COMPRESSIVE MICROWAVE RADAR HOLOGRAPHY

The Pennsylvania State University

The Graduate School

College of Engineering

COMPRESSIVE MICROWAVE

RADAR HOLOGRAPHY

A Thesis in

Electrical Engineering

by

Scott A. Wilson

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Master of Science

December 2014

The thesis of Scott A. Wilson was reviewed and approved* by the following:

Ram M. NarayananProfessor of Electrical EngineeringThesis Advisor

Timothy J. KaneProfessor of Electrical Engineering

Kultegin AydinProfessor of Electrical EngineeringHead of Electrical Engineering Department

* Signatures are on file in the Graduate School

ii

Abstract

Radar holography has been established as an effective image reconstruction process

by which the measured diffraction pattern across an aperture provides information of

a three-dimensional target scene of interest. Compressive sensing has emerged as a

new paradigm in applications involving large amounts of data acquisition and storage.

The fusion of these two fields of research has had only limited consideration in radar

applications. Typically, full sets of data are collected at the Nyquist rate only to be

compressed at some later point, where information-bearing data are retained and in-

consequential data are discarded. However, under sparse conditions, it is possible to

collect data at random sampling intervals less than the Nyquist rate and still gather

enough meaningful data for accurate signal reconstruction. In the research presented

in this thesis, we employ sparse sampling techniques in the recording of digital mi-

crowave holograms over a two-dimensional scanning aperture. Using a simple and

fast non-linear interpolation scheme prior to image reconstruction, we show that the

reconstituted image quality is well-retained with limited perceptual loss.

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Table of Contents

List of Figures v

Acknowledgments viii

1 Introduction 1

2 Microwave Holography 32.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Fourier Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Holographic Image Reconstruction . . . . . . . . . . . . . . . . . . . 9

3 Compressive Sensing 113.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Sparsity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Simulation Procedure and Results 164.1 Simulation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Single-Frequency Image Reconstruction . . . . . . . . . . . . . . . . . 164.3 Wideband Image Reconstruction . . . . . . . . . . . . . . . . . . . . 18

5 Experimental Procedure and Results 205.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Single-Frequency Image Reconstruction . . . . . . . . . . . . . . . . . 215.3 Wideband Image Reconstruction . . . . . . . . . . . . . . . . . . . . 22

5.3.1 Image Reconstruction of a Simple Target . . . . . . . . . . . . 225.3.2 Image Reconstruction of a Single Concealed Object . . . . . . 255.3.3 Image Reconstruction of Multiple Concealed Objects . . . . . 31

5.4 Noise and Filtering Considerations . . . . . . . . . . . . . . . . . . . 385.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Conclusions and Future Work 44

A Additional Experimental Data 45

References 59

iv

List of Figures

1 Configuration for interferometric recording. . . . . . . . . . . . . . . . 72 Geometric configuration for holographic imaging system. . . . . . . . 103 Hologram produced by simulated target letter ’H’ and corresponding

discrete Wavelet transform (DWT) coefficients; thresholding of near-zero coefficients and reconstruction of hologram using inverse discreteWavelet transform (IDWT) from non-zero coefficients. . . . . . . . . . 14

4 (a) Simulated single-frequency hologram, (b) reconstructed image at z= 1.5 m plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Normal holographic reconstruction method versus reconstruction frominterpolated data subset. . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Holographic imaging process; (a) Distributed target at [xp, yp, z0] isilluminated with step-frequency waveform at discrete intervals along[x, y, z = 0]. (b) Sparsely sampled wideband hologram record. (c)Complete wideband hologram record approximation via interpolation.(d) Reconstructed three-dimensional image scene from interpolateddataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 Experimental test setup for holographic imaging system. . . . . . . . 208 Comparison of holographic reconstruction processes for normally sam-

pled dataset and randomly undersampled dataset. . . . . . . . . . . . 219 Holographic imaging and sparse reconstruction procedure of a prop

gun target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310 Wideband holographic reconstruction of target at a distance of 1.5

meters in cluttered environment; (a) reconstructed image from fulldataset, (b) reconstructed image from only 10% random samples (Cross-correlation = 0.995, PSNR = 33.6 dB); (c) Progression of reconstructedimage quality as number of measurement samples is reduced. . . . . . 24

11 Holographic imaging and sparse reconstruction at z = 0.63 m of aconcealed knife inside stuffed animal. Sparsely-sampled holograms atdifferent compression ratios and corresponding image reconstructions. 26

12 Holographic imaging and reconstruction of prop gun concealed withinpurse 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

13 Holographic imaging and reconstruction of prop gun concealed withinpurse 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

14 Acoustic guitar with concealed weapon, hologram record at 12 GHz,and reconstituted image. . . . . . . . . . . . . . . . . . . . . . . . . . 29

15 Image reconstruction of hollow-body acoustic guitar with concealedweapon at various compression ratios. . . . . . . . . . . . . . . . . . . 30

16 Experimental setup and holographic image reconstruction of purse andconcealed weapon inside of stuffed animal. . . . . . . . . . . . . . . . 32

17 Holographic imaging and reconstruction of concealed weapon with bookinside of purse 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

18 Sparse holographic image reconstruction of backpack containing sweat-shirt and concealed knife. . . . . . . . . . . . . . . . . . . . . . . . . . 34

v

19 Holographic reconstruction of concealed objects within cardboard box1; image of wrench reconstituted at z = 0.364 m; image of prop gunreconstituted at z = 0.540 m. . . . . . . . . . . . . . . . . . . . . . . 36

20 Holographic reconstruction of concealed objects within cardboard box2; image of wrench reconstituted at z = 0.375 m; image of knife recon-stituted at z = 0.493 m; image of prop gun reconstituted at z = 0.623m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

21 Sparse holographic image reconstruction of wrench (z = 0.364 m) andprop gun (z = 0.540 m) inside cardboard box 1 for different compressionratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

22 Filtering process in order to reduce noise artifacts introduced by mea-surement system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

23 Reconstructed images of prop gun with various noise levels added priorto reconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

24 Comparison of image quality with various noise levels added prior toreconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

25 Comparison of interpolation methods; SNR, MSE, and SAD versuscompression ratio averaged over N = 14 independent datasets. . . . . 43

26 Comparison of interpolation methods applied prior to reconstruction;PSNR, L1-norm, and L2-norm versus compression ratio. . . . . . . . 45













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vii

Acknowledgments

I would first like to give thanks to my thesis advisor, Dr. Ram M. Narayanan, for all

of the guidance and wisdom he has provided me. Dr. Narayanan has been a great

role model and educator. Under his supervision, Dr. Narayanan has made it possible

for me to achieve my academic goals and has paved the way for me to launch into a

successful career in Electrical Engineering.

Secondly, I would like to give gratitude to Dr. Eric Lenzing from Applied Research

Laboratory for providing some of the measurement hardware that made it possible for

me to conduct my experimental research. I also wish to thank Dr. Timothy Kane for

serving on my thesis committee and providing very useful comments and suggestions.

Furthermore, I would also like to thank all of the fellow student researchers in the

Radar and Communications Lab group for providing a fun and stimulating work

environment.

Lastly, I would like to give my greatest thanks to my family for everything they

have done for me, as well as for allowing me to discover and pursue my passions. I

wouldn’t be where I am today without all of their love, encouragement and support

they have provided me over the years.

viii

1 Introduction

Over the last 50 years, principles established in optical holography have been demon-

strated in the microwave regime. In the early stages of development, microwave

holograms were recorded on film or other types of analog media and image reconsti-

tution could be performed only by physically illuminating the hologram with a coher-

ent optical source. Advancements in data acquisition systems and signal processing

algorithms have made it possible to numerically reconstruct a target scene from a

digitally-recorded hologram. Most recently, fast Fourier transform (FFT) methods

have been able to demonstrate high-resolution image reconstruction of a digitally-

recorded hologram in nearly real-time [1]. Using a coherent microwave radar system,

interference patterns produced by a scattering target can be measured and recorded

as a digital hologram, after which the original target scene may be reconstructed using

an FFT-based reconstruction algorithm. For systems operating at single-frequency,

the reconstruction algorithm provides the ability to re-focus a two-dimensional image

to any single focal depth. Wideband systems yield an additional dimension for range

resolution, thereby extending the focal depth to where scattering fields produced by

three-dimensional geometries can be resolved into a well-focused surface volume.

Compared to optical imaging systems, advantages of the wideband holographic

radar include the ability to penetrate common materials and structures, retain ac-

curate phase and amplitude information of scattered fields, as well as being able to

measure and numerically reconstruct target information in multiple dimensions [2].

For these reasons, a simple wideband holographic radar system has been developed

for measuring the backscattered fields from an illuminated target scene. The FFT-

based reconstruction algorithm has been demonstrated on a basic simulation model,

as well as on experimental data for simple targets, as well as concealed objects. These

results were then evaluated under sparse sampling conditions in order to demonstrate

that the traditional sampling theorem is not an absolute requirement in image recon-

1

struction of sparse target scenes. The concept has been established for applications in

optical holography [3,4], yet this approach has not yet been demonstrated for X-band

radar applications. The findings of this research have indicated that there are defi-

nite benefits to the combined application of compressed sensing and wideband radar

holography.

2

2 Microwave Holography

In 1948, the concept of optical holographic imaging was first established by Gabor [5],

which he first termed as “wavefront reconstruction,” but later came to be known as

holography. The term holography is of Greek origin meaning “whole recording.”

The innovation in Gabor’s work was in that by capturing the interference pattern in

both amplitude and phase between a coherent reference wave and backscattered light

waves, reconstruction of the original target scene was possible. As a pioneer in this

field of holographic imaging, Gabor’s findings paved the way for the development of

holographic imaging in both the optical and microwave spectrum.

2.1 Background

Although the principles of holography had been discovered as early as the late 1930’s,

novel applications and techniques continue to advance this popular field of study.

Some of the earliest work in this field was conducted by scientists and researchers

such as Gabor [5–8], Leith [9], Dooley [10], and Anderson [11] to name a few. After

preliminary work in optical holography had been established, one of the first ex-

tensions of this concept into the microwave regime was presented in a publication

by Dooley [10] in 1965, where the diffraction pattern from a scattering target was

recorded using a voltage probe and an oscilloscope to create an amplitude hologram.

The hologram record obtained in this experimental work contained sufficient infor-

mation to reconstruct the original target scene. However, since the system was only

able to capture amplitude measurements, phase information was not obtained and

therefore range information was lost in the process and image reconstruction quality

was adversely affected as a result.

One of the most notable early achievements in the development of microwave

holography was presented in a 1971 paper published by Leith [9], where the relation-

3

ship between radar and holography was established. From this work, it was shown

that amplitude and phase could be recorded using microwave electronics and without

the need of an incident reference wave upon the recording medium. Also in that

same year, Dennis Gabor was formally recognized for his early achievements and was

awarded the Nobel prize in physics for his invention of holographic microscopy [2].

One practical application of holographic radar systems has been developed for

subsurface imaging and concealed object detection [1, 12–19]. A group from the

Bauman Moscow State Technical University has developed several subsurface radar

systems, known as RASCAN, for applications including mine detection, utility map-

ping, non-destructive testing of building structures, and concealed weapon detec-

tion [12–14, 16, 17]. These systems are capable of operating at single frequency or

multi-frequency in the ranges of 1.6 - 2.0 GHz, 3.6 - 4.0 GHz, and 6.4 - 6.8 GHz [20].

The major disadvantage of these microwave imaging systems is in their limited band-

width, greatly limiting the range resolution. Additionally, scan time could be im-

proved through application of sparse sampling, allowing the systems to produce im-

ages of reasonable quality with many fewer samples than required by their current

design.

Similar types of scanning systems have been developed for active microwave holo-

graphic imaging. Using transmitted and reflected signals, Amineh et al. [21] proposed

a system which uses two parallel mechanical scanners on both sides of the inspection

region for measurement of forward and backward-scattering fields. Their frequency

range of operation selected was 3.0 - 10.0 GHz and only utilized 27 frequencies, provid-

ing a maximum ambiguous range of only 70 mm. Despite the impractical performance

parameters, the imaging capability of this system was satisfactory. However, the cost

and complexity of a system requiring two single-element scanners is prohibitive in

many ways, primarily in scan time and possible synchronization error.

Microwave holographic techniques have recently been applied in the commercial

4

automotive market. These systems have aimed to collect and provide information for

ground speed, acceleration, road condition recognition, pre-crash detection, automatic

cruise control (ACC), parking aid, and back-up warning [22]. One design proposal

from a research group at Toyota is for development of a holographic radar with

antenna switching for synthesis of a 1-D holographic antenna array. The proposed

interferometric system would be capable of resolving targets with angular resolution

of less than 2 degrees and an azimuthal field of view of more than 20 degrees [23]. The

major considerations of this system are low cost and low form-factor. Development

of commercial holographic radar products will certainly drive methods further for

reducing cost in system design. The concept of compressed sensing could certainly

have a role in achieving this important goal.

One of the most useful applications of microwave holographic imaging has been

in personnel screening and airport security [19, 24]. The concern for chemical and

biological warfare agents has risen as constant threats of terrorism have become a

grim reality. Research and development of a holographic imaging system intended

for detecting various types of threats has been ongoing at Pacific Northwest National

Laboratory over the last 20 years and has yielded some impressive results. The

systems demonstrated in [1,19,24–26] are generally operated with bandwidths around

10 GHz for high resolution imaging applications. One of the physical limitations in the

design of these systems is that it has not been possible to meet the Nyquist criterion

in regards to antenna spacing due to the physical size of the radiating elements,

introducing the possibility of aliasing for high spatial frequency target scenes [25,26].

This problem may be mitigated by introducing the random sampling and interpolation

techniques demonstrated in the following sections of this document. Furthermore,

system design cost and complexity may also be reduced by considering the simple

methods presented for sparse holographic image reconstruction.

5

2.2 Fourier Optics

For near-field imaging, the theory of Fourier optics is used in the modeling of spherical

wave scattering and image reconstruction from the recorded interference pattern of

such. The idea is that a curved wavefront may be regarded as a superposition of plane

waves, thereby allowing the use of Fourier analysis and diffraction theory to derive a

forward scattering model. A standard technique for recording phase and amplitude

on an analog recording medium is interferometry [2]. By adding a mutually coherent

reference wave to the object wave at the interface of the recording medium, the

intensity of the sum of these two complex fields then depends on both amplitude and

phase of the unknown scattered electric field. This concept is depicted in Figure 1. To

understand this technique, we can represent the complex wavefront of the scattered

field as

a(x, y) = |a(x, y)| exp[jφ(x, y)] (1)

and the reference wave of known phase and amplitude with which a(x, y) interferes

with as

A(x, y) = |A(x, y)| exp[jψ(x, y)]. (2)

By adding 1 and 2, the intensity measured at the recording medium interface can be

expressed as

I(x, y) = |A(x, y)|2 + |a(x, y)|2 + 2|A(x, y)||a(x, y)| cos[ψ(x, y)− φ(x, y)] (3)

where the first two terms of the expression depend only on amplitude, while the last

term depends on the phase relationship between the “object” wave and the interfering

“reference” wave. This “total record” of both phase and amplitude for the scattered

wave a(x, y) can be regarded as a hologram.

6

Figure 1: Configuration for interferometric recording.

One of the most important parameters in any imaging application is resolution.

For three-dimensional imaging applications, we must consider both lateral resolution,

or cross-range resolution, as well as range resolution. For the proposed aperture

system, the lateral resolution is diffraction limited. Bragg’s law determines that the

cross-range resolution can be expressed as

δx =λR

2D(4)

where λ is the center frequency of the transmitted waveform, R is the range to the

target, and D is the size of the aperture. Lateral resolution will be the same in the y-

direction. Range resolution is solely dependent on the temporal frequency bandwidth

of the transmitted waveform and can be expressed as

δz =c

2B(5)

where c is the speed of light through the measurement environment, and B is the

transmit signal bandwidth.

7

As previously mentioned, a hologram is produced by recording the interference

pattern, both amplitude and phase, between a coherent reference wave and diffracted

waves produced by a target scatterer. Since supplying this coherent reference wave

is impractical, the major advantage to microwave holography over the traditional

optical holographic setup is in the capability to supply the interfering reference wave

using microwave circuit elements. In microwave holography, the phase and amplitude

can be measured directly using simple configurations of basic microwave components,

such as oscillators and mixers. A local oscillator (LO) can effectively be used as the

reference wave in the same way as in the case of optical holography. Using the LO

reference, the received signal can be mixed down to baseband then digitally sampled

and reconstructed on a computer through numerical synthesis of the original LO

frequency. The added benefit of this digital holographic technique is that the reference

signal can be numerically synthesized for image reconstruction, thus requiring no

physical source for reconstitution as in the case of optical holograms.

Well-defined algorithms have been applied to effectively shift the aperture focal

plane to the location of the target, producing a focused image of the original tar-

get scene based off of the target’s reflectivity function [1, 18, 25–27]. Although this

method is similar to conventional synthetic aperture radar imaging techniques, it dif-

fers in imaging geometry and in that it requires no field approximations [27]. For

near-field imaging, a common reconstruction technique assumes no far-field approx-

imations and only relies on the FFT algorithm. This reconstruction process was

originally formulated for two-dimensional synthetic aperture radar image reconstruc-

tion and has been extended to accommodate three-dimensional wideband holographic

reconstruction [1]. Results are presented in Chapters 4 and 5 to demonstrate both

reconstruction methods.

8

2.3 Holographic Image Reconstruction

The process of numerically reconstructing a two-dimensional hologram is relatively

straightforward. Considering the measurement configuration used in Figure 1, a

transceiver is scanned over the two-dimensional aperture, illuminating the scatter-

ing target and measuring the amplitude and phase of the backscattered field. The

response measured by the transceiver can be represented as

s(x′, y′, ω) =

∫∫f(x, y, z)e−j2k

√(x−x′)2+(y−y′)2+(z−z0)2dxdy (6)

where each point of the target’s reflectivity function f(x, y, z) is simply multiplied

by a round trip phase factor and summed at each sample location (x′, y′, z0). The

convention used in Figure 2 is that primed coordinates relate to the aperture plane,

while unprimed coordinates correspond to the target plane located at z = 0. The

phase factor is also a function of wavenumber, denoted as k = ω/c, where ω is

the angular frequency and c is the speed of light. The total measured response

s(x, y, ω) over the aperture represents a complex hologram record for each frequency

. As derived in [1], the FFT-based reconstruction algorithms for 2-D and 3-D image

reconstruction of a target f(x, y) at z0 are given, respectively, by

f(x, y) = FT−12D [FT2D[s(x, y)]e−j√

4k2−k2x−k2yz0 ] (7)

f(x, y, z) = FT−13D [FT2D[s(x, y, ω)]e−j√

4k2−k2x−k2yz0 ] (8)

where FT−13D denotes the three-dimensional inverse-FFT operation, FT2D and FT−12D

indicate the two-dimensional FFT and inverse-FFT operations, respectively, and kx,

ky and kz are planar wavenumbers in the x, y, and z directions, respectively, which

span the range between −2k and +2k. We drop the distinction between primed and

unprimed coordinates because at the plane of reconstruction, these coordinate systems

9

Figure 2: Geometric configuration for holographic imaging system.

intersect [1]. In the wideband holographic system, a set of linear step frequencies fi

are transmitted between f0 and fn, related to wavenumber by ki = c/(2πfi). Further-

more, because the data are discretely sampled in x, y, and ω, a geometric correction

must be applied before applying the wideband reconstruction of equation (8) by re-

sampling FT2D[s(x, y, ω)] uniformly in the kz dimension. This is accomplished by

nearest neighbor interpolation across holograms at each source wavenumber between

k0 and kn.

Current multi-dimensional holographic imaging methods require large amounts

of data be collected over an entire two-dimensional aperture at sampling intervals

less than λ/2. This can be done using a two-dimensional antenna array as well

as a scanning aperture configuration. The collected data can then be used to form

focused images at different planes along the target scene. When recording holographic

datasets of sparse target scenes, it is important to consider whether all of these

measurement data are actually necessary in order to reconstruct the original target

scene. This concern will be discussed in the following chapter.

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3 Compressive Sensing

It has been recognized since the early days of information theory that data is often

highly redundant, and is therefore compressible [28–30]. This redundancy of data

holds implications that Shannon-Nyquist criterion may be misleading, as it may be

possible to possess the same amount of information from a fewer number of observa-

tions than the conventional sampling theory suggests [31–33]. A variety of compres-

sion techniques have been developed in order to reduce the amount of data after it has

been collected [34,35]. However, this paradigm has recently been challenged with the

field of compressed sensing, where minimization of data is desired at the collection

step rather than after the data is obtained [31]. This section covers the concepts be-

hind sparse sampling and considers the requirements and theoretical limits of signal

reconstruction from undersampled datasets.

3.1 Background

The field of compressive sensing (CS) has most recently attracted much attention for

its potential benefits in applications performing large-scale data acquisition. Com-

pression techniques have long been in use in order to remove an amount of redundancy

from data, such as seen in audio and image files [35]. To extend this idea of data reduc-

tion, it was questioned whether it would be possible to simply collect the meaningful

data without wasting resources to obtain the unsubstantial data. In the mid-to-late-

2000’s, the first set of proofs were derived by Candes [4, 32] and Donoho [31, 36, 37]

which provided the theoretical foundation for compressed sensing and signal recon-

struction from incomplete data. This work opened the doors to a vast amount of

related research in application-based compressed sensing.

Using the CS framework outlined in [4,31–33], a variety of reconstruction methods

have been developed. Among them, Chen and Donoho saw application of their Basis

11

Pursuit method [38] as a means of minimizing the l1-norm for incomplete datasets.

One of the most popular algorithms used in solving ill-posed problems is known as

total-variation (TV) regularization [39]. This method has been a well-established

iterative method for converging upon solutions to ill-posed problems, but does not

meet performance needs for converging upon high-order solutions [40]. Another pop-

ular algorithm developed primarily for image restoration is the two-step iterative

shrinkage/thresholding (TwIST), which uses a TV-minimization approach in solving

ill-conditioned or ill-posed inverse problems [40].

Most recently, these methods have been applied in optical and millimeter-wave

compressive holography [3,41,42]. Cull et al. [41] used a 94 GHz source with a scan-

ning probe to measure sparse holographic data and reconstruct the undersampled

data using iterative minimization techniques. Similarly, Brady et al. used both the

TV-minimization and TwIST algorithms to reconstruct holographic data produced

in the optical spectrum (λ = 632.8 nm). In both cases, adequate solutions were

converged upon using these iterative methods. However, the cost of processing time

in these algorithms can be prohibitive in real-time imaging applications. Therefore,

the simple and effective method outlined in this document is intended to provide a

compromise in computational cost and image reconstruction quality. Our proposed

method allows for relatively accurate estimation and reconstruction from sparse holo-

graphic data on the order of seconds, as opposed to hours as noted in [3]. The only

knowledge required to implement the non-iterative estimation is that the target scene

is sparse in some arbitrary transform space (i.e. Fourier or Wavelet). This simple

interpolation technique is demonstrated and discussed in Chapter 5.

3.2 Sparsity

There are many applications which utilize sparsity in some form or another, whether

in measurement systems, antenna array design, or in system characterization. De-

12

velopment of a precise theory for random sampling has only truly been approached

on a statistical basis [37]. Compressed sensing (CS) theory has pushed the envelope

in challenging the notion of Nyquist’s minimum sampling theory and suggesting that

under certain conditions, sufficient information about a signal can be obtained using

fewer measurements than what has been traditionally considered necessary. As de-

scribed by Donoho [31], the CS framework suggests that if a signal can be sparsely

represented in some orthonormal basis Ψ (i.e. Haar-Wavelet transform), accurate re-

construction of a K-sparse signal can be obtained from M randomly chosen samples

for M < N total samples, M satisfying

M ≥ C ·K · logN (9)

where the value of C is a small positive empirical value [3]. The sensing operation

can be understood by the system of linear equations, written in matrix form, as

g = Φf = ΦΨα (10)

where g is the observable set of measurements, Φ is known as the sensing operator,

and f is the signal (image) to be determined. A sparsifying operator Ψ is used to

transform the object function into an arbitrary domain (i.e. wavelet domain), pro-

ducing a K-sparse coefficients vector α. Two common techniques are used to estimate

the original function f , namely, l1-norm minimization and total variation (TV) opti-

mization [4,40]. The problem with both of these approaches is that they are iterative

processes, requiring a significant amount of time to determine a fit solution. This may

be appropriate for recovering lost or corrupt data, but these methods are relatively

ineffective in a real-time holographic imaging system. Our method implements a sim-

ple interpolation technique prior to image reconstruction, which is effective due to the

fact that the target scene to be reconstructed is sparse in some arbitrary dimension.

13

Figure 3: Hologram produced by simulated target letter ’H’ and corresponding dis-crete Wavelet transform (DWT) coefficients; thresholding of near-zero coefficientsand reconstruction of hologram using inverse discrete Wavelet transform (IDWT)from non-zero coefficients.

Due to this sparsity, interpolation can be applied to the randomly undersampled holo-

gram before holographic image reconstruction with relatively low mean squared error

(MSE). This property is shown both qualitatively and quantitatively in Chapters 4

and 5, with additional results presented in Appendix A.

We know that there are many cases where data is compressible, i.e. JPEG files,

MP3 files, and streaming video. In general, we have recognized that this reduction

of data does not result in a significant reduction of meaningful information. A signal

is considered to be sparse if it can be represented in some orthonormal basis with

only K non-zero terms, K << N . It has been shown in papers by Donoho [31] that

accurate reconstruction can be achieved from only M < N samples. As an example,

sparsity is exemplified in Figure 3 by applying the Wavelet transform to a complex

hologram, revealing that many of the Wavelet coefficients are near zero. This image

14

is said to be K-sparse when there are only K << N significant coefficients. Using

(9) and assuming a value of C = 1, the minimum number of random samples needed

for sparse signal representation is M = K · logN = 4678 samples. From N = 16384

of the original samples, the compressed signal has a sparse representation requiring

only K = 1110 non-zero terms. Thus, it has been shown that accurate reconstruction

is possible with many fewer samples than normally required by Nyquist’s theorem.

15

4 Simulation Procedure and Results

4.1 Simulation Process

Simulated data are produced in MATLAB by importing a binary image and treating

every non-zero pixel as an isotropic radiating element. For the scanned aperture in

Figure 2, the total response measured at each detector pixel is simply the summed am-

plitude and phase from each radiating element of the target model, which is captured

by equation 6.

4.2 Single-Frequency Image Reconstruction

In this section, we discuss the simulation results from sparsely-sampled digital holo-

grams produced by a single frequency. We use the principles discussed in Chapter 2

to numerically produce a hologram via computer simulation.

The simulated hologram of Figure 4(a) is 128 × 128 samples and was generated

from a single source frequency of 9.0 GHz. The sampling step size in both x- and y-

directions is ∆x = ∆y = 8.3 mm, resulting in a scanned aperture dimension of 1.06 m

× 1.06 m. The target’s simulated distance from the aperture plane was z = 1.5 m. The

complex-valued hologram data were then used to numerically reconstruct the original

Figure 4: (a) Simulated single-frequency hologram, (b) reconstructed image at z =1.5 m plane.

16

target image using equation (2). The same process is used in the X-band holographic

radar systems discussed in [12–18]. However, we have extended our method to include

a random sampling process with interpolation prior to reconstruction, significantly

reducing the sampling and data acquisition requirements.

The inclusion of the random sampling process can be seen in the simulation ex-

periment of Figure 5. A single-frequency hologram is produced from a simulated

target scene of two humans, one carrying a concealed weapon. Random sampling is

performed on the original hologram dataset to produce a subset of data containing

only 10% of the original number of samples. After applying a simple cubic-spline

interpolator to the subsampled dataset, a reconstructed image is produced with a

cross-correlation value against the original reconstructed image of ρ = 0.98.

Figure 5: Normal holographic reconstruction method versus reconstruction from in-terpolated data subset.

17

4.3 Wideband Image Reconstruction

In a real-world holographic imaging system where scan time and data storage

requirements are system limitations, our results show that the random sampling

paradigm can be very advantageous for X-band applications. Applying these con-

cepts to wideband datasets has been investigated in simulation as well as through

an experimental test setup. Given a distributed, three-dimensional target, wideband

holography provides greater depth of field for better imaging resolution [1]. As an ex-

ample, a randomly sampled wideband hologram is measured from a simulated target

existing within three-dimensional space, as seen in Figure 6(a). Three-dimensional

holographic reconstruction is performed only after interpolation to produce a full 3-D

datacube of amplitude and phase information for the target in 3-D space. For sparse

target scenes, many fewer measurements can be made while still providing enough

information for three-dimensional reconstruction, as seen in Figure 6(d). The amount

by which data collection may be reduced depends upon the sparsity of the target to

be imaged.

18

Figure 6: Holographic imaging process; (a) Distributed target at [xp, yp, z0] is il-luminated with step-frequency waveform at discrete intervals along [x, y, z = 0].(b) Sparsely sampled wideband hologram record. (c) Complete wideband hologramrecord approximation via interpolation. (d) Reconstructed three-dimensional imagescene from interpolated dataset.

19

5 Experimental Procedure and Results

5.1 Experimental Setup

Our experimental setup for holographic measurements primarily consists of a two-

dimensional X-Y scanner and network analyzer, as shown in Figure 7. For the data

presented in this section, holograms were recorded over the scanned aperture using

a horn antenna with an azimuthal beamwidth = 25◦, operating in different modes

between 2.0 - 12.0 GHz, 8.4 11.4 GHz, and 8.4 - 17.4 GHz depending on the desired

range resolution and sampling step size. The sampling interval spacing is dictated by

the shortest wavelength used in the step-frequency waveform. We used λ/4 sampling

intervals. An Agilent N5225A 50 GHz PNA Network Analyzer was used to collect

S11 parameters over the scanned aperture, which has a maximum translation length

of 90 cm in both x and y directions.

Figure 7: Experimental test setup for holographic imaging system.

20

5.2 Single-Frequency Image Reconstruction

The first measurement target was of a letter ‘E’ measuring 25 cm wide × 40 cm tall.

The target was built out of wood and wrapped in aluminum foil for better reflectivity.

For our system, a raster scan of 16,384 sample points over the entire aperture took

approximately 40 minutes.

Figure 8 was produced from a single-frequency slice (11.4 GHz) taken from the

wideband datacube. The target object was imaged at a distance of z = 1.5 m from the

aperture plane, which measured 128×128 samples across an 84 cm × 84 cm scan area.

Because the data were collected in a cluttered environment, background measurement

data were taken and subtracted from the hologram to improve the signal-to-clutter

(SCR) ratio for the single frequency reconstruction. In practical applications, this is

not always possible. However, applying this reconstruction process to the wideband

Figure 8: Comparison of holographic reconstruction processes for normally sampleddataset and randomly undersampled dataset.

21

dataset improves image quality significantly. Despite this, image reconstruction with

high correlation is achieved with compression ratios even greater than 10:1, i.e. using

less than 10% of the original number of samples.

Larger bandwidths and a greater number of frequencies can provide better range

discrimination between target and clutter objects, allowing for better quality recon-

struction even in cluttered environments. This is demonstrated in the following sec-

tion.

5.3 Wideband Image Reconstruction

In order to produce an image of reasonable quality in a cluttered environment, it is

necessary not only to meet sampling requirements for spatial resolution in the x− y

plane, but is also necessary to have sufficient range resolution in the z-direction.

For this reason, wideband imaging is particularly effective in cluttered environments,

where discrimination between multiple targets can be accomplished with the proper

bandwidth and frequency sampling specifications. In all of the presented results, a

basic linear stepped-frequency waveform was used. This waveform was chosen be-

cause of its simplicity, low instantaneous bandwidth, and yet high overall bandwidth

provided for stretch-processing.

5.3.1 Image Reconstruction of a Simple Target

Preliminary experimentation aimed to demonstrate basic holographic image recon-

struction of a metallic object. Measurements were taken in an open indoor environ-

ment. For every set of measurement data, a background clutter measurement was

also recorded for background subtraction in order to improve image reconstruction

quality. For any practical system, background subtraction is a simple and effective

solution at reducing clutter imposed by the stationary measurement environment.

In order to demonstrate the wideband image reconstruction process, two targets

22

Figure 9: Holographic imaging and sparse reconstruction procedure of a prop guntarget.

were constructed out of wood and wrapped in aluminum foil for higher reflectivity

and thus higher imaging contrast. The first target was a prop handgun cutout,

measuring approximately 12 cm × 19 cm. The gun was placed on a low-dielectric

pedestal 1.03 m from the antenna transceiver. RF absorber material was placed

behind the target in order to attenuate signal reflected off of the back wall, as well as

multipath reflections. A scanning aperture of 25 cm × 40 cm was used to measure

the backscattered electric field pattern. The results for sparse holographic image

reconstruction of the prop handgun are presented in Figure 9. After collecting the

23

full dataset, sparse random sampling was applied at varying compression ratios and

cubic-spline interpolation method was used upon these sparse datasets in order to

produce hologram approximations to be used for image reconstruction. As a side

note, our choice of the cubic-spline interpolation scheme versus other methods is

justified in the following section.

Figure 10: Wideband holographic reconstruction of target at a distance of 1.5 metersin cluttered environment; (a) reconstructed image from full dataset, (b) reconstructedimage from only 10% random samples (Cross-correlation = 0.995, PSNR = 33.6 dB);(c) Progression of reconstructed image quality as number of measurement samples isreduced.

24

The second target used to demonstrate simple target image reconstruction was

constructed in the shape of the capital letter ‘E.’ This target measured approximately

40 cm × 25 cm. The target was imaged over an aperture size of 86.7 cm × 86.7 cm

with a step-frequency waveform operating between 8.4 GHz to 11.4 GHz, providing

a bandwidth of 3 GHz and range resolution of 5 cm. Results are shown in Figure

10. Under the environmental conditions for this test setup, 3 GHz of bandwidth was

sufficient for imaging of an isolated target. As clutter and other targets of interest are

introduced at close proximity, higher signal bandwidth is required in order to prevent

spill-over between range bins.

5.3.2 Image Reconstruction of a Single Concealed Object

Holographic imaging has been used in identifying concealed threat objects [1,19,24].

It is pertinent that compressed sensing be applied to the detection process where

scanning time may be reduced or 2D antenna arrays may be designed in a more

simple, cost-effective manner. In order to demonstrate the benefits of reduced random

sampling in this application, threat objects were concealed and imaged at X-band.

The first case shown in Figure 11 is of an X-band holographic image (8.4 - 11.4

GHz) of a knife measuring approximately 20 cm concealed within a stuffed animal.

The knife blade is made of stainless steel with a plastic composite handle, and the

stuffed animal is made up of polyester with cotton stuffing. Measurements were

made with the target at a distance of approximately 50 cm. For this particular target

setup, it can clearly be seen by the reconstructed images that the concealed threat

object can be detected using the holographic imaging technique. Furthermore, a

reduction of samples taken of the sparse target scene does not noticeably diminish

the recognizability of the concealed weapon in the reconstituted images.

The next set of measurements were intended to demonstrate threat object detec-

tion when concealed inside of a purse. Two different purses were used to show the

25

Figure 11: Holographic imaging and sparse reconstruction at z = 0.63 m of a concealedknife inside stuffed animal. Sparsely-sampled holograms at different compressionratios and corresponding image reconstructions.

26

variability of image reconstruction. In Figure 12, the prop gun was placed inside of a

thick leather purse and measured at X-band (8.4 - 11.4 GHz) with a target distance

of approximately 100 cm. The holographic image was reconstituted at 99 cm with

relatively good recognizability.

In Figure 13, the bandwidth was extended to 9 GHz (8.4 - 17.4 GHz) in order

to improve image resolution. The target was placed at approximately 100 cm, and

image reconstitution was best formed at 104 cm as shown in the diagram. It can be

seen that reconstructed image from the inside of the second purse reveals a strong

semblance of a possible threat object. For both datasets, further consideration of the

effectiveness of sparse sampling was made, and corresponding results are contained

within Appendix A.

In order to demonstrate that this imaging technique may be used on a variety of

Figure 12: Holographic imaging and reconstruction of prop gun concealed withinpurse 1.

27

Figure 13: Holographic imaging and reconstruction of prop gun concealed withinpurse 2.

common objects, detection was also tested on a hollow-body acoustic guitar contain-

ing a concealed threat object. For the data seen in Figure 14, the prop gun was placed

inside of the hollow-body acoustic guitar located at approximately 50 cm. In order to

improve penetration through the wooden body and maintain fair imaging resolution,

the transmit signal used was a linear stepped-frequency waveform between 2.0 - 12.0

GHz. A total of 401 frequency measurements were made per x−y position in order to

resolve a maximum ambiguous range of 600 cm, which prevented aliasing of primary

reflections from approximately 450 cm off of the back wall inside of the measurement

chamber. The scan aperture dimensions measured 87.5 cm × 55 cm with a measure-

ment step size of 6.25 mm in both x and y directions. Results from sparse sampling

and interpolation of the 3D hologram dataset are presented in Figure 15, showing

that even for a complex target scene, our sparse sampling reconstruction technique

provides reasonable image quality for threat object recognition and detection.

28

Figure 14: Acoustic guitar with concealed weapon, hologram record at 12 GHz, andreconstituted image.

29

Figure 15: Image reconstruction of hollow-body acoustic guitar with concealedweapon at various compression ratios.

30

5.3.3 Image Reconstruction of Multiple Concealed Objects

Beyond single-target configurations, the sparse holographic imaging technique may

be used in complex target scene configurations. In many cases, it is critical to be able

to resolve possible threat objects within a volume container. Using a waveform with

a large bandwidth provides the necessary range resolution to discriminate between

multiple target locations within a 3D volume. This is critical in applications such as

airport security scanners [24], non-destructive structural testing [14], and automotive

radar [22,23].

The first case that is considered is a multiple-target setup where a concealed

weapon is located inside of a stuffed animal. A purse containing a book and a set

of keys is collocated in the z-dimension so as to place both targets within the same

range-bin slice. The setup and imaging results can be seen in Figure 16. The target

scene was measured at frequencies 14.0 - 17.0 GHz across an aperture at z = 0.0

cm with scanned area dimensions of 84.7 cm × 42.4 cm. Although it is unknown to

the naked eye, holographic image reconstruction at z = 67.8 cm reveals a concealed

weapon hidden inside the stuffed animal. Reflectivity of the book and keys was much

lower than the knife, so it is difficult to make these shapes out. Also, the purse made

of thick leather reduces the amount of transmitted/reflected signal and distorts the

image reconstruction due to diffuse scattering. The higher frequency waveform was

used to improve spatial resolution, however, penetration was likely reduced, making

it difficult to image items inside the purse.

The second experiment in this section probes a purse with concealed items inside,

including a thick textbook and the prop gun. The size of the measurement aperture

in this case was 40 cm × 35 cm. This dataset was collected at 401 discrete frequencies

between 8.4 - 17.4 GHz for higher range resolution in order to make it possible to

discriminate the gun from the book at different ranges. The thickness of the prop

gun measured only to approximately 2 cm, so a range resolution of ∆R = 1.67 cm

31

Figure 16: Experimental setup and holographic image reconstruction of purse andconcealed weapon inside of stuffed animal.

32

Figure 17: Holographic imaging and reconstruction of concealed weapon with bookinside of purse 3.

33

Figure 18: Sparse holographic image reconstruction of backpack containing sweatshirtand concealed knife.

34

ensures that the range of the gun does not fall into the same range slice as the book.

The images shown in Figure 17 include the contents to be imaged, the hologram

recording, and the reconstructed image produced at a distance of z = 42.0 cm. Since

wavelength at the center frequency 12.9 GHz is λ = 2.3 cm, the spatial resolution of

the reconstructed image is approximately δx = 1.2 cm, making it possible to resolve

finer details about the object, such as the trigger cutout.

The third experiment in this section tests sparse holographic image reconstruction

for concealed weapon detection on a backpack containing a sweatshirt and a knife.

By visual inspection, it is unapparent whether or not a threat object is contained

inside the backpack. The images shown in Figure 18 demonstrate how the sparse

holographic imaging technique is effective in revealing objects that are concealed and

which may pose a threat. The transmit waveform used was 14.0 - 17.0 GHz with 201

discrete frequencies. The higher frequency content of the waveform requires a finer

spatial sampling resolution, dx = dy = 4.4 mm, which inevitably requires a larger

number of samples within a given aperture size. The measurement data was made

up of 90 × 132 pixels over an aperture size of 39.7 cm × 58.3 cm, requiring a total of

N = 2,387,880 samples for the complete dataset. At a constant sampling interval of

λ/4, this is the amount of data that would typically be expected to produce an image

of such a target scene given the waveform parameters. However, it is demonstrated

in an empirical sense by the images reconstructed at different compression ratios. We

see that even at 5% sampling, the presence of the concealed knife is still known. This

effectively allows for higher resolution images without requiring an extensive amount

of data to be collected.

The last experiments were conducted in order to demonstrate that concealed tar-

gets located at different ranges can be resolved and identified. A cardboard box filled

with paper measuring 20 cm W × 30 cm L × 25 cm H was used to conceal metallic

objects placed in different locations in space. Targets used include a steel wrench

35

Figure 19: Holographic reconstruction of concealed objects within cardboard box 1;image of wrench reconstituted at z = 0.364 m; image of prop gun reconstituted at z= 0.540 m.

measuring approximately 25 cm long, a pocket knife measuring approximately 15 cm

long, and the prop gun measuring approximately 20 cm long.

In Figure 20, the three objects were spaced 15 cm apart from one another in a stag-

gered pattern to avoid shadowing. In Figure 19, the wrench and prop gun were spaced

approximately 20 cm apart in the z-direction at opposite sides of the box. For good

separation in the range dimension, 401 frequency samples from a stepped-frequency

waveform between 2.0 - 12.0 GHz was used. This frequency range was chosen as a

compromise in required spatial sampling step size. Provided better high-frequency

equipment, high waveform frequencies may provide better target imaging resolution

at the cost of smaller sampling step sizes. Sparse holographic image reconstruction

of targets in 3D space demonstrated in Figure 21 demonstrates that high resolution

images may be obtained with many fewer samples.

36

Figure 20: Holographic reconstruction of concealed objects within cardboard box 2;image of wrench reconstituted at z = 0.375 m; image of knife reconstituted at z =0.493 m; image of prop gun reconstituted at z = 0.623 m.

37

Figure 21: Sparse holographic image reconstruction of wrench (z = 0.364 m) andprop gun (z = 0.540 m) inside cardboard box 1 for different compression ratios.

5.4 Noise and Filtering Considerations

In any imaging system, it is important to consider the effects of noise on image

quality. Noise can be introduced into the measurements and reconstructed images in

a variety of different ways. The first and most obvious source of noise would be the

noise present within the measurement instrumentation used in the microwave imaging

system. The Agilent N5225A 50 GHz PNA Network Analyzer used has a noise floor

of -117 dBm at 10 Hz IF bandwidth. For our near-field imaging system operating at

0 dBm transmit power, this level of noise is essentially negligible.

One of the primary sources of noise that affected measurement quality resulted

from the mechanical scanning action. To reduce vibrations, the scanner was config-

ured to run continuous line scans, as opposed to a point-by-point acquisition scheme

38

Figure 22: Filtering process in order to reduce noise artifacts introduced by measure-ment system.

which induced a large amount of vibration every time the scanner stopped in position.

Provided that the scan speed is not excessive, Doppler shift between the target scene

and antenna is negligible. The effect of this mechanically-induced noise is highlighted

in Figure 22, where the hologram dataset is processed using a disc-shaped 2D median

filter to mitigate some of the measurement noise artifacts. This effectively smooths

out any sharp discontinuities, providing a better approximation of the electric field

distribution over the aperture.

In a sparse holographic imaging system, missing pixel information prevents the use

of common spatial filtering techniques. However, the step in which pixel values are

determined via interpolation effectively serves as a lowpass filter process. To evaluate

39

Figure 23: Reconstructed images of prop gun with various noise levels added prior toreconstruction.

40

Figure 24: Comparison of image quality with various noise levels added prior toreconstruction.

the effects of noise induced on sparse measurement data prior to reconstruction, three

sets of data were evaluated. The data shown in Figure 23 represents part of the

analysis presented in Figure 24. For each dataset, SNR was measured at different

compression ratios for various levels of noise added to the hologram (40 dB, 30 dB,

and 20 dB noise floor). When a large amount of data pixels were collected, the SNR

was affected at the same scale as the amount of noise introduced. As the hologram

samples became more sparse, the noise had less influence on the overall SNR of the

reconstructed image. In one respect, the sparser the dataset, the less influence the

additive noise has on the reconstructed image. Values of missing pixels are estimated

41

via interpolation of a cubic-spline function between existing data and are therefore

not subjected to high frequency noise.

5.5 Discussion

Our results suggest that for sparse signals (target scenes), many fewer samples are

needed for accurate reconstruction compared to the amount that would traditionally

be required. In contrast to the common alternative methods for signal estimation

(e.g. l1-norm minimization), our processing scheme takes merely seconds, as opposed

to hours [3]. Although the full dataset would not need to be collected in practice,

quantitative evaluation of the effectiveness of this modified reconstruction process

requires a reference signal (image) to provide a baseline for comparative quality met-

rics. In our evaluation, we compared sum of absolute differences (SAD, also known

as l1-norm), mean-squared error (MSE, also known at l2-norm), and SNR versus

compression ratio for three different interpolation schemes: linear, cubic-spline, and

nearest neighbor. Compression ratio is simply the inverse of the number of samples

used in the undersampled subset.

The performance of each interpolation scheme was considered for each of the 14

experimental datasets (comprehensive results in Appendix A) and was averaged over

the 14 datasets to produce the plots seen in Figure 25. From these results, it is fair to

conclude that the cubic-spline interpolation scheme is most effective in sparse holo-

graphic image reconstruction. However, depending on the sparsity of the target scene,

satisfactory image reconstruction quality may vary with compression ratio. With this

in mind, a system may be designed using the sparse holographic reconstruction tech-

nique to meet desired image resolution and noise floor specifications while benefiting

from the drastic reduction in sampling requirements.

42

Figure 25: Comparison of interpolation methods; SNR, MSE, and SAD versus com-pression ratio averaged over N = 14 independent datasets.

43

6 Conclusions and Future Work

Holographic imaging is an effective means for measuring information about three-

dimensional target scenes. Reconstruction methods that are commonly used in com-

pressed sensing applications are not necessarily useful in real-time applications due

to the computational burden of iterative optimization techniques. Through simula-

tion and experimentation, we have verified that for low-frequency hologram records,

simple interpolation of sparsely-sampled target scenes provides a quick and reliable

way to reconstruct sparse datasets for accurate image reconstruction.

For scanning radar applications, data collection time can be drastically reduced

through application of sparse sampling. This reduced scan time will typically benefit

a real-time system by allowing improvements in processing and decision-making al-

gorithms. Furthermore, the reduction of required data storage as earned value may

afford better design specifications in other aspects of a holographic measurement sys-

tem. When considering the design of a two-dimensional antenna array, sparse holo-

graphic reconstruction may also provide several advantages in certain applications.

By reducing the number of detector elements, a direct decrease in material cost is

achieved, as well as a reduction in complexity for the transmission line network and

supporting elements.

This study has considered the significance of interpolation techniques applied to

digital holograms with low spatial frequency prior to image reconstruction. Results

have demonstrated that this is highly effective when imaging target scenes which can

be represented in some sparse domain. Future work could potentially include analysis

of reconstruction performance under different clutter scenarios, noise levels and target

complexity. Future system development may consider different transceiver antennae,

signal bandwidths, as well as possible fusion of sparse sampling, interpolation and l1-

norm minimization techniques for holographic processing and image reconstruction.

44

A Additional Experimental Data

Figure 26: Comparison of interpolation methods applied prior to reconstruction;PSNR, L1-norm, and L2-norm versus compression ratio.

45


46


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48


49


50


51


52


53


54


55


56


57


58

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COMPRESSIVE MICROWAVE RADAR HOLOGRAPHY

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