e.,

dk = −∇F (xk), (A.3)

and the next step (iterate) is given by

xk+1 = xk + αkdk, (A.4)

where the step size αk is determined by finding the minimizer of the objective function


at xk and dk. That is a line search procedure is carried out to find αk that minimize

the objective function along a line through

αk = argminαF (xk + αdk). (A.5)

The above steps ( A.3– A.5) are repeated until we get an approximate solution.

There are many methods and algorithms that have been recently used for com-

pressed sensing problems [189–202]. There are also a number of toolboxes and codes

developed by many groups (see for example [203–205]).

The gradient projection method by Figueiredo et al. [104,206], that we have used

for image reconstruction in chapter 2, can be used to solve the following unconstrained

optimization problem

minx

1

2||y −Ax||22 + τ ||x||1 , (A.6)

which includes a quadratic (ℓ2) error term combined with a sparseness-inducing (ℓ1)

regularization term. Here x ∈ Rn, y ∈ R

k, andA is an k×nmatrix, τ is a nonnegative

parameter, and ||v||p = (∑

i |vi|p)1/p denotes the ℓp norm.

If we make the following substitution for x:

x = u− v, u ≥ 0, v ≥ 0, (A.7)

the optimization problem (Eqn. A.6) can be reformulated as a bound-constrained


quadratic program:

minu,v

1

2||y −A(u− v)||22 + τ1T

nu+ τ1Tnv, (A.8)

s.t. u ≥ 0

v ≥ 0.

Here ui = (xi)+ and vi = (−xi)+ for all i = 1, 2, ..., n, where (.)+ denotes the positive-

part operator defined as (x)+ = max{0, x} and 1n = [1, 1, ..., 1]T is the vector consist-

ing of n ones. In more standard form (see Eqn. A.2), the bound-constrained quadratic

program (Eqn. A.8) becomes

minz

cTz+1

2zTBz ≡ F (z), (A.9)

s.t. z ≥ 0,

where z =

[u

v

], c = τ12n +

[−ATy

ATy

], and B =

[ATA −ATA

−ATA ATA

].

The gradient projection method is then applied to Eqn. A.9. After an initial guess

(z(0)), the next iterate z(k+1) is determined through the following steps. We set, for

some scalar parameter α(k) > 0, which is determined using a line search similar to

Eqn. A.5,

w(k) = (z(k) − α(k)∇F (z(k)))+. (A.10)


After picking a second scalar λ(k) ∈ [0, 1], we determine the next iterate through

z(k+1) = z(k) + λ(k)(w(k) − z(k)). (A.11)

Explicit expressions for α(k) and λ(k) are given in [104]. A convergence test is per-

formed and the above steps are repeated until an approximate solution is found.

Another program that we have used is called CVX [207]. CVX is a freely available

Matlab-based program for various optimization problems. In chapter 1, we have

described how different ℓ1 and ℓ2 minimization are using a numerical example. Here

we have the optimization problem:

minx

||x||1 subject to y = Ax, (A.12)

The following snippet is used to do ℓ1 minimization to generate Fig. 1.3(a) in

chapter 1.

cvx_begin

variable xp(n);

minimize(norm(xp, 1));

subject to

A*xp==y;

cvx_end

Image Reconstruction and Discrimination at Low Light Levels by Petros Zerom

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