Image reconstruction in quantitative X-ray phase...

Image reconstruction in quantitativeX-ray phase-contrast imaging employing

multiple measurements

Cheng-Ying Chou, Yin Huang, Daxin Shi, and Mark A. AnastasioDepartment of Biomedical Engineering, Medical Imaging Research Center,

Illinois Institute of Technology, Chicago, IL 60616

[email protected]

Abstract: X-ray phase-contrast imaging is a technique that aims toreconstruct the projected absorption and refractive indexdistributions of anobject. One common feature of reconstruction formulas for phase-contrastimaging is the presence of isolated Fourier domain singularities, whichcan greatly amplify the noise levels in the estimated Fourier domain andlead to noisy and/or distorted images in spatial domain. In this article, wedevelop a statistically optimal reconstruction method that employs multiple(> 2) measurement states to mitigate the noise amplification effects due tosingularities in the reconstruction formula. Computer-simulation studies arecarried out to quantitatively and systematically investigate the developedmethod, within the context of propagation-based X-ray phase-contrastimaging. The reconstructed images are shown to possess dramaticallyreduced noise levels and greatly enhanced imaging contrast.

© 2007 Optical Society of America

OCIS codes:(110.7440) X-ray imaging; (100.5070) Phase retrieval; (170.3010) Image recon-struction techniques.

References and links1. A. Krol, A. Ikhlef, J.-C. Kieffer, D. Bassano, C. C. Chamberlain, Z. Jiang, H. Pepin, and S. C. Prasad, “Laser-

based microfocused X-ray source for mammography: feasibility study,” Med. Phys.24, 725–732 (1997).2. R. Waynant, “Toward practical coherent x-ray sources: Potential medical applications,” IEEE J. Quantum Elec-

tron.6, 1465–1469 (2000).3. H. Yamada, “Novel x-ray source based on a tabletop synchrotron and its unique features,” Nucl. Instrum. Methods

Phys. Res. B199, 509–516 (2003).4. R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol.49,

3573–3583 (2004).5. W. Thomlinson, P. Suortti, and D. Chapman, “Recent advancesin synchrotron radiation medical research,” Nucl.

Instrum. Methods Phys. Res. A543, 288–296 (2005).6. T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing

materials using hard X-rays,” Nature (London)373, 335–338 (1996).7. K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard

x-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).8. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli,

and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol.42, 2015–2025 (1997).9. C. J. Kotre and I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med.

Biol. 44, 2853–2866 (1999).10. X. Wu and H. Liu, “Clinical implementation of X-ray phase-contrast imaging: Theoretical foundations and design

considerations,” Med. Phys.30, 2169–2179 (2003).11. M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A.

Anastasio, and C. Muehleman, “Multiple-image radiography,”Phys. Med. Biol.48, 3875–3895 (2003).

#79616 - $15.00 USD Received 5 Feb 2007; revised 7 May 2007; accepted 20 May 2007; published 25 Jul 2007

(C) 2007 OSA 6 August 2007 / Vol. 15, No. 16 / OPTICS EXPRESS 10002

12. E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys.30, 2292–2296 (2003).

13. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging withlow-brilliance X-ray sources,” Nature Phys.2, 258–261 (2006).

14. B.L. Henke, E.M. Gullikson, and J.C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, andreflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables54, 181–342 (1993).

15. F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo,A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, andF. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol.43, 2845–2852 (1998).

16. F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, M. Fabrizioli, R. Longo,R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba,A. Vacchi, E. Vallazza, and F. Zanconati, “Mammography with Synchrotron Radiation: Phase-Detection Tech-niques,” Radiology215, 286–293 (2000).

17. M. Z. Kiss, D. E. Sayers, Z. Zhong, C. Parham, and E. D. Pisano, “Improved image contrast of calcifications inbreast tissue specimens using diffraction enhanced imaging,” Phys. Med. Biol.49, 3427–3439 (2004).

18. E. Pisano, R. Johnston, D. Chapman, J. Geradts, M. Iacocca, C. Livasy, D. Washburn, D. Sayers, Z. Zhong,M. Kiss, and W. Thomlinson, “Human Breast Cancer Specimens: Diffraction-enhanced Imaging with HistologicCorrelation-Improved Conspicuity of Lesion Detail Comparedwith Digital Radiography,” Radiology214, 895–901 (2000).

19. T. Tanaka, C. Honda, S. Matsuo, K. Noma, H. Ohara, N. Nitta,S. Ota, K. Tsuchiya, Y. Sakashita, A. Yamada,M. Yamasaki, A. Furukawa, M. Takahashi, and K. Murata, “The first trial of phase contrast imaging for digitalfull-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol.40, 385–396 (2005).

20. C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissueof the foot and ankle,” J. Anat.208, 115–124 (2006).

21. J. G. Brankov, M. N. Wernick, Y. Yang, J. Li, C. Muehleman, Z. Zhong, and M. A. Anastasio, “A computedtomography implementation of multiple-image radiography,” Med. Phys.33, 278–289 (2006).

22. A. Bravin, “Exploiting the x-ray refraction contrast with an analyser: the state of the art,” J. Phys. D36, A24–A29(2003).

23. P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomographywith high energy x-rays,” Phys. Med. Biol.44, 741–749 (1999).

24. P. Cloetens, “Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard SynchrotronRadiation: Principles, Implementation and Applications,” Ph.D. thesis, Vrije Universiteit Brussel (1999).

25. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,”Rev. Sci. Instrum.68, 2774–2782 (1997).

26. M. Born and E. Wolf,Principles of Optics(Cambridge University Press, 1999).27. D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, andM. Kitchen, “Quantitative phase retrieval using

coherent imaging systems with linear transfer functions,” Opt. Commun.234, 87–105 (2004).28. P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. Landuyt,J. P. Guigay, and M. Schlenker, “Holotomography:

Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys.lett. 75, 2912–2914 (1999).

29. A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A19, 472–480(2002).

30. T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest,and S. W. Wilkins, “Phase-and-amplitude computertomography,” Appl. Phys. Lett.89, 034102 (2006).

31. M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analyticimage reconstruction in local phase-contrast tomog-raphy,” Phys. Med. Biol.49, 121–144 (2004).

32. Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhancedx-ray imaging of weak objects,” J. Phys. D37, 1262–1274 (2004).

33. Y. I. Nesterets, T. E. Gureyev, K. M. Pavlov, D. M. Paganin, and S. W. Wilkins, “Combined analyser-based andpropagation-based phase-contrast imaging of weak objects,” Opt. Commun.259, 19–31 (2006).

34. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins,“Linear algorithms for phase retrieval in the Fresnelregion,” Opt. Commun.231, 53–70 (2004).

35. D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. Theeffects of noise,” J. Microsc.214, 51–61 (2003).

36. Y. Huang and M. A. Anastasio, “Statistically principleduse of in-line measurements in intensity diffractiontomography,” J. Opt. Soc. Am. A24, 626–642 (2007).

37. D. M. Paganin,Coherent X-Ray Optics(Oxford University Press, 2006).38. T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase

retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006).39. Y. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-

based phase-contrast imaging,” J. Phys. D38, 4259–4271 (2005).40. K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with



slowly varying transfer functions and their application toX-ray phase-contrast imaging,” J. Phys. D37, 2746–2750 (2004).

41. J. W. Goodman,Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2004).42. J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik49, 121–

125 (1977).43. T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. Paganin, K.M. Pavlov, and S. W. Wilkins, “Stability and locality

of amplitude and phase contrast tomographies,” Proc. SPIE6318, 63180V (2006).44. S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: Statistical properties,” J. Opt.

Soc. Am.60, 1478–1483 (1970).45. W. D. Stanley, G. R. Dougherty, and R. Dougherty,Digital Signal Processing(Reston Publishing Company, Inc.,

1984).

1. Introduction

Due to the advent and accessibility of synchrotron and laboratory X-ray sources [1, 2, 3]that possess high degrees of coherence, new imaging methods, broadly characterized asX-rayphase-contrastimaging methods [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] are being developed that havedramatic advantages over conventional radiographic and tomographic X-ray imaging systems.Unlike conventional radiographic methods, phase-contrast imaging methods exploit a contrastmechanism based on differences in the complex-valued X-rayrefractive index values of tissue.At diagnostic X-ray energies, variations in the real component of the refractive index of tis-sues are several orders of magnitude larger than are variations in the imaginary component, orequivalently, the X-ray attenuation coefficient[14]. Consequently, phase-contrast imaging maypermit the visualization and quantification of tissues thathave identical, or very similar, X-rayabsorption properties. Additionally, unlike absorption-contrast, the phase-contrast mechanismpersists at relatively high energies, which permits low-dose imaging[15]. Experimental stud-ies of X-ray phase-contrast imaging have revealed significantly enhanced contrast and tissuediscriminability over conventional radiographic methodsin applications including cancer de-tection [16, 17, 18, 19] and cartilage imaging [20, 21].

Most experimental implementations of X-ray phase-contrast imaging [4] that hold promisefor clinical imaging are broadly characterized as being either analyzer-based or propagation-based. In analyzer-based methods [22, 11], the object is irradiated with a quasi-monochromatic,parallel-beam, X-ray wavefield, and an analyzer crystal or other X-ray diffracting element [13]is present between the object and detector. The analyzer crystal diffracts the components ofthe transmitted wavefield traveling at or near the crystal’sBragg angle, thereby rejecting allwavefield components traveling outside a narrow angular range. This angular filtering of thetransmitted wavefield produces a phase-contrast enhancement in the recorded intensity image.Propagation-based methods [23, 6, 24, 25], also known as in-line methods, do not require theuse of diffractive X-ray elements, and employ a classic in-line holographic measurement ge-ometry that is essentially similar to magnification radiography [9]. The object is irradiated witha plane-wave or paraxial X-ray wavefield possessing a sufficient degree of spatial coherence,and the intensity of the transmitted wavefield is recorded bya detector placed at some non-zerodistance behind the object. In this case, the measured intensity distribution represents an in-line(Gabor) hologram [26] that contains coded information about the refractive properties of theobject. The phase shifts that are introduced into the probing wavefield by the refractive indexvariations within the object are transferred into intensity variations in the final measurement bythe process of free-space (Fresnel) wave propagation between the object and detector.

Quantitative X-ray phase-contrast imaging methods seek toreconstruct separate images thatdepict the object’s projected absorption and real-valued refractive index distributions, which re-flect two distinct and complementary intrinsic object properties. To achieve this, the measuredintensity data must generally be recorded at two or more distinct “states” of the imaging sys-tem [27]. For example, in analyzer-based methods, different measurement states correspond to



distinct orientations of the analyzer crystal, while in propagation-based methods they could cor-respond to distinct object-to-detector distances. Quantitative phase-contrast imaging methodsare computed-imaging methods, and require use of reconstruction algorithms for image for-mation. When implemented in tomographic mode, an estimate ofthe three-dimensional (3D)complex refractive index distribution can be reconstructed by use of phase-contrast tomographymethods [28, 29, 30, 31, 21].

Under certain assumptions regarding the transmitted wavefield, phase-contrast imaging sys-tems can be modeled as linear, shift-invariant, imaging systems. Assuming measurements areacquired at two distinct states of the imaging system, Fourier-based reconstruction formulascan be derived [27, 32, 33, 34] from knowledge of the imaging system’s optical transfer func-tion. A common feature of these reconstruction formulas, however, is the presence of isolatedFourier domain singularities. In practice, the Fourier components of the projected object prop-erties residing near the singularities can contain greatlyamplified noise levels, resulting innoisy and distorted images. Because the locations of the singularities are determined by thechosen two measurement states, their effects can be mitigated by acquiring intensity measure-ments at multiple (> 2) measurement states [28, 35]. For example, different pairs of intensitymeasurements can be utilized to reconstruct different Fourier components of the object prop-erties, which provides the opportunity to potentially avoid the singularities. However, suchsimple strategies do not exploit fully the statistically complementary information contained inthe available measurement data, and can result in images with suboptimal statistical properties.

In this work, we propose and investigate a methodology for image reconstruction in quantita-tive phase-contrast imaging when multiple (> 2) intensity measurements are available. Linearestimators are proposed that combine the available intensity measurements to produce statis-tical estimates of the projected object properties whose Fourier components possess optimallyreduced variances. This general strategy is inspired by a recent study of intensity diffractiontomography [36] by our group. Explicit forms of the reconstruction formulas are derived forpropagation-based phase-contrast imaging, where a general noise model and finite-samplingeffects are considered. Computer-simulation studies are conducted to demonstrate the efficacyof the proposed method.

2. Background

In this section, the salient features of X-ray phase-contrast imaging are reviewed. We referthe reader to the monograph by Paganin [37] for a comprehensive treatment of phase-contrastimage formation.

2.1. Interaction of X-ray wavefield with object

As depicted in Fig. 1, consider that an object is irradiated by a monochromatic X-ray wavefieldUi with wavelengthλ , which is traveling along the positivez-axis. The effects of imperfectwavefield coherence will not be considered, but can be addressed as in [38, 39]. The object ischaracterized by its complex-valued refractive index distribution

n(~r) ≡ 1−δ (~r)+ jβ (~r), (1)

where~r = (x,y,z), andδ (~r) andβ (~r) quantify the X-ray refractive and absorption properties ofthe object. The quantityβ (~r) is related to the linear X-ray attenuation coefficientµ(~r) as

µ(~r) = 2kβ (~r), (2)

wherek ≡ 2πλ is the wavenumber. Note that classic radiographic methods are sensitive only

to variations inβ (~r), while phase-contrast methods are sensitive to variationsin bothδ (~r) and



β (~r). The wavefieldUo(x,y) on the object plane, which has been transmitted through the object,is given by

Uo(x,y) = T(x,y)Ui (3)

whereT(x,y) is the transmission function that can be expressed generally as

T(x,y) = M(x,y)exp[ jφ(x,y)]. (4)

The amplitude modulusM(x,y) = exp[−A(x,y)] and the phase shiftφ(x,y) describe how theamplitude and wavefront (i.e., phase), respectively, of the probing wavefield are perturbed bythe presence of the object. They are related to imaginary andreal components, respectively, ofn(~r) as

A(x,y) = k∫

dzβ (~r) (5a)

φ(x,y) = −k∫

dzδ (~r) (5b)

where the line-integrals are computed over the support of the object.

Fig. 1. A schematic of a generic X-ray phase-contrast imaging system.

2.2. Linear shift-invariant X-ray phase-contrast imagingsystems

Consider again the generic X-ray phase-contrast imaging system shown in Fig. 1. LetUm(x,y)denote the transmitted wavefield on a detector plane of constantz, which is downstream fromthe object plane. The integer-valued subscriptm is employed to denote the state of the imagingsystem. For many analyzer- and propagation-based phase-contrast imaging systems,Um(x,y)andUo(x,y) can be regarded as the output and input, respectively, of a linear and shift-invariantcoherent imaging system [40, 41]. In this case,

Um(x,y) = Gm(x,y)∗Uo(x,y), (6)

where∗ denotes a two-dimensional (2D) convolution andGm(x,y) describes the impulse re-sponse of the system in statem. In propagation-based imaging, for example,Gm(x,y) describesthe Fresnel propagator [26], with distinct values ofm corresponding to different object-to-detector distances. Alternatively, in analyzer-based imaging, Gm(x,y) describes the coherentimpulse response of the analyzer crystal or other diffractive element(s) employed, with distinctvalues ofm corresponding to different orientations of the analyzer. In hybrid systems [39],Gm(x,y) describes the net effect of both.



On the detector plane, the intensityIm(x,y) = |Um(x,y)|2 is recorded, which represents aradiograph with mixed absorption- and phase-contrast. From knowledge of the measured inten-sity Im(x,y), a modified data function can be defined as

Km(x,y) ≡ 1−Im(x,y)

Ii, (7)

whereIi = |Ui |2 is the intensity of the incident X-ray beam. LetKm(u,v) denote the 2D Fourier

transform (FT) ofKm(x,y) defined as

Km(u,v) =

∫∫ ∞

−∞dxdy Km(x,y)exp[− j2π(ux+vy)]. (8)

In biomedical imaging applications, the conditions of|A(x,y)|<< 1 and slowly varyingφ(x,y)can often be met [10, 42]. Under such conditions, it can be shown that [43]

Km(u,v) = 2Gam(u,v)A(u,v)+2Gp

m(u,v)φ(u,v) (9)

whereGam(u,v) is the amplitude transfer function (ATF):

Gam(u,v) =

12[Gm(u,v)+ G∗

m(−u,−v)], (10)

andGpm(u,v) is the phase transfer function (PTF):

Gpm(u,v) =

12 j

[Gm(u,v)− G∗m(−u,−v)]. (11)

Here,Gm(u,v), A(u,v), andφ(u,v) are the 2D FTs ofGm(x,y), A(x,y), andφ(x,y), respectively,andG∗

m(·, ·) denotes the complex conjugate ofGm(·, ·). The interested reader is referred to Ref.[43] for a detailed derivation of the imaging model in Eq. (9).

Equation (9) relates the measured intensityIm(x,y), or equivalentlyKm(x,y), to the 2D FTs ofthe sought-after quantitiesA(x,y) andφ(x,y). If an additional measurementIn(x,y) is obtainedwhen the imaging system is in staten 6= m, A(u,v) andφ(u,v) can be determined algebraicallyas

A(u,v) =Gp

n(u,v)Km(u,v)− Gpm(u,v)Kn(u,v)

2[Ga

m(u,v)Gpn(u,v)− Ga

n(u,v)Gpm(u,v)

] (12a)

φ(u,v) =−Ga

n(u,v)Km(u,v)+ Gam(u,v)Kn(u,v)

2[Ga


n(u,v)Gpm(u,v)

] . (12b)

Equations (12a) and (b) represent reconstruction formulasfor quantitative phase-contrast imag-ing. From the determinedA(u,v) andφ(u,v), A(x,y) andφ(x,y) are computed by application ofthe inverse 2D FT. Note that Eqs. (12a) and (b) contain isolated poles at frequency components(u,v) for which

Gam(u,v)Gp

n(u,v)− Gan(u,v)Gp

m(u,v) = 0. (13)

When Eq. (12) is applied to noisy, or otherwise inconsistent,measurement data, the Fouriercomponents(u,v) of A(u,v) andφ(u,v) residing near the poles will contain greatly amplifiednoise levels. This can result in noisy and/or distorted images. In the remainder of this arti-cle, we describe a statistically principled method for circumventing this when measurementscorresponding to multiple (> 2) states of the system are available.



3. Variance reduction in quantitative X-ray phase-contrast imaging

We consider that intensity measurementsIm(x,y) are acquired at three distinct statesm= 1,2,3of the system. The results below are generalized to the case of an arbitrary number of measure-ments in the Appendix. The intensity data functionIm(x,y) is interpreted as a stochastic pro-cess, which reflects that the measurements are contaminatedby stochastic errors such as de-tector noise. Our goal is to exploit the statistically complementary information in the availablemeasurements to reduce the variance of the estimatedA(u,v) andφ(u,v), and thereby mitigatethe large amplification of noise due to poles in the reconstruction formulas.

Because the reconstruction formulas in Eq. (12) require intensity measurements at two dis-

tinct states,(

N2

)estimates ofA(u,v) andφ(u,v) can be computed from knowledge of measure-

ments obtained atN system states. When reconstructed from noisy measurements,these esti-mates will be generally distinct. The notationAm,n(u,v) and φm,n(u,v), m 6= n = 1,2,3, willbe employed to describe the estimates for the caseN = 3, where the subscripts denote thatmeasurementsIm(x,y) andIn(x,y) were employed. Because the locations of poles in Eq. (12)depend on the choice of measurement states, the components of Am,n(u,v) or φm,n(u,v) that arehighly contaminated by noise will be determined by the measurement state pair (m,n).

A natural strategy for mitigating noise amplification is to combine theAm,n(u,v) or φm,n(u,v)in a way that attempts to cancel the poles in each two-state estimate [36], to form final estimatesA(u,v) or φ(u,v), respectively, that possess reduced variances for all(u,v). However, it shouldbe noted that only two of the three available estimates are independent in the sense that

φm,n(u,v) = αφl ,m(u,v)φ l ,m(u,v)+αφ

l ,n(u,v)φ l ,n(u,v) (14)

Am,n(u,v) = αal ,m(u,v)Al ,m(u,v)+αa

l ,n(u,v)Al ,n(u,v) (15)

with l 6= m 6= n = 1,2,3. In other words, any estimate can be expressed as a linear combinationof the remaining two. The coefficients in Eqs. (14) and (15) are frequency-dependent and givenby

αφl ,m(u,v) =

−Gan(u,v)

[Ga

l (u,v)Gpm(u,v)− Ga

m(u,v)Gpl (u,v)

]

Gal (u,v)

[Ga


n(u,v)Gpm(u,v)

] , (16a)

αφl ,n(u,v) =

Gam(u,v)

[Ga

l (u,v)Gpn(u,v)− Ga

n(u,v)Gpl (u,v)

]

Gal (u,v)

[Ga


n(u,v)Gpm(u,v)

] , (16b)

αal ,m(u,v) =

−Gpn(u,v)

[Ga

l (u,v)Gpm(u,v)− Ga

m(u,v)Gpl (u,v)

]

Gpl (u,v)

[Ga


n(u,v)Gpm(u,v)

] , (16c)

αal ,n(u,v) =

Gpm(u,v)

[Ga

l (u,v)Gpn(u,v)− Ga

n(u,v)Gpl (u,v)

]

Gpl (u,v)

[Ga


n(u,v)Gpm(u,v)

] . (16d)

It can be verified thatαφ

l ,m(u,v)+αφl ,n(u,v) ≡ 1 (17a)

αal ,m(u,v)+αa

l ,n(u,v) ≡ 1. (17b)

Accordingly, final estimates ofA(u,v) andφ(u,v) that exploit statistically complementary in-formation in the three intensity measurements can be formedas

φ(u,v) = ωφ1,2(u,v)φ1,2(u,v)+ωφ

1,3(u,v)φ1,3(u,v) (18)



A(u,v) = ωa1,2(u,v)A1,2(u,v)+ωa

1,3(u,v)A1,3(u,v), (19)

where the combination coefficients satisfy

ωφ1,2 +ωφ

1,3 = 1 (20a)

ωa1,2 +ωa

1,3 = 1. (20b)

Equations (18) and (19) represent linear estimators for producing unbiased estimates ofφ(u,v)and A(u,v), respectively. Because the combination coefficientsωφ

m,n(u,v) and ωam,n(u,v) are

frequency-dependent, they can be designed to cancel poles present in theφm,n(u,v) and

Am,n(u,v), respectively. Moreover, as described next, they can be designed to optimally re-duce the variance ofφ(u,v) andA(u,v). We consider first the problem of producing estimatesφ(u,v) having reduced variances. The following notation will be employed:

σ2m,n(u,v) ≡ Var{φm,n(u,v)} (21)

ρ(r)k,l ;m,n(u,v)+ jρ(i)

k,l ;m,n(u,v) ≡ Cov{φ k,l (u,v), φm,n(u,v)} (22)

Rm,n(u,v)+ jIm,n(u,v) ≡ ωφm,n(u,v), (23)

where ‘Var’ and ‘Cov’ denote the variance and covariance, respectively of a random process.The variance ofφ(u,v) is given by

Var{φ(u,v)} = |ωφ1,2(u,v)|2σ2

1,2 + |ωφ1,3(u,v)|2σ2

1,3

+2Re[ωφ

1,2(u,v)[ωφ1,3(u,v)]∗Cov{φ1,2(u,v), φ1,3(u,v)}

](24)

where the superscript * denotes the complex conjugate. In order to minimize Var{φ(u,v)}, weneed that

∂Var{φ}∂R1,2

∣∣∣R(op)

1,2

= 0 (25a)

∂Var{φ}∂ I1,2

∣∣∣I (op)1,2

= 0, (25b)

whereR1,2 andI1,2 are the real and imaginary components ofωφ1,2, respectively. The solution

of these equations yields

R(op)1,2 (u,v) =

σ21,3−ρ(r)

1,2;1,3

σ21,2 +σ2

1,3−2ρ(r)1,2;1,3

(26a)

I (op)1,2 (u,v) =

ρ(i)1,2;1,3

σ21,2 +σ2

1,3−2ρ(r)1,2;1,3

. (26b)

The optimal choice ofωφ1,3(u,v) is subsequently determined by Eq. (20a).

The specification of the combination coefficientsωa1,2(u,v) andωa

1,3(u,v) that optimally re-

duce the variance ofA(u,v) via Eq. (19) are also determined by Eq. (26) when the quantitiesin Eqs. (21)-(23) are redefined appropriately. A heuristic method for choosing the combinationcoefficients that can effectively mitigate noise amplification when the noise model is not knownis described later.

As described by Eq. 12, the reconstruction formulas for determining the phase functionφ(x,y) and attenuation functionA(x,y) are described by simple algebraic forms in the Fourier



domain. Consequently, the large amplification of noise due to poles in the reconstruction for-mulas can be mitigated in a mathematically straightforwardand physically understandable wayin the Fourier domain. Reducing the Fourier domain varianceof the phase and attenuation es-timates generally leads to spatial domain estimates that possess reduced variances. This can beunderstood by noting that

∫∫R2 dxdyVar{φ(x,y)}=

∫∫R2 dudvVar{φ(u,v)} The left hand-side of

this equation defines the global variance of the sought-after phase function. This indicates thata phase function possessing a reduced global variance can beobtained from an estimateφ(u,v)with a reduced variance. Because Var{φ(x,y)} is nonnegative, a lower global variance suggests,in general, lower local variances in the determined phase function. The same observation holdstrue forA(x,y).

4. Application to multi-plane propagation-based imaging

In the remainder of the article, the general methodology described in Section 3 is investigatedwithin the context of propagation-based X-ray phase-contrast imaging. In this section, the ex-plicit forms of the optimal estimators in Eqs. (18) and (19) are derived in their continuousforms. The effects of finite sampling are examined in Section5.

4.1. Two-state reconstruction formulas

In propagation-based imaging, the different system statescan correspond to different object-to-detector distances. We consider that three intensity measurementsIm(x,y) are acquired ondistinct detector planesz= zm, wherem= 1,2,3. The impulse response in Eq. (6) correspondsto the Fresnel propagator

Gm(x,y) =exp[ jkzm]

jλzmexp

[jπ

x2 +y2

λzm

], (27)

and its 2D FT yields the transfer function

Gm(u,v) = exp[ jkzm− jπλzm(u2 +v2)]. (28)

The corresponding reconstruction formulas [24] are found by use of Eq. (12):

φm,n(u,v) =−cos(πλzn f 2)Km(u,v)+cos(πλzm f 2)Kn(u,v)

Dm,n(29)

Am,n(u,v) =−sin(πλzm f 2)Kn(u,v)+sin(πλzn f 2)Km(u,v)

Dm,n, (30)

where f 2 ≡ u2 +v2 andDm,n(u,v) ≡ 2sin

(πλ f 2(zm−zn)

). (31)

As before, the indicesm, n satisfym= 1,2, n = 2,3 with n > m.Note thatDm,n = 0 is equivalent to Eq. (13), and specifies the locations of poles in the re-

construction formulas. Equation (29) contains poles corresponding to frequencies(u,v) thatsatisfy

u2 +v2 =l

λ (zm−zn), (32)

where l is an integer. One such pole is located at zero-frequencyu = v = 0, indicating thatthe low-frequency components ofφ(u,v) will contain highly amplified noise levels. The exis-tence of additional poles, away from the origin of the Fourier space, depends on the detectorresolution and the detector pair spacing. Let (uM,vM) denote the maximum spatial frequencies



recorded by the detector. Additional poles in the reconstruction formulas described by Eqs. (29)and (30) will be present when

u2M +v2

M ≥1

λ (zm−zn). (33)

Equation (33) indicates that for a fixed detector resolution, additional poles inφm,n(u,v) willemerge when the detector spacing(zm− zn) is sufficiently large. Likewise, this discussion ofpoles also applies toAm,n(u,v), with the exception that the pole atu = v = 0 is not present dueto a cancellation.

4.2. Second-order statistics for determination of optimalcombination coefficients

In order to determine the optimal combination coefficientsωφ1,2(u,v) andωφ

1,3(u,v) that mini-

mize the varianceφ(u,v) [via Eq. (18)], the variance and covariance information in Eqs. (21)and (22) must be determined. Knowledge of the analogous quantities involving A(u,v) is re-quired for determination of the optimal combination coefficients ofωa

1,2(u,v) andωa1,3(u,v) that

minimize the variance ofA(u,v).From Eqs. (29) and (30), one finds readily that

Var{φm,n(u,v)} =cos2(πλzn f 2)Var{Im(u,v)}+cos2(πλzm f 2)Var{In(u,v))}

D2m,n

(34)

Var{Am,n(u,v)} =sin2(πλzn f 2)Var{Im(u,v)}+sin2(πλzm f 2)Var{In(u,v)}

D2m,n

. (35)

The joint covariances of the Fourier data are computed as

Cov{φ1,2(u,v), φ1,3(u,v)} =cos(πλz2 f 2)cos(πλz3 f 2)Var{I1(u,v)}

D1,2D1,3(36)

Cov{A1,2(u,v), A1,3(u,v)} =sin(πλz2 f 2)sin(πλz3 f 2)Var{I1(u,v)}

D1,2D1,3. (37)

It should be noted that Eqs. (36) and (37) are real-valued, and therefore the imaginary compo-nents of the optimal combination coefficients will vanish [see Eq. (26b)].

4.3. Heuristic determination of combination coefficients

When the second-order statistics in Section 4.2 are not known, the optimal combinations coef-ficientsωa

m,n(u,v) andωφm,n(u,v) cannot be computed. However, the large noise amplification

due to poles inφm,n(u,v) andAm,n(u,v) can still be effectively mitigated by suitable heuristicspecification of the combination coefficients. From Eqs. (34)-(35), near the locations of poles,we find that

Var{φm,n(u,v)} ∝1

D2m,n(u,v)

(38)

Var{Am,n(u,v)} ∝1

D2m,n(u,v)

. (39)

Heuristic combination coefficientsωheurm,n (u,v) should have a (u,v)-dependence that is inversely

proportional to that indicated in Eqs. (38)-(39). Moreover, at locations of poles we should have



ωheurm,n (u,v) ≡ 0. Based on these requirements, theωheur

m,n (u,v) for use in estimatingφ(u,v) viaEq. (18) can be chosen as

ωheur1,2 (u,v) =

D21,2 +αφ

1,2D22,3

D21,2 +D2

1,3 +D22,3

(40a)

ωheur1,3 (u,v) =

D21,3 +αφ

1,3D22,3

D21,2 +D2

1,3 +D22,3

, (40b)

whereαφ1,2 andαφ

1,3 are defined in Eq. (16). When estimatingA(u,v), αφ1,2 andαφ

1,3 are replaced

by αa1,2 andαa

1,3, respectively. Due to their construction,ωheur1,2 (u,v) andωheur

1,3 (u,v) satisfy thenormalization condition in Eq. (20).

5. Computation of optimal combination coefficients with considerationof finite sampling

Below, the second-order statistics described in Section 4.2, and subsequently the optimal com-bination coefficientsωa

m,n(u,v) and ωφm,n(u,v) are computed explicitly with consideration of

finite sampling effects.

5.1. Noise model and finite sampling considerations

The discretely sampled intensity data on a measurement plane z= zm is denoted as

Im[r,s] = Im(x,y)∣∣∣x=r∆x,y=s∆y

, (41)

wherer ands are integer-valued indices that reference detector elements, and∆x = ∆y denotesthe element dimension in a square detector array of dimension L×L. Equation (41) assumesidealized (Dirac delta) sampling, namely, the averaging effects of sampling aperture are not con-sidered. However, the analysis follows can be generalized to address such effects. The squarebracket, ‘[·]’, is introduced to represent the functions whose argumentsare discretely sampled.We assume the noise model satisfies [35, 44]

Im[r,s] = I0m[r,s]+nm[r,s], (42)

whereI0m[r,s] denotes the noiseless intensity data and the signal-dependent noisenm(r,s) has

zero-mean andVar{nm[r,s]} =

(I0m[r,s]

)2 σ2(zm), (43)

where the real-valued quantityσ2(zm) can depend on the detector locationzm [35]. We alsoassume

Cov{nm[r,s],nm′ [r ′,s′]} = Var{nm[r,s]}δrr ′δss′δmm′ , (44)

whereδmn denotes the Kronecker delta function.

5.2. Second-order statistics of discrete data functions

In order to compute the second-order statistics described in Section 4.2, knowledge of the vari-ance ofIm(u,v) is required. To compute this from knowledge of discretely sampled data, thecontinuous 2D FT will be approximated by use of the discrete Fourier transform (DFT) [45].

The 2D DFT [45] of Eq. (42) can be written as follows

Im[p,q] = I0m[p,q]+ nm[p,q] (45)



where

Im[p,q] =N−1

∑r=0

N−1

∑s=0

Im[r,s]exp

[− j

2πN

(pr +qs)

](46)

I0m[p,q] =

N−1

∑r=0

N−1

∑s=0

I0m[r,s]exp

[− j

2πN

(pr +qs)

](47)

nm[p,q] =N−1

∑r=0

N−1

∑s=0

nm[r,s]exp

[− j

2πN

(pr +qs)

], (48)

with p, qdenoting the integer-valued Fourier indices that are conjugate to[r,s], andN specifyingthe number of detector elements in each dimension of the square 2D detector. The variance ofnm[p,q] is computed as

Var{nm[p,q]} =N−1

∑r,r ′=0

N−1

∑s,s′=0

exp

[− j

2πN

(p(r − r ′)+q(s−s′)

)]Cov{nm[r,s],nm[r ′,s′]}

=N−1

∑r=0

N−1

∑s=0

E{(nm[r,s])2} =N−1

∑r=0

N−1

∑s=0

(I0m[r,s]

)2 σ2(zm), (49)

where ‘E{·}’ denotes the statistical expectation operator. The continuous and discrete FTs ofthe intensity data are related as [45]

Im(u,v)∣∣∣u=p∆u,v=q∆v

≈L2

N2 Im[p,q], (50)

where∆u = ∆v = 1L are the frequency domain sampling intervals alongu andv axes. By use of

Eqs. (49) and (50), we find that

Var{Im(u,v)}∣∣∣u=p∆u,v=q∆v

≈L4

N4 Var{Im[p,q]} =L4

N4 σ2(zm)N−1

∑r=0

N−1

∑s=0

(I0m[r,s]

)2. (51)

Finally, using Eq. (51) with the results of Section 4.2, we arrive at

Var{φm,n(u,v)}∣∣∣u=p∆u,v=q∆v

≈

L4

N4

cos2(πλzn f 2)∑N−1r=0 ∑N−1

s=0

(I0[r,s;zm]

)2 σ2(zm)+cos2(πλzm f 2)∑N−1r=0 ∑N−1

s=0

(I0[r,s;zn]

)2 σ2(zn)

D2m,n

(52)

Var{Am,n(u,v)}∣∣∣u=p∆u,v=q∆v

≈

L4

N4

sin2(πλzn f 2)∑N−1r=0 ∑N−1

s=0

(I0[r,s;zm]

)2 σ2(zm)+sin2(πλzm f 2)∑N−1r=0 ∑N−1

s=0

(I0[r,s;zn]

)2 σ2(zn)

D2m,n

,

(53)where, as before,m= 1,2, n = 2,3 with n > m, and

Cov{φ1,2, φ1,3} ≈L4

N4

cos(πλz2 f 2)cos(πλz3 f 2)∑N−1r=0 ∑N−1

s=0

(I0[r,s;z1]

)2 σ2(z1)

D1,2D1,3

∣∣∣u=p∆u,v=q∆v

(54)

Cov{A1,2, A1,3} ≈L4

N4

sin(πλz2 f 2)sin(πλz3 f 2)∑N−1r=0 ∑N−1

s=0

(I0[r,s;z1]

)2 σ2(z1)

D1,2D1,3

∣∣∣u=p∆u,v=q∆v

.

(55)



5.3. Explicit forms of optimal combination coefficients

By use of Eqs. (52)-(55) and Eq. (26), we arrive at the optimalcombination coefficients for usein estimatingφ(u,v) or A(u,v) via Eq. (18) or (19), respectively:

ωφ1,2(u,v) =

Kφ1 −Kφ

2

Kφ1 +Kφ

3 −2Kφ2

(56)

ωa1,2(u,v) =

Ka1 −Ka

2

Ka1 +Ka

3 −2Ka2, (57)

where

Kφ1 = D2

1,2

[cos2(πλz3 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z1]

)2 σ21 +cos2(πλz1 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z3]

)2 σ23

]

(58a)

Kφ2 = D1,2D1,3

[cos(πλz2 f 2)cos(πλz3 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z1]

)2 σ21

](58b)

Kφ3 = D2

1,3

[cos2(πλz2 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z1]

)2 σ21 +cos2(πλz1 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z2]

)2 σ22

],

(58c)and

Ka1 = D2

1,2

[sin2(πλz3 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z1]

)2 σ21 +sin2(πλz1 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z3]

)2 σ23

]

(59a)

Ka2 = D1,2D1,3

[sin(πλz2 f 2)sin(πλz3 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z1]

)2 σ21

](59b)

Ka3 = D2

1,3

[sin2(πλz2 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z1]

)2 σ21 +sin2(πλz1 f 2)

N−1

∑r=0

N−1

∑s=0

(I0[r,s;z2]

)2 σ22

],

(59c)whereσ2

m ≡ σ2(zm). The corresponding forms ofωφ1,3(u,v) andωa

1,3(u,v) are determined byEq. (20).

If ∑N−1r=0 ∑N−1

s=0

(I0[r,s;zm]

)2does not vary significantly as a function ofm (i.e., the detector

locationzm), Eq. (56) can be expressed in the somewhat simplified form:ωφ

1,2(u,v) ≈

D21,2

[cos2(πλz3 f 2)σ2

1 +cos2(πλz1 f 2)σ23

]−D1,2D1,3

[cos(πλz2 f 2)cos(πλz3 f 2)σ2

1

]

∑3m,n=2m6=n

D21,m

[cos2(πλzn f 2)σ2

1 +cos2(πλz1 f 2)σ2n

]−2D1,2D1,3

[cos(πλz2 f 2)cos(πλz3 f 2)σ2

1

] .

(60)

6. Numerical Studies

A preliminary computer-simulation study of propagation-based X-ray phase-contrast imagingwas conducted to corroborate and quantitatively investigate the proposed reconstruction meth-ods.



6.1. Numerical phantom and in-line measurement geometry

The in-line measurement geometry shown in Fig. 2 was assumed. A monochromatic X-rayplane-wave with wavelengthλ = 1×10−10 m, propagated along thez-axis and irradiated anobject. Three detector planes located atz= zm, m= 1,2,3, were considered to be behind theobject. The detector contained 1024×1024 elements of dimension of 1µm2, and was assumedto have otherwise idealized physical properties. Two measurement geometries were considered,which will be referred to as Geometry ‘A’ and Geometry ‘B’. InGeometry ‘A’, the detectorplanes were positioned atz1 = 19 mm,z2 = 96 mm,z3 = 182 mm, while the correspondingpositions in Geometry ‘B’ werez1 = 12 mm,z2 = 38 mm, andz3=72 mm.

Fig. 2. The measurement geometry of propagation-based X-ray phase-contrast imagingemploying multiple detector-planes.

A 3D mathematical phantom comprised of five uniform ellipsoids possessing differentcomplex-valued X-ray refractive index values representative of soft tissue was employed torepresent the object to-be-imaged. The semi-axes of the largest ellipsoid were 188.416µm,163.840µm, and 141.312µm. From knowledge of the phantom, the projected object proper-tiesφ(x,y) andA(x,y) were computed according to Eq. (5) and are displayed in Fig. 3.

0

1

2

3

4

5x 10

−3

(a) A(x,y)

−0.04

−0.02

0

0.02

0.04

0.06

(b) φ(x,y)

Fig. 3. Images of the true object properties (a)A(x,y) and (b)φ(x,y).

6.2. Measurement data and simulation studies

The simulated intensity measurements were computed as follows. From knowledge ofφ(x,y)andA(x,y), the transmitted wavefieldUo(x,y) on the object plane was computed according toEqs. (3) and (4). Subsequently, sampled values of the wavefield Um(x,y) on each detector planez = zm, m = 1,2,3, was computed by use of Eq. (6) withGm(x,y) specified by the Fresnelpropagator in Eq. (27). The convolution in Eq. (6) was computed by use of the 2D fast Fourier



transform (FFT) algorithm. The intensity dataIm[r,s] were then computed as the square of thewavefield modulus on each detector plane.

Noisy measurement dataIm[r,s] were computed according to Eq. (42). The noise processnm[r,s] was described by a uncorrelated Gaussian distribution whose variance was determinedby Eq. (43) withσ2(zm) = 0.05%. Ensembles of 1000 noisy realizations ofIm[r,s] were com-puted form= 1,2,3.

6.3. Image reconstruction

Estimatesφm,n(u,v) and Am,n(u,v) were computed from each pair of noisy intensity data byuse of Eqs. (29) and (30), respectively. The presence of poles can pose considerable difficultyin determining these estimates. Simply settingφm,n(u,v) = 0 or Am,n(u,v) = 0 at the locationsof poles will lead to inaccuracy in the resulted estimates. Additionally, even if the poles areavoided, the data errors can be greatly amplified in the vicinity of poles, where the denominatorsof Eqs. (29) and (30) take on small values. In current studies, the reconstruction formulas wereregularized by setting the estimates ofφm,n(u,v) andAm,n(u,v) to zeros in the vicinity of poleswhenDm,n ≤ 2× 10−7. These estimates were combined, according to Eqs. (18) and (19), toform final estimatesφ(u,v) andA(u,v) that possess optimally reduced variances. The requiredcombination coefficients were computed according to Eqs. (56) and (57). Becauseσ2(zm) wasfixed at a constant value, it can be verified that, in this special case, the optimal combinationcoefficients given in Eqs. (56) and (57) are identical to the heuristic ones defined by Eq. (40).Corresponding estimates ofφ(x,y) andA(x,y) were computed by application of the 2D inverseFFT algorithm. The variances of the reconstructed object properties in both the Fourier andspatial domains were estimated empirically.

6.4. Numerical results

Numerical studies were conducted to corroborate the noise analysis described in Section 5.Specifically, the theoretically predicted Var{φ(u,v)}, as stated in Eq. (24), was compared to anempirically determined estimate. The same was done for Var{A(u,v)}. When computing thetheoretically predicted Var{φ(u,v)}, Eqs. (34), (36), and (56) were employed with Eq. (24).Similarly, Eqs. (35), (37), and (57) were employed to determine the theoretically predictedVar{A(u,v)}. The empirical estimate of Var{φ(u,v)} was determined as follows. Firstly, em-pirical estimates of the two-state variance and covariancefunctions in Eqs. (21) and (22) werecomputed from the ensembles of noisy intensity data. Secondly, these quantities were employedto determine estimates of the optimal combination coefficientsωφ

m,n(u,v) via Eq. (26). Lastly,an empirical estimate of Var{φ(u,v)} was determined from an ensemble of noisy images re-constructed by use of Eq. (18). The same procedure was followed for determining the empiricalestimates of Var{A(u,v)}. Figures 4 and 6 display the determined variance maps correspondingto Geometry ‘A’ and Geometry ‘B’, respectively, which have been logarithmically transformedfor display purposes. In each figure, subfigures (a) and (b) display the theoretically predicted

and empirically determined images of log[Var{A(u,v)}

], respectively, while the corresponding

images of log[Var{φ(u,v)}

]are contained in subfigures (c) and (d), respectively. The theoreti-

cally predicted and empirical variance maps appear nearly identical. This is confirmed by Figs.5 and 7, in which horizontal profiles through the centers of the theoretically predicted andempirical variance maps are superimposed, respectively. These results demonstrate excellentagreement for all Fourier components.

Figure 8 displays estimates ofAm,n(x,y) reconstructed from noisy intensity data measured inGeometry ‘A’ by use of detector planes (a) (1,2), (b) (1,3), (c) (2,3). The ‘optimal’ estimate of



−3.5

−3

−2.5

−2

−1.5

−1

−0.5

(a)

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

(b)

0

5

10

15

(c)

0

5

10

15

(d)

Fig. 4. Images of theoretical and empirical estimates of Var{A(u,v)} measured in Geome-try ‘A’ are displayed logarithmically in subfigures (a)-(b), respectively. The correspondingvariance maps ofφ(u,v) are contained in subfigures (c)-(d), respectively.

2.5 3 3.5 4 4.5 5

x 105

10−2

10−1

100

101

v (1/m)

Var{

A[0

,v]}

(a)

2.5 3 3.5 4 4.5 5

x 105

10−2

10−1

100

v (1/m)

Var{

φ[0

,v]}

(b)

Fig. 5. Variance profiles of images in Fig. 4. Subfigure (a) contains the theoretically andempirically determined variance profiles ofA(u,v), which are depicted by solid and dashedcurves, respectively. The corresponding variance profiles ofφ(u,v) are shown in subfigure(b).



−3.5

−3

−2.5

−2

−1.5

−1

−0.5

(a)

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

(b)

0

5

10

15

(c)

0

5

10

15

(d)

Fig. 6. Images of theoretical and empirical estimates of Var{A(u,v)} measured in Geome-try ‘B’ are displayed logarithmically in subfigures (a)-(b), respectively. The correspondingvariance maps ofφ(u,v) are contained in subfigures (c)-(d), respectively.

2.5 3 3.5 4 4.5 5

x 105

10−2

10−1

100

v (1/m)

Var{

A[0

,v]}

(a)

2.5 3 3.5 4 4.5 5

x 105

10−2

10−1

100

v (1/m)

Var{

φ[0

,v]}

(b)

Fig. 7. Variance profiles of images in Fig. 6. Subfigure (a) contains the theoretically andempirically determined variance profiles ofA(u,v), which are depicted by solid and dashedcurves, respectively. The corresponding variance profiles ofφ(u,v) are shown in subfigure(b).



A(x,y) obtained by use of Eq. (19), which employs all three intensity measurements, is shownin (d). The corresponding estimates ofφm,n(x,y) andφ(x,y) are shown in subfigures (e)-(g) and(h), respectively. The excessively noisy appearances of the Am,n(x,y) in subfigures (a)-(c) aredue to the low absorption contrast of the object. In this measurement geometry, the relativelylarge detector spacings result in the occurrence of extra poles away from the origin in Fourierspace [see Eq. (33)]. This creates severe noise amplification in the high-frequency componentsof the two-state reconstructionsAm,n(x,y) andφm,n(x,y). The optimal estimates ofA(x,y) andφ(x,y) shown in subfigures (d) and (h), respectively, contain obviously reduced noise levels ascompared to the two-state reconstructions. The estimates of φm,n(x,y) andφ(x,y) appear to becontaminated by low-frequency noise, as evident by their lumpy background appearances. Thisis explained by the fact that all three estimates ofφm,n(x,y) have a pole at the origin of Fourierspace and therefore this pole cannot be removed by the estimator in Eq. (18). Note that takinga simple average of the available two-state estimates ofφm,n(x,y) or Am,n(x,y) would not be aneffective reconstruction strategy because it does not mitigate large noise amplifications due topoles in the reconstruction formulas away from the origin ofFourier space.

Variance data for the images contained in Fig. 8 are shown in Fig. 9. Figures 9(a) and (b)display empirical variance estimates of the reconstructedFourier components. Each subfigurecontains profiles corresponding to the Fourier variance estimated by use of detector pair (1,2)(dashed curve), detector pair (1,3) (dashdotted curve), detector (2,3) (dotted curve), and theoptimal ones reconstructed by use of Eqs. (18) or (19) (solidcurve). As expected, the optimalestimates possess variances that are lower than any of the two-state estimates, for all frequencycomponents. Subfigures (c) and (d) display empirical estimates of the corresponding images inthe spatial domain. The labeling of the profiles is the same asdescribed above. The variancesof the optimally estimated images are seen to be lower than the two-state reconstructions, forall values of(x,y). This reflects that estimates having reduced Fourier variances will generallyhave reduced variances in the spatial domain.

The corresponding results obtained with Geometry ‘B’ are shown in Figs. 10 and 11. FromFig. 10, we find again that the optimal estimates ofA(x,y) andφ(x,y) contain obviously reducednoise levels as compared to the two-state reconstructions.The quantitative variance data in Fig.11 confirms that the optimal estimates ofA(x,y) andφ(x,y) contain lower variances than thetwo-state reconstructions.

In practice, the detector locations cannot be determined with perfect precision. The uncer-tainty in the detector positions will introduce inconsistencies between the measured data andassumed imaging model. In order to investigate the effects of this, images were reconstructedfrom the noisy intensity measurements corresponding to Geometry ‘A’, where imperfect knowl-edge of the detector positions was assumed. The assumed detector positions ˆzm were related tothe true locationszm by zm = zm + ε(zm), with m = 1,2,3, whereε(zm) represents the posi-tioning errors. The three sets of errorsε(zm) contained in Table 1 were considered. The cor-responding estimates ofA(x,y) and φ(x,y) reconstructed by the optimal estimation methodare shown in Fig. 12. Subfigures (a)-(c) contain the optimal estimates ofA(x,y) for detectorposition error levels (1), (2), and (3), respectively. The corresponding estimates ofφ(x,y) arecontained in subfigures (d)-(f). Despite the geometry uncertainties, the reconstructedA(x,y)andφ(x,y) closely resemble the corresponding images in Fig. 8(d) and (h), which were recon-structed from the same noisy data but assuming perfect knowledge of the imaging geometry.The mean square error (MSE) values of the estimates ofA(x,y) displayed in Fig. 12(a)-(c) are1.26×10−7, 1.27×10−7, and 1.32×10−7, respectively. The MSE values forφ(x,y) shown inFig. 12(d)-(f) are 3.37×10−4, 9.34×10−4, and 3.37×10−4, respectively. As a reference, theMSE values for the estimates ofA(x,y) andφ(x,y) shown in Figs. 8(d) and (h) were 1.25×10−7

and 4.65×10−4, respectively. Note that these MSE values were computed by assuming a dis-



−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

(a) A1,2

−1

−0.5

0

0.5

1

(b) A1,3

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

(c) A2,3

−1

0

1

2

3

4

5

x 10−3

(d) Aoptm

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(e) φ1,2

−0.2

−0.1

0

0.1

0.2

(f) φ1,3

−0.3

−0.2

−0.1

0

0.1

0.2

(g) φ2,3

−0.04

−0.02

0

0.02

0.04

0.06

(h) φoptm

Fig. 8. Estimates ofA(x,y) reconstructed from noisy intensity data measured in Geome-try ‘A’ by use of detector-planes (a)(1,2), (b)(1,3), (c)(2,3),and (d) an optimally-weightedcombination of all three detector-planes. The corresponding estimates of φ(x,y) are shownin subfigures (e)-(h).

crete (Cartesian grid) representation of the true functions A(x,y) andφ(x,y), and do not reflectan average over the distribution of the measurement noise. These observations suggest that ourmulti-plane reconstruction method may be robust to geometry errors under certain conditions.However, a detailed investigation of the effects of geometry errors on the proposed image re-construction methods remains an important topic for futureresearch.

Table 1. Error levels in the detector positions in Geometry ‘A’.

Geometry ‘A’ ε(z1) ε(z2) ε(z3)

error level 1 -3 mm +2 mm -5 mm

error level 2 -5 mm +4 mm +10 mm

error level 3 -19 mm +12 mm +17 mm



−5 0 5

x 105

10−2

100

102

104

v (1/m)

Var{

A[0

,v]}

(a)

−5 0 5

x 105

10−2

100

102

104

106

108

1010

v (1/m)

Var{

φ[0

,v]}

(b)

300 400 500 600 70010

−8

10−6

10−4

10−2

100

Pixel

Var{

A[0

,y]}

(c)

300 400 500 600 70010

−4

10−3

10−2

10−1

Pixel

Var{

φ[0

,y]}

(d)

Fig. 9. Empirical variance profiles ofA(u,v) andφ(u,v) measured in Geometry ‘A’ are dis-played in subfigures (a)-(b), respectively. Each subfigure contains profiles correspondingto the Fourier variances estimated by use of detector-planes (1,2) (dashed curve), detector-planes (1,3) (dashdotted curve), detector-planes (2,3) (dotted curve), and the optimal one(solid curve). Subfigures (c) and (d) display empirical variance profiles of the correspond-ing images in the spatial domain.

7. Summary and conclusions

From knowledge of independent intensity measurements (i.e., phase-contrast radiographs) ac-quired at distinct states of the imaging system, quantitative phase-contrast X-ray imaging meth-ods seek to reconstruct the projected X-ray absorption and refractive index distributions of anobject. If the interaction of the X-ray wavefield with the imaging system is described as alinear, shift-invariant, coherent imaging system, Fourier-based reconstruction formulas can bederived from knowledge of the imaging system’s optical transfer function. A common featureof these reconstruction formulas, however, is the presenceof isolated Fourier domain singular-ities. These poles can greatly amplify noise levels in the estimated Fourier components, whichcan lead to noisy and/or distorted images in the spatial domain. If intensity measurements areacquired at three or more distinct states of the imaging system, it is sometimes possible to miti-gate the noise amplification due to the poles. However, thereremains a need for the developmentof statistically principled reconstruction methods to achieve this.

In this work, we proposed and investigated a methodology forimage reconstruction in quanti-tative phase-contrast imaging when multiple (> 2) intensity measurements are available. Linearestimators were proposed that combine the available intensity measurements in such a way thatthe Fourier components of the desired object properties have reduced variances. These estima-tors involve the use of combination coefficients that shouldbe chosen to cancel poles present inestimates obtained by use of any two measurements. From knowledge of the measurement noisemodel, optimal forms of the combination coefficients are derived that minimize the Fouriervariance of the final estimates. When such information is not available, we demonstrated that



−0.05

0

0.05

(a) A1,2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

(b) A1,3

−0.1

−0.05

0

0.05

0.1

(c) A2,3

−1

0

1

2

3

4

5

x 10−3

(d) Aoptm

−0.2

−0.1

0

0.1

(e) φ1,2

−0.2

−0.1

0

0.1

0.2

(f) φ1,3

−0.4

−0.2

0

0.2

0.4

(g) φ2,3

−0.1

−0.05

0

0.05

(h) φoptm

Fig. 10. Estimates ofA(x,y) reconstructed from noisy intensity data measured in Geometry‘B’ by use of detector-planes (a)(1,2), (b)(1,3), (c)(2,3), and (d) an optimally-weightedcombination of all three detector-planes. The corresponding estimates of φ(x,y) are shownin subfigures (e)-(h).

−5 0 5

x 105

10−2

100

102

104

v (1/m)

Var{

A[0

,v]}

(a)

−5 0 5

x 105

10−2

100

102

104

106

108

1010

v (1/m)

Var{

φ[0

,v]}

(b)

300 400 500 600 70010

−10

100

1010

1020

Pixel

Var{

A[0

,y]}

(c)

300 400 500 600 70010

−5

100

105

1010

1015

1020

Pixel

Var{

φ[0

,y]}

(d)

Fig. 11. Empirical variance profiles ofA(u,v) and φ(u,v) measured in Geometry ‘B’ aredisplayed in subfigures (a)-(b), respectively. Each subfigure contains profiles correspondingto the Fourier variances estimated by use of detector-planes (1,2) (dashed curve), detector-planes (1,3) (dashdotted curve), detector-planes (2,3) (dotted curve), and the optimal one(solid curve). Subfigures (c) and (d) display empirical variance profiles of the correspond-ing images in the spatial domain.



−1

0

1

2

3

4

5

x 10−3

(a)

−1

0

1

2

3

4

5

x 10−3

(b)

−1

0

1

2

3

4

5

x 10−3

(c)

−0.04

−0.02

0

0.02

0.04

0.06

(d)

−0.04

−0.02

0

0.02

0.04

0.06

(e)

−0.04

−0.02

0

0.02

0.04

0.06

(f)

Fig. 12. The optimally determined estimates ofA(x,y) are reconstructed from noisy inten-sity data measured in Geometry ‘A’ with detector position uncertainty of (a) error level 1,(b) error level 2, and (c) error level 3. The corresponding estimates ofφ(x,y) are containedin subfigures (d)-(f).

the large noise amplification due to the poles can still be mitigated by an appropriate heuristicchoice of the combination coefficients.

The developed reconstruction methods were investigated, in detail, within the context ofpropagation-based X-ray phase-contrast imaging. Explicit forms of the optimal estimators werederived in both their continuous and discrete forms. Preliminary computer-simulation studieswere conducted to demonstrate the efficacy of the proposed reconstruction methods, and cor-roborate our theoretical analysis.

As quantitative X-ray phase-contrast imaging is in its infancy, there remain numerous im-portant topics for future investigation. One important topic is the experimental and theoreticalinvestigation of the proposed reconstruction methods within the context of clinically feasibleX-ray sources for phase-contrast imaging. This will require generalization of the methods tocompensate for non-ideal physical factors such as partial coherence effects. The maximumnoise level for which a useful image can be reconstructed by use of the proposed methods isnot easily answered by a single number or rule. It will depend, in a non-trivial way, on themeasurement geometry, which determines the locations of poles in the reconstruction formu-las, the explicit nature of the object’s refractive index distribution, and the ultimate use of theimage. It will be important to conduct a detailed investigation of the statistical properties of thereconstructed images and their influence on various task-based detectability measures.

Appendix: Generalization to≥3 measurement states

Consider a measurement geometry that consists ofM measurement states 1, 2, 3,· · · , M. Esti-mates of the desired object properties can be computed by useof any measurement-state pair;thus there will be(M

2 ) distinct estimates available for theM-state system. Consequently, thefinal estimate ofφ(u,v), for example, can be obtained from a weighted summation of all the



available estimates. Letφm,n(u,v) denote the phase estimate computed by use of the intensitydata acquired on measurement pair(m,n). A final unbiased estimate that possesses a potentiallyreduced variance can be written as

φ(u,v) =M−1

∑m=1

M

∑n=m+1

ωφm,n(u,v)φm,n(u,v), (A1)

whereωφm,n(u,v) are generally the complex-valued coefficients that satisfy

M−1

∑m=1

M

∑n=m+1

ωφm,n(u,v) = 1. (A2)

As described in Section 3, however, any estimateφm,n(u,v) is a linearly combination of any

other two estimatesφ1,m(u,v) andφ1,n(u,v), wherem, n are integer-valued indices. Therefore,Eq. (A1) can be simplified to

φ(u,v) =M

∑m=2

ωφ1,m(u,v)φ1,m(u,v). (A3)

The variance of the final estimateφ(u,v) can be obtained readily as

Var{φ(u,v)} =M

∑l=2

|ωφ1,l (u,v)|2Var{φ1,l (u,v)}

+2Re

[M−1

∑m=2

M

∑n=m+1

ωφ1,m(u,v)ωφ∗

1,n(u,v)Cov{φ1,m(u,v)φ1,n(u,v)}

], (A4)

wherel = 2,3, · · · ,M in the first summation term, andm= 2,3, · · · ,M−1, n= 3,4, · · · ,M withn > m in the second summation term.

We introduce the following notation:

σ21,m(u,v) ≡ Var{φ1,m[u,v]}, (A5)

ρ(r)1,m;1,n(u,v)+ jρ(i)

1,m;1,n(u,v) ≡ Cov{φ1,m[u,v], φ1,n[u,v]}, (A6)

R1,m(u,v)+ jI1,m(u,v) ≡ ωφ1,m(u,v). (A7)

Thus the variance of the final Fourier estimate can be obtained,

Var{φ} =M

∑l=2

(R21,l + I2

1,l )σ21,l

+2

{M−1

∑m=2

M

∑n=m+1

[ρ(r)

1,m;1,n(R1,mR1,n + I1,mI1,n)−ρ(i)1,m;1,n(R1,nI1,m−R1,mI1,n)

]}. (A8)

In order to reach an optimal final estimate, variance needs tobe specified such that,

∂Var{φ}∂R1,m

∣∣∣R(op)

1,m

= 0

∂Var{φ}∂ I1,m

∣∣∣I (op)1,m

= 0, (A9)



whereR(op)1,m andI (op)

1,m specify the value of the optimal combination coefficient forωφ1,m with m=

2,3, · · · ,M. Analytic formula for determination of the combination coefficients can be derivedas follows. From knowledge of Eq. (A8) and taking use of Eq. (A9), a (2M −4)× (2M −4)system is formed, which can be expressed by a matrix equation.

Hx = b (A10)

where

H =

h11 h12 · · · h1,2M−4...

.... ..

...h2M−4,1 h2M−4,2 · · · h2M−4,2M−4

, (A11)

x =(

R(op)1,2 I (op)

1,2 R(op)1,3 I (op)

1,3 · · · R(op)1,M−1 I (op)

1,M−1

)T, (A12)

andb =

(b1 b2 · · · b2M−4

)T, (A13)

where them-th equation in the system specifies the partial derivative of Var{φ} with respectto them-th component of Eq. (A12). AssumingH is nonsingular, inversion of Eq. (A10) isaccomplished readily:

xm(u,v) =det(Hm)

det(H)m= 1,2,3, · · · ,2M−4 (A14)

whereHm is the matrix obtained by replacing them-th column ofH with b. If we specify thecoefficientωφ

1,m(u,v) to be real-valued, the termI1,m(u,v) will vanish, yielding a(M − 2)×

(M−2) system.Analogous to Section 4.3, the heuristically determined combination coefficientωheur

1,m for M−detector system can be computed readily as:

ωheur1,m (u,v) =

D21,m+α1,m∑M

n=2n6=m

D2m,n

∑M−1l=1 ∑M

m=l+1D2l ,m

(A15)

Acknowledgment

This research was supported in part by National Institutes of Health research grant EB004507and a National Science Foundation CAREER Award 0546113.



Image reconstruction in quantitative X-ray phase...

Documents

Transcript of Image reconstruction in quantitative X-ray phase...