## Image reconstruction in quantitative X-ray phase-contrast imaging employing multiple measurements

Optics Express, Vol. 15, Issue 16, pp. 10002-10025 (2007)

http://dx.doi.org/10.1364/OE.15.010002

Acrobat PDF (7916 KB)

### Abstract

X-ray phase-contrast imaging is a technique that aims to reconstruct the projected absorption and refractive index distributions of an object. One common feature of reconstruction formulas for phase-contrast imaging is the presence of isolated Fourier domain singularities, which can greatly amplify the noise levels in the estimated Fourier domain and lead to noisy and/or distorted images in spatial domain. In this article, we develop a statistically optimal reconstruction method that employs multiple (>2) measurement states to mitigate the noise amplification effects due to singularities in the reconstruction formula. Computer-simulation studies are carried out to quantitatively and systematically investigate the developed method, within the context of propagation-based X-ray phase-contrast imaging. The reconstructed images are shown to possess dramatically reduced noise levels and greatly enhanced imaging contrast.

© 2007 Optical Society of America

## 1. Introduction

1. 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). [CrossRef] [PubMed]

2. R. Waynant, “Toward practical coherent x-ray sources: Potential medical applications,” IEEE J. Quantum Electron. **6**, 1465–1469 (2000). [CrossRef]

*X-ray*

*phase-contrast*imaging methods [4

4. R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. **49**, 3573–3583 (2004). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef] [PubMed]

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

14. B.L. Henke, E.M. Gullikson, and J.C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables **54**, 181–342 (1993). [CrossRef]

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, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. **43**, 2845–2852 (1998). [CrossRef] [PubMed]

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 Techniques,” Radiology **215**, 286–293 (2000).

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

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 Histologic Correlation-Improved Conspicuity of Lesion Detail Compared with Digital Radiography,” Radiology **214**, 895–901 (2000). [PubMed]

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 digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. **40**, 385–396 (2005). [CrossRef] [PubMed]

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

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

4. R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. **49**, 3573–3583 (2004). [CrossRef] [PubMed]

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). [CrossRef]

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

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

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). [CrossRef]

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). [CrossRef] [PubMed]

27. D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. **234**, 87–105 (2004). [CrossRef]

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). [CrossRef]

29. A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. **A 19**, 472–480 (2002). [CrossRef]

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

31. M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. **49**, 121–144 (2004). [CrossRef] [PubMed]

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

27. D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. **234**, 87–105 (2004). [CrossRef]

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

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

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). [CrossRef]

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

36. Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. **A 24**, 626–642 (2007). [CrossRef]

## 2. Background

37. D. M. Paganin, *Coherent X-Ray Optics* (Oxford University Press, 2006). [CrossRef]

### 2.1. Interaction of X-ray wavefield with object

*U*with wavelength λ, which is traveling along the positive

_{i}*z*-axis. The effects of imperfect wavefield coherence will not be considered, but can be addressed as in [38

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). [CrossRef]

*β*(

*δ*(

*β*(

*U*(

_{o}*x*,

*y*) on the object plane, which has been transmitted through the object, is given by

*T*(

*x*,

*y*) is the transmission function that can be expressed generally as

### 2.2. Linear shift-invariant X-ray phase-contrast imaging systems

*U*(

_{m}*x*,

*y*) denote the transmitted wavefield on a detector plane of constant

*z*, which is downstream from the object plane. The integer-valued subscript

*m*is employed to denote the state of the imaging system. For many analyzer- and propagation-based phase-contrast imaging systems,

*U*(

_{m}*x*,

*y*) and

*U*(

_{o}*x*,

*y*) can be regarded as the output and input, respectively, of a linear and shift-invariant coherent imaging system [40, 41]. In this case,

*G*(

_{m}*x*,

*y*) describes the impulse response of the system in state

*m*. In propagation-based imaging, for example,

*G*(

_{m}*x*,

*y*) describes the Fresnel propagator [26], with distinct values of

*m*corresponding to different object-to-detector distances. Alternatively, in analyzer-based imaging,

*G*(

_{m}*x*,

*y*) describes the coherent impulse response of the analyzer crystal or other diffractive element(s) employed, with distinct values of

*m*corresponding to different orientations of the analyzer. In hybrid systems [39],

*G*(

_{m}*x*,

*y*) describes the net effect of both.

*I*(

_{m}*x*,

*y*)=|

*U*(

_{m}*x*,

*y*)|

^{2}is recorded, which represents a radiograph with mixed absorption- and phase-contrast. From knowledge of the measured intensity

*I*(

_{m}*x*,

*y*), a modified data function can be defined as

*I*=|

_{i}*U*|

_{i}^{2}is the intensity of the incident X-ray beam. Let

*K*̃

*(*

_{m}*u*,

*v*) denote the 2D Fourier transform (FT) of

*K*(

_{m}*x*,

*y*) defined as

*A*(

*x*,

*y*)|≪1 and slowly varying

*ϕ*(

*x*,

*y*) can often be met [10

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). [CrossRef] [PubMed]

*G*̃

*(*

^{a}_{m}*u*,

*v*) is the amplitude transfer function (ATF):

*G*̃

*(*

^{p}_{m}*u*,

*v*) is the phase transfer function (PTF):

*G*̃

*(*

_{m}*u*,

*v*),

*Ã*(

*u*,

*v*), and

*ϕ*̃(

*u*,

*v*) are the 2D FTs of

*G*(

_{m}*x*,

*y*),

*A*(

*x*,

*y*), and

*ϕ*(

*x*,

*y*), respectively, and

*G*̃*

*(·, ·) denotes the complex conjugate of*

_{m}*G*̃

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

_{m}*I*(

_{m}*x*,

*y*), or equivalently

*K*(

_{m}*x*,

*y*), to the 2D FTs of the sought-after quantities

*A*(

*x*,

*y*) and

*ϕ*(

*x*,

*y*). If an additional measurement

*I*(

_{n}*x*,

*y*) is obtained when the imaging system is in state

*n*≠

*m*,

*A*̃(

*u*,

*v*) and

*ϕ*̃(

*u*,

*v*) can be determined algebraically as

*A*̃(

*u*,

*v*) and

*ϕ*̃(

*u*,

*v*),

*A*(

*x*,

*y*) and

*ϕ*(

*x*,

*y*) are computed by application of the inverse 2D FT. Note that Eqs. (12a) and (b) contain isolated poles at frequency components (

*u*,

*v*) for which

*u*,

*v*) of

*A*̃(

*u*,

*v*) and

*ϕ*̃(

*u*,

*v*) residing near the poles will contain greatly amplified noise levels. This can result in noisy and/or distorted images. In the remainder of this article, we describe a statistically principled method for circumventing this when measurements corresponding to multiple (>2) states of the system are available.

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

*I*(

_{m}*x*,

*y*) are acquired at three distinct states

*m*=1,2,3 of the system. The results below are generalized to the case of an arbitrary number of measurements in the Appendix. The intensity data function

*I*(

_{m}*x*,

*y*) is interpreted as a stochastic process, which reflects that the measurements are contaminated by stochastic errors such as detector noise. Our goal is to exploit the statistically complementary information in the available measurements to reduce the variance of the estimated

*A*̃(

*u*,

*v*) and

*ϕ*̃(

*u*,

*v*), and thereby mitigate the large amplification of noise due to poles in the reconstruction formulas.

*Ã*(

*u*,

*v*) and

*ϕ*̃(

*u*,

*v*) can be computed from knowledge of measurements obtained at

*N*system states. When reconstructed from noisy measurements, these estimates will be generally distinct. The notation

*Ã*

_{m,n}(

*u*,

*v*) and

*ϕ*̃

_{m,n}(

*u*,

*v*),

*m*≠

*n*=1,2,3, will be employed to describe the estimates for the case

*N*=3, where the subscripts denote that measurements

*I*(

_{m}*x*,

*y*) and

*I*(

_{n}*x*,

*y*) were employed. Because the locations of poles in Eq. (12) depend on the choice of measurement states, the components of

*Ã*

_{m,n}(

*u*,

*v*) or

*ϕ*̃

_{m,n}(

*u*,

*v*) that are highly contaminated by noise will be determined by the measurement state pair (

*m*,

*n*).

*Ã*

_{m,n}(

*u*,

*v*) or

*ϕ*̃

_{m,n}(

*u*,

*v*) in a way that attempts to cancel the poles in each two-state estimate [36

36. Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. **A 24**, 626–642 (2007). [CrossRef]

*Ã*(

*u*,

*v*) or

*ϕ*̃(

*u*,

*v*), respectively, that possess reduced variances for all (

*u*,

*v*). However, it should be noted that only two of the three available estimates are independent in the sense that

*l*≠

*m*≠

*n*=1,2,3. In other words, any estimate can be expressed as a linear combination of the remaining two. The coefficients in Eqs. (14) and (15) are frequency-dependent and given by

*Ã*(

*u*,

*v*) and

*ϕ*̃(

*u*,

*v*) that exploit statistically complementary information in the three intensity measurements can be formed as

*ϕ*̃(

*u*,

*v*) and

*Ã*(

*u*,

*v*), respectively. Because the combination coefficients

*ω*

^{ϕ}_{m,n}(

*u*,

*v*) and

*ω*

^{a}_{m,n}(

*u*,

*v*) are frequency-dependent, they can be designed to cancel poles present in the

*ϕ*̃

_{m,n}(

*u*,

*v*) and

*Ã*

_{m,n}(

*u*,

*v*), respectively. Moreover, as described next, they can be designed to optimally reduce the variance of

*ϕ*̃(

*u*,

*v*) and

*Ã*(

*u*,

*v*). We consider first the problem of producing estimates

*ϕ*̃(

*u*,

*v*) having reduced variances. The following notation will be employed:

*ϕ*̃(

*u*,

*v*) is given by

*ϕ*̃(

*u*,

*v*)}, we need that

*R*

_{1,2}and

*I*

_{1,2}are the real and imaginary components of

*ω*

^{ϕ}_{1,2}, respectively. The solution of these equations yields

*ω*

^{a}_{1,2}(

*u*,

*v*) and

*ω*

^{a}_{1,3}(

*u*,

*v*) that optimally reduce the variance of

*Ã*(

*u*,

*v*) via Eq. (19) are also determined by Eq. (26) when the quantities in Eqs. (21)–(23) are redefined appropriately. A heuristic method for choosing the combination coefficients that can effectively mitigate noise amplification when the noise model is not known is described later.

*ϕ*(

*x*,

*y*) and attenuation function

*A*(

*x*,

*y*) are described by simple algebraic forms in the Fourier domain. Consequently, the large amplification of noise due to poles in the reconstruction formulas can be mitigated in a mathematically straightforward and physically understandable way in the Fourier domain. Reducing the Fourier domain variance of the phase and attenuation estimates generally leads to spatial domain estimates that possess reduced variances. This can be understood by noting that

*ϕ*̃(

*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 holds true for

*A*(

*x*,

*y*).

## 4. Application to multi-plane propagation-based imaging

### 4.1. Two-state reconstruction formulas

*I*(

_{m}*x*,

*y*) are acquired on distinct detector planes

*z*=

*z*, where

_{m}*m*=1,2,3. The impulse response in Eq. (6) corresponds to the Fresnel propagator

*f*

^{2}≡

*u*

^{2}+

*v*

^{2}and

*m*,

*n*satisfy

*m*=1,2,

*n*=2,3 with

*n*>

*m*.

*D*

_{m,n}=0 is equivalent to Eq. (13), and specifies the locations of poles in the reconstruction formulas. Equation (29) contains poles corresponding to frequencies (

*u*,

*v*) that satisfy

*l*is an integer. One such pole is located at zero-frequency

*u*=

*v*=0, indicating that the low-frequency components of

*ϕ*̃(

*u*,

*v*) will contain highly amplified noise levels. The existence of additional poles, away from the origin of the Fourier space, depends on the detector resolution and the detector pair spacing. Let (

*u*,

_{M}*v*) 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

_{M}*ϕ*̃

_{m,n}(

*u*,

*v*) will emerge when the detector spacing (

*z*-

_{m}*z*) is sufficiently large. Likewise, this discussion of poles also applies to

_{n}*Ã*

_{m,n}(

*u*,

*v*), with the exception that the pole at

*u*=

*v*=0 is not present due to a cancellation.

### 4.2. Second-order statistics for determination of optimal combination coefficients

*ω*

^{ϕ}_{1,2}(

*u*,

*v*) and

*ω*

^{ϕ}_{1,3}(

*u*,

*v*) that minimize the variance

*ϕ*̃(

*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 required for determination of the optimal combination coefficients of

*ω*

^{a}_{1,2}(

*u*,

*v*) and

*ω*

^{a}_{1,3}(

*u*,

*v*) that minimize the variance of

*A*̃(

*u*,

*v*).

### 4.3. Heuristic determination of combination coefficients

*ω*

^{a}_{m,n}(

*u*,

*v*) and

*ω*

^{ϕ}_{m,n}(

*u*,

*v*) cannot be computed. However, the large noise amplification due to poles in

*ϕ*̃

_{m,n}(

*u*,

*v*) and

*Ã*

_{m,n}(

*u*,

*v*) can still be effectively mitigated by suitable heuristic specification of the combination coefficients. From Eqs. (34)–(35), near the locations of poles, we find that

*ω*

^{heur}

_{m,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

*ω*

^{heur}

_{m,n}(

*u*,

*v*)≡0. Based on these requirements, the

*ω*

^{heur}

_{m,n}(

*u*,

*v*) for use in estimating

*ϕ*̃(

*u*,

*v*) via Eq. (18) can be chosen as

*α*

^{ϕ}_{1,2}and

*α*

^{ϕ}_{1,3}are defined in Eq. (16). When estimating

*Ã*(

*u*,

*v*),

*α*

^{ϕ}_{1,2}and

*α*

^{ϕ}_{1,3}are replaced by

*α*

^{a}_{1,2}and

*α*

^{a}_{1,3}, respectively. Due to their construction,

*ω*

^{heur}

_{1,2}(

*u*,

*v*) and

*ω*

^{heur}

_{1,3}(

*u*,

*v*) satisfy the normalization condition in Eq. (20).

## 5. Computation of optimal combination coefficients with consideration of finite sampling

*ω*

^{a}_{m,n}(

*u*,

*v*) and

*ω*

^{ϕ}_{m,n}(

*u*,

*v*) are computed explicitly with consideration of finite sampling effects.

### 5.1. Noise model and finite sampling considerations

*z*=

*z*is denoted as

_{m}*r*and

*s*are integer-valued indices that reference detector elements, and Δ

*x*=Δ

*y*denotes the element dimension in a square detector array of dimension

*L*×

*L*. Equation (41) assumes idealized (Dirac delta) sampling, namely, the averaging effects of sampling aperture are not considered. However, the analysis follows can be generalized to address such effects. The square bracket, ‘[·]’, is introduced to represent the functions whose arguments are discretely sampled. We assume the noise model satisfies [35

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

44. S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: Statistical properties,” J. Opt. Soc. Am. **60**, 1478–1483 (1970). [CrossRef]

*I*

^{0}

*[*

_{m}*r*,

*s*] denotes the noiseless intensity data and the signal-dependent noise

*n*(

_{m}*r*,

*s*) has zero-mean and

*σ*

^{2}(

*z*) can depend on the detector location

_{m}*z*[35

_{m}35. D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. **214**, 51–61 (2003). [CrossRef]

*δ*denotes the Kronecker delta function.

_{mn}### 5.2. Second-order statistics of discrete data functions

*Ĩ*(

*u*,

*v*) is required. To compute this from knowledge of discretely sampled data, the continuous 2D FT will be approximated by use of the discrete Fourier transform (DFT) [45].

*p*,

*q*denoting the integer-valued Fourier indices that are conjugate to [

*r*,

*s*], and

*N*specifying the number of detector elements in each dimension of the square 2D detector. The variance of

*ñ*[

_{m}*p*,

*q*] is computed as

*u*and

*v*axes. By use of Eqs. (49) and (50), we find that

*m*=1,2,

*n*=2,3 with

*n*>

*m*, and

### 5.3. Explicit forms of optimal combination coefficients

*ϕ*̃(

*u*,

*v*) or

*Ã*(

*u*,

*v*) via Eq. (18) or (19), respectively:

*σ*

^{2}

*≡*

_{m}*σ*

^{2}(

*z*). The corresponding forms of

_{m}*ω*

^{ϕ}_{1,3}(

*u*,

*v*) and

*ω*

^{a}_{1,3}(

*u*,

*v*) are determined by Eq. (20).

^{N-1}

_{r=0}∑

^{N-1}

_{s=0}(

*I*

^{0}[

*r*,

*s*;

*z*])

_{m}^{2}does not vary significantly as a function of

*m*(i.e., the detector location

*z*), Eq. (56) can be expressed in the somewhat simplified form:

_{m}## 6. Numerical Studies

### 6.1. Numerical phantom and in-line measurement geometry

*λ*=1×10

^{-10}m, propagated along the

*z*-axis and irradiated an object. Three detector planes located at

*z*=

*z*,

_{m}*m*=1,2,3, were considered to be behind the object. The detector contained 1024×1024 elements of dimension of 1

*µ*m

^{2}, and was assumed to have otherwise idealized physical properties. Two measurement geometries were considered, which will be referred to as Geometry ‘A’ and Geometry ‘B’. In Geometry ‘A’, the detector planes were positioned at

*z*

_{1}=19 mm,

*z*

_{2}=96 mm,

*z*

_{3}=182 mm, while the corresponding positions in Geometry ‘B’ were

*z*

_{1}=12 mm,

*z*

_{2}=38 mm, and

*z*

_{3}=72 mm.

*µ*m, 163.840

*µ*m, and 141.312

*µ*m. From knowledge of the phantom, the projected object properties

*ϕ*(

*x*,

*y*) and

*A*(

*x*,

*y*) were computed according to Eq. (5) and are displayed in Fig. 3.

### 6.2. Measurement data and simulation studies

*ϕ*(

*x*,

*y*) and

*A*(

*x*,

*y*), the transmitted wavefield

*U*(

_{o}*x*,

*y*) on the object plane was computed according to Eqs. (3) and (4). Subsequently, sampled values of the wavefield

*U*(

_{m}*x*,

*y*) on each detector plane

*z*=

*z*,

_{m}*m*=1,2,3, was computed by use of Eq. (6) with

*G*(

_{m}*x*,

*y*) specified by the Fresnel propagator in Eq. (27). The convolution in Eq. (6) was computed by use of the 2D fast Fourier transform (FFT) algorithm. The intensity data

*I*[

_{m}*r*,

*s*] were then computed as the square of the wavefield modulus on each detector plane.

### 6.3. Image reconstruction

*ϕ*̃

_{m,n}(

*u*,

*v*) and

*Ã*

_{m,n}(

*u*,

*v*) were computed from each pair of noisy intensity data by use of Eqs. (29) and (30), respectively. The presence of poles can pose considerable difficulty in determining these estimates. Simply setting

*ϕ*̃

_{m,n}(

*u*,

*v*)=0 or

*Ã*

_{m,n}(

*u*,

*v*)=0 at the locations of poles will lead to inaccuracy in the resulted estimates. Additionally, even if the poles are avoided, the data errors can be greatly amplified in the vicinity of poles, where the denominators of Eqs. (29) and (30) take on small values. In current studies, the reconstruction formulas were regularized by setting the estimates of

*ϕ*̃

_{m,n}(

*u*,

*v*) and

*Ã*

_{m,n}(

*u*,

*v*) to zeros in the vicinity of poles when

*D*

_{m,n}≤2×10

^{-7}. These estimates were combined, according to Eqs. (18) and (19), to form final estimates

*ϕ*̃(

*u*,

*v*) and

*Ã*(

*u*,

*v*) that possess optimally reduced variances. The required combination coefficients were computed according to Eqs. (56) and (57). Because

*σ*

^{2}(

*z*) was fixed at a constant value, it can be verified that, in this special case, the optimal combination coefficients given in Eqs. (56) and (57) are identical to the heuristic ones defined by Eq. (40). Corresponding estimates of

_{m}*ϕ*(

*x*,

*y*) and

*A*(

*x*,

*y*) were computed by application of the 2D inverse FFT algorithm. The variances of the reconstructed object properties in both the Fourier and spatial domains were estimated empirically.

### 6.4. Numerical results

*ϕ*̃(

*u*,

*v*)}, as stated in Eq. (24), was compared to an empirically determined estimate. The same was done for Var{

*Ã*(

*u*,

*v*)}. When computing the theoretically 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 predicted Var{

*Ã*(

*u*,

*v*)}. The empirical estimate of Var{

*ϕ*̃(

*u*,

*v*)} was determined as follows. Firstly, empirical estimates of the two-state variance and covariance functions in Eqs. (21) and (22) were computed from the ensembles of noisy intensity data. Secondly, these quantities were employed to determine estimates of the optimal combination coefficients

*ω*ϕ

_{m,n}(

*u*,

*v*) via Eq. (26). Lastly, an empirical estimate of Var{

*ϕ*̃(

*u*,

*v*)} was determined from an ensemble of noisy images reconstructed by use of Eq. (18). The same procedure was followed for determining the empirical estimates of Var{

*Ã*(

*u*,

*v*)}. Figures 4 and 6 display the determined variance maps corresponding to Geometry ‘A’ and Geometry ‘B’, respectively, which have been logarithmically transformed for display purposes. In each figure, subfigures (a) and (b) display the theoretically predicted and empirically determined images of log [Var{

*Ã*(

*u*,

*v*)}], respectively, while the corresponding images of log [Var{

*ϕ*̃(

*u*,

*v*)}] are contained in subfigures (c) and (d), respectively. The theoretically 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 and empirical variance maps are superimposed, respectively. These results demonstrate excellent agreement for all Fourier components.

*A*

_{m,n}(

*x*,

*y*) reconstructed from noisy intensity data measured in Geometry ‘A’ by use of detector planes (a) (1,2), (b) (1,3), (c) (2,3). The ‘optimal’ estimate of

*A*(

*x*,

*y*) obtained by use of Eq. (19), which employs all three intensity measurements, is shown in (d). The corresponding estimates of

*ϕ*

_{m,n}(

*x*,

*y*) and

*ϕ*(

*x*,

*y*) are shown in subfigures (e)–(g) and (h), respectively. The excessively noisy appearances of the

*A*

_{m,n}(

*x*,

*y*) in subfigures (a)–(c) are due to the low absorption contrast of the object. In this measurement geometry, the relatively large detector spacings result in the occurrence of extra poles away from the origin in Fourier space [see Eq. (33)]. This creates severe noise amplification in the high-frequency components of the two-state reconstructions

*A*

_{m,n}(

*x*,

*y*) and

*ϕ*

_{m,n}(

*x*,

*y*). The optimal estimates of

*A*(

*x*,

*y*) and

*ϕ*(

*x*,

*y*) shown in subfigures (d) and (h), respectively, contain obviously reduced noise levels as compared to the two-state reconstructions. The estimates of

*ϕ*

_{m,n}(

*x*,

*y*) and

*ϕ*(

*x*,

*y*) appear to be contaminated by low-frequency noise, as evident by their lumpy background appearances. This is explained by the fact that all three estimates of

*ϕ*

_{m,n}(

*x*,

*y*) have a pole at the origin of Fourier space and therefore this pole cannot be removed by the estimator in Eq. (18). Note that taking a simple average of the available two-state estimates of

*ϕ*

_{m,n}(

*x*,

*y*) or

*A*

_{m,n}(

*x*,

*y*) would not be an effective reconstruction strategy because it does not mitigate large noise amplifications due to poles in the reconstruction formulas away from the origin of Fourier space.

*x*,

*y*). This reflects that estimates having reduced Fourier variances will generally have reduced variances in the spatial domain.

*A*(

*x*,

*y*) and

*ϕ*(

*x*,

*y*) contain obviously reduced noise levels as compared to the two-state reconstructions. The quantitative variance data in Fig. 11 confirms that the optimal estimates of

*A*(

*x*,

*y*) and

*ϕ*(

*x*,

*y*) contain lower variances than the two-state reconstructions.

## 7. Summary and conclusions

## Appendix: Generalization to ≥3 measurement states

*M*measurement states 1, 2, 3, ⋯,

*M*. Estimates of the desired object properties can be computed by use of any measurement-state pair; thus there will be (

*M*-state system. Consequently, the final estimate of

*ϕ*̃(

*u*,

*v*), for example, can be obtained from a weighted summation of all the available estimates. Let

*ϕ*̃

_{m,n}(

*u*,

*v*) denote the phase estimate computed by use of the intensity data acquired on measurement pair (

*m*,

*n*). A final unbiased estimate that possesses a potentially reduced variance can be written as

*ω*̂

^{ϕ}_{m,n}(

*u*,

*v*) are generally the complex-valued coefficients that satisfy

*ϕ*̃

_{m,n}(

*u*,

*v*) is a linearly combination of any other two estimates

*ϕ*̃

_{1,m}(

*u*,

*v*) and

*ϕ*̃

_{1,n}(

*u*,

*v*), where

*m*,

*n*are integer-valued indices. Therefore, Eq. (A1) can be simplified to

*ϕ*̃(

*u*,

*v*) can be obtained readily as

*l*=2,3, ⋯,

*M*in the first summation term, and

*m*=2,3, ⋯,

*M*-1,

*n*=3,4, ⋯,

*M*with

*n*>

*m*in the second summation term.

*R*

^{(op)}

_{1,m}and

*I*

^{(op)}

_{1,m}specify the value of the optimal combination coefficient for

*ω*

^{ϕ}_{1,m}with

*m*=2,3, ⋯,

*M*. Analytic formula for determination of the combination coefficients can be derived as follows. From knowledge of Eq. (A8) and taking use of Eq. (A9), a (2

*M*-4)×(2

*M*-4) system is formed, which can be expressed by a matrix equation.

*m*-

*th*equation in the system specifies the partial derivative of Var{

*ϕ*̃} with respect to the

*m*-

*th*component of Eq. (A12). Assuming

**H**is nonsingular, inversion of Eq. (A10) is accomplished readily:

**H**

*is the matrix obtained by replacing the*

_{m}*m*-

*th*column of

**H**with

**b**. If we specify the coefficient

*ω*

^{ϕ}_{1,m}(

*u*,

*v*) to be real-valued, the term

*I*

_{1,m}(

*u*,

*v*) will vanish, yielding a (

*M*-2)×(

*M*-2) system.

*ω*

^{heur}

_{1,m}for

*M*- detector system can be computed readily as:

## Acknowledgment

## References and links

1. | 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. |

2. | R. Waynant, “Toward practical coherent x-ray sources: Potential medical applications,” IEEE J. Quantum Electron. |

3. | H. Yamada, “Novel x-ray source based on a tabletop synchrotron and its unique features,” Nucl. Instrum. Methods Phys. Res. |

4. | R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. |

5. | W. Thomlinson, P. Suortti, and D. Chapman, “Recent advances in synchrotron radiation medical research,” Nucl. Instrum. Methods Phys. Res. |

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) |

7. | K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. |

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. |

9. | C. J. Kotre and I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. |

10. | X. Wu and H. Liu, “Clinical implementation of X-ray phase-contrast imaging: Theoretical foundations and design considerations,” Med. Phys. |

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. |

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. |

13. | F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. |

14. | B.L. Henke, E.M. Gullikson, and J.C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables |

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, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. |

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 Techniques,” Radiology |

17. | M. Z. Kiss, D. E. Sayers, Z. Zhong, C. Parham, and E. D. Pisano, “Improved image contrast of calcifications in breast tissue specimens using diffraction enhanced imaging,” Phys. Med. Biol. |

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 Histologic Correlation-Improved Conspicuity of Lesion Detail Compared with Digital Radiography,” Radiology |

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 digital full-field mammography using a practical molybdenum x-ray tube,” Invest. Radiol. |

20. | C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, “Multiple-image radiography for soft tissue of the foot and ankle,” J. Anat. |

21. | J. G. Brankov, M. N. Wernick, Y. Yang, J. Li, C. Muehleman, Z. Zhong, and M. A. Anastasio, “A computed tomography implementation of multiple-image radiography,” Med. Phys. |

22. | A. Bravin, “Exploiting the x-ray refraction contrast with an analyser: the state of the art,” J. Phys. |

23. | P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. |

24. | P. Cloetens, “Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation: 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. |

26. | M. Born and E. Wolf, |

27. | D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. |

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. |

29. | A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. |

30. | T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. |

31. | M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. |

32. | Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak objects,” J. Phys. |

33. | Y. I. Nesterets, T. E. Gureyev, K. M. Pavlov, D. M. Paganin, and S. W. Wilkins, “Combined analyser-based and propagation-based phase-contrast imaging of weak objects,” Opt. Commun. |

34. | T. E. Gureyev, A. Pogany, D.M. Paganin, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. |

35. | D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. |

36. | Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. |

37. | D. M. Paganin, |

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. |

39. | Y. I. Nesterets, T. E. Gureyev, and S.W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Phys. |

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 to X-ray phase-contrast imaging,” J. Phys. |

41. | J. W. Goodman, |

42. | J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik |

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. SPIE |

44. | S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: Statistical properties,” J. Opt. Soc. Am. |

45. | W. D. Stanley, G. R. Dougherty, and R. Dougherty, |

**OCIS Codes**

(000.0000) General : General

(100.5070) Image processing : Phase retrieval

(110.7440) Imaging systems : X-ray imaging

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: February 5, 2007

Revised Manuscript: May 7, 2007

Manuscript Accepted: May 20, 2007

Published: July 25, 2007

**Virtual Issues**

Vol. 2, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Cheng-Ying Chou, Yin Huang, Daxin Shi, and Mark A. Anastasio, "Image reconstruction in quantitative X-ray phase-contrast imaging employing multiple measurements," Opt. Express **15**, 10002-10025 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-10002

Sort: Year | Journal | Reset

### References

- A. Krol, A. Ikhlef, J.-C. Kieffer, D. Bassano, C. C. Chamberlain, Z. Jiang, H. Pepin, and S. C. Prasad, "Laserbased microfocused X-ray source for mammography: feasibility study," Med. Phys. 24, 725-732 (1997). [CrossRef] [PubMed]
- R. Waynant, "Toward practical coherent x-ray sources: Potential medical applications," IEEE J. Quantum Electron. 6, 1465-1469 (2000). [CrossRef]
- H. Yamada, "Novel x-ray source based on a tabletop synchrotron and its unique features," Nucl. Instrum.Methods Phys. Res. B 199, 509-516 (2003).
- R. Lewis, "Medical phase contrast x-ray imaging: current status and future prospects," Phys. Med. Biol. 49, 3573-3583 (2004). [CrossRef] [PubMed]
- W. Thomlinson, P. Suortti, and D. Chapman, "Recent advances in synchrotron radiation medical research," Nucl. Instrum. Methods Phys. Res. A 543, 288-296 (2005).
- 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).
- 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). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- C. J. Kotre and I. P. Birch, "Phase contrast enhancement of x-ray mammography: a design study," Phys. Med. Biol. 44, 2853-2866 (1999). [CrossRef] [PubMed]
- X. Wu and H. Liu, "Clinical implementation of X-ray phase-contrast imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169-2179 (2003). [CrossRef] [PubMed]
- 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). [CrossRef]
- E. F. Donnelly, R. R. Price, and D. R. Pickens, "Characterization of the phase-contrast radiography edgeenhancement effect in a cabinet x-ray system," Med. Phys. 30, 2292-2296 (2003). [CrossRef] [PubMed]
- F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature Phys. 2, 258-261 (2006). [CrossRef]
- B.L. Henke, E.M. Gullikson, and J.C. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92," At. Data Nucl. Data Tables 54, 181-342 (1993). [CrossRef]
- 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, and F. Zanconati, "Low-dose phase contrast x-ray medical imaging," Phys. Med. Biol. 43, 2845-2852 (1998). [CrossRef] [PubMed]
- 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 Techniques," Radiology 215, 286-293 (2000).
- M. Z. Kiss, D. E. Sayers, Z. Zhong, C. Parham, and E. D. Pisano, "Improved image contrast of calcifications in breast tissue specimens using diffraction enhanced imaging," Phys. Med. Biol. 49, 3427-3439 (2004). [CrossRef] [PubMed]
- 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 Histologic Correlation-Improved Conspicuity of Lesion Detail Compared with Digital Radiography," Radiology 214, 895- 901 (2000). [PubMed]
- 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 digital full-field mammography using a practical molybdenum x-ray tube," Invest. Radiol. 40, 385-396 (2005). [CrossRef] [PubMed]
- C. Muehleman, J. Li, Z. Zhong, J. G. Brankov, and M. N. Wernick, "Multiple-image radiography for soft tissue of the foot and ankle," J. Anat. 208, 115-124 (2006). [CrossRef] [PubMed]
- J. G. Brankov, M. N. Wernick, Y. Yang, J. Li, C. Muehleman, Z. Zhong, and M. A. Anastasio, "A computed tomography implementation of multiple-image radiography," Med. Phys. 33, 278-289 (2006). [CrossRef] [PubMed]
- A. Bravin, "Exploiting the x-ray refraction contrast with an analyser: the state of the art," J. Phys. D 36, A24-A29 (2003).
- P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, "In-line holography and phase-contrast microtomography with high energy x-rays," Phys. Med. Biol. 44, 741-749 (1999). [CrossRef] [PubMed]
- P. Cloetens, "Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation: Principles, Implementation and Applications," Ph.D. thesis, Vrije Universiteit Brussel (1999).
- 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). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
- D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, "Quantitative phase retrieval using coherent imaging systems with linear transfer functions," Opt. Commun. 234, 87-105 (2004). [CrossRef]
- 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). [CrossRef]
- A. V. Bronnikov, "Theory of quantitative phase-contrast computed tomography," J. Opt. Soc. Am. A 19, 472-480 (2002). [CrossRef]
- T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006). [CrossRef]
- M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, "Analytic image reconstruction in local phase-contrast tomography," Phys. Med. Biol. 49, 121-144 (2004). [CrossRef] [PubMed]
- Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W. Wilkins, "Quantitative diffraction-enhanced x-ray imaging of weak objects," J. Phys. D 37, 1262-1274 (2004).
- Y. I. Nesterets, T. E. Gureyev, K. M. Pavlov, D. M. Paganin, and S. W. Wilkins, "Combined analyser-based and propagation-based phase-contrast imaging of weak objects," Opt. Commun. 259, 19-31 (2006). [CrossRef]
- T. E. Gureyev, A. Pogany, D.M. Paganin, and S.W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004). [CrossRef]
- D. Paganin, A. Barty, P. J. Mcmahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2003). [CrossRef]
- Y. Huang and M. A. Anastasio, "Statistically principled use of in-line measurements in intensity diffraction tomography," J. Opt. Soc. Am. A 24, 626-642 (2007). [CrossRef]
- D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006). [CrossRef]
- 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). [CrossRef]
- Y. I. Nesterets, T. E. Gureyev, and S.W. Wilkins, "Polychromaticity in the combined propagation-based/analyserbased phase-contrast imaging," J. Phys. D 38, 4259-4271 (2005).
- 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 to X-ray phase-contrast imaging," J. Phys. D 37, 2746- 2750 (2004).
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2004).
- J.-P. Guigay, "Fourier transform analysis of Fresnel diffraction patterns and in-line holograms," Optik 49, 121- 125 (1977).
- 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. SPIE 6318, 63180V (2006).
- S. Lowenthal and H. Arsenault, "Image formation for coherent diffuse objects: Statistical properties," J. Opt. Soc. Am. 60, 1478-1483 (1970). [CrossRef]
- W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing (Reston Publishing Company, Inc., 1984).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1. |
Fig. 2. |
Fig. 3. |

Fig. 4. |
Fig. 5. |
Fig. 6. |

Fig. 7. |
Fig. 8. |
Fig. 9. |

Fig. 10. |
Fig. 11. |
Fig. 12. |

OSA is a member of CrossRef.