## Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging

Optics Express, Vol. 17, Issue 17, pp. 14466-14480 (2009)

http://dx.doi.org/10.1364/OE.17.014466

Acrobat PDF (419 KB)

### Abstract

Quantitative in-line X-ray phase-contrast imaging methods seek to reconstruct separate images that depict an object’s projected absorption and refractive properties. An understanding of the statistical properties of the reconstructed images can facilitate the identification of optimal imaging parameters for specific diagnostic tasks. However, the statistical properties of quantitative X-ray phase-contrast imaging remain largely unexplored. In this work, we derive analytic expressions that describe the second-order statistics of the reconstructed absorption and phase images. Concepts from statistical decision theory are applied to demonstrate how the statistical properties of images corresponding to distinct imaging geometries can influence signal detectability.

© 2009 Optical Society of America

## 1. Introduction

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

7. R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. **49(16)**, 3573–3583 (2004). URL http://stacks.iop.org/0031-9155/49/3573. [CrossRef]

8. 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(10)**, 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845. [CrossRef]

17. 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]

22. T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express **16(5)**, 3223–3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223. [CrossRef]

25. H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. **90**, 9758–9765 (1993). [CrossRef] [PubMed]

24. H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A **7**, 1266–1278 (1990). [CrossRef] [PubMed]

26. K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, and G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A **2**, 1752–1759 (1985). [CrossRef] [PubMed]

27. C. K. Abbey and M. P. Eckstein, “Classification images for simple detection and discrimination tasks in correlated noise,” J. Opt. Soc. Am. A **24**, B110–B124 (2007). [CrossRef]

## 2. Background: in-line X-ray phase-contrast imaging

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

### 2.1. Measurement model

*n*(

*r*⃗)≡1-

*δ*(

*r*⃗)+

*jβ*(

*r*⃗),

*j*≡√-1, is irradiated by monochromatic and time-harmonic X-ray plane wave

*U*with wavelength

_{i}*λ*. The irradiating wave propagates in the direction of the positive

*z*-axis. Immediately behind the object on the object plane

*z*=0, the transmitted wavefield

*U*(

_{t}*x,y*) can be described as

*L*denotes the paths of X-ray beams, which are approximated to be parallel with the

*z*-axis.

*U*(

_{m}*x,y*) denote the transmitted wavefield on a detector plane

*z*=

*z*, which is down-stream from the object plane. Note that the integer-valued subscript

_{m}*m*is employed to index the object-to-detector distance

*z*. The wavefield intensity data recorded on this detector plane are given by

_{m}*h*(

_{m}*x,y*) denotes the associated Fresnel propagator kernel and *

_{2}indicates a two-dimensional (2D) convolution operation. From knowledge of the measured intensity

*I*(

_{m}*x,y*), we define the data function

*I*=|

_{i}*U*|

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

*ϕ*(

*x,y*) and

*A*(

*x,y*) from knowledge of two or more intensity measurements

*I*(

_{m}*x,y*) acquired at distinct object-to-detector distances. More generally, the intensity measurements could be acquired on a fixed detector plane and an alternative degree of freedom in the imaging system could be varied [28

28. T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays,” Phys. Rev. Lett. **86**, 5827–5830 (2001). [CrossRef] [PubMed]

### 2.2. Image reconstruction based on the contrast transfer function (CTF)

*ϕ*(

*x,y*) and

*A*(

*x,y*). We assume that the two intensity measurements

*I*(

_{m}*x,y*) and

*I*(

_{n}*x,y*) are acquired on the detector planes

*z*=

*z*and

_{m}*z*=

*z*, respectively, and their 2D FTs will be denoted as

_{n}*Ĩ*(

_{m}*u,v*) and

*Ĩ*(

_{n}*u,v*). Similarly, the 2D FT of the data function

*K*(

_{m}*x,y*) will be denoted as

*K*̃

_{m}(

*u,v*).

12. M. Langer, P. Cloetens, J. P Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. **35(10)**, 4556–66 (2008). [CrossRef]

*ϕ*(

*x,y*) and

*A*(

*x,y*) can be determined as

*f*≡

_{2}*u*+

_{2}*v*and the subscripts (

_{2}*m,n*) have been added to

*ϕ*̃(

*u,v*) and

*A*̃(

*u,v*) to denote that they have been determined by the intensity measurements

*I*(

_{m}*x,y*) and

*I*(

_{n}*x,y*). Subsequently, estimates of

*ϕ*(

_{m,n}*x,y*) and Am,n(

*x,y*) can be obtained via the inverse 2D FT. In many applications, including medical imaging [2

2. 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]

*πλ f*

^{2}(

*z*)]≈

_{n}-z_{m}*πλ f*

^{2}(

*z*) and cos[

_{n}-z_{m}*πλ f*

^{2}

*z*]≈cos[

_{m}*πλ f*

^{2}

*z*]≈1. For simplicity we will assume such conditions, under which Eqs. (8) and (9) can be simplified as

_{n}## 3. Second-order statistics of image estimates: Continuous case

*I*(

_{m}*x,y*), and hence the data function

*K*(

_{m}*x,y*) and reconstructed images, will be interpreted as stochastic processes, reflecting that in practice they are inherently random quantities. We will let Cov{

*A,B*} and Var{

*A*} denote the covariance and variance of the stochastic processes

*A*and

*B*[23, 31].

### 3.1. Autocovariance of the Fourier estimates

**Phase image:**By use of the definition of the autocovariance [31] and the phase retrieval formula in Eq. (10), it can be verified that

*f*′

^{2}≡

*u*′

^{2}+

*v*′

^{2}and

*δ*(·) and

*δ*′ denote the 1D Dirac and Kronecker delta functions, and therefore Eq. (12) can be expressed as

_{m,m}**Absorption image:**In an analogous way, by use of Eq. (11) the Fourier autocovariance of the absorption image can be determined as

### 3.2. Autocovariance of the image estimates

## 4. Second-order statistics of image estimates: Discrete case

### 4.1. Finite sampling considerations and discrete noise model

*z*=

*z*as

_{m}*r*and

*s*are integer-valued detector-element indices and Δ

*x*=Δ

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

*L*×

*L*. Equation (20) 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. Here and elsewhere, a square bracket, ‘[·]’, is employed to describe discretely sampled quantities, in order to distinguish them from functions of continuous variables.

17. 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]

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

*I*[

_{m}*r, s*] and

*I*

^{0}

*[*

_{m}*r, s*] denote the noise-contaminated and noiseless measurement data, respectively. The noise term

*n*[

_{m}*r, s*] is assumed to have a zero mean and could be signal-dependent. We also assume the noise satisfies

*σ*

^{2}[

*r, s; z*]≡Var{

_{m}*n*[

_{m}*r, s*]}. This implies that the autocovariance of the sampled data function

*K*[

_{m}*r, s*]=

*I*[

_{m}*r, s*]/

*I*-1 satisfies

_{i}*I*of the incident beam as a deterministic quantity.

_{i}### 4.2. Second-order statistics

*K*̃

*(*

_{m}*u,v*) is required. To compute this in the case of of discretely sampled measurement data, the continuous 2D FT will be approximated by use of the discrete Fourier transform (DFT) [33].

*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. Analogously, the 2D DFT of the sampled data function

*K*[

_{m}*r, s*] will be denoted as

*K*̃

*[*

_{m}*p,q*].

*u*and

*v*axes. In order for this approximation to be useful, we consider the effects of aliasing to be negligible. A systematic investigation of the role of pre-sampling blur on image statistics and signal detectability in phase-contrast image formation remains an important topic for future studies [34

34. A. R. Pineda and H. H. Barrett, “What does DQE say about lesion detectability in digital radiography?” in Proc. of SPIE , vol. **4320**, pp. 561–569 (2001). [CrossRef]

## 5. Task-based image quality: A signal detection study

35. R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. **30(6)**, 489–518 (1985). URL http://stacks.iop.org/0031-9155/30/489. [CrossRef]

25. H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. **90**, 9758–9765 (1993). [CrossRef] [PubMed]

36. W. E. Smith and H. H. Barrett, “Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A **3(5)**, 717–725 (1986). URL http://josaa.osa.org/abstract.cfm?URI=josaa-3-5-717. [CrossRef]

*ϕ*[

_{m,n}*r, s*] or

*A*[

_{m,n}*r, s*], where

*N*is the dimension of the digital image. The matrix

**K**will denote the image autocovariance E{(

_{g}**g**-

**g**̄)(

**g**-

**g**̄)†}, where † is the conjugate transpose (adjoint) operation, E{·} denotes the statistical expectation operator associated with the ensemble of noisy images, and

**g**̄≡E{

**g**} is the mean image. Note that the elements of

**K**are specified by Eqs. (31) or (33).

_{g}### 5.1. Signal detection task and the Hotelling discriminant

**g**

_{1}and

**g**

_{2}, and Δ

**g**̄≡

**g**̄

**-**

_{1}**g**̄

**. The Hotelling observer computes the linear discriminant**

_{2}**w**† is the adjoint of the Hotelling template

**K**

_{g}^{-1}in Eq. (35) can present difficulties due to the large dimensions of

**K**. For example, the 128×128 element detector considered in our simulation studies corresponds to a matrix

_{g}**K**of dimension 16384×16384. To circumvent difficulties with inverting this matrix, a subregion Hotelling observer [37

_{g}37. M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express **11(5)**, 460–475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460. [CrossRef]

38. A. H. Baydush, D. M. Catarious, C. K. Abbey, and C. E. Floyd, “Computer aided detection of masses in mammography using subregion Hotelling observers,” Med. Phys. **30(7)**, 1781–1787 (2003). URL http://link.aip.org/link/?MPH/30/1781/1. [CrossRef]

37. M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express **11(5)**, 460–475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460. [CrossRef]

38. A. H. Baydush, D. M. Catarious, C. K. Abbey, and C. E. Floyd, “Computer aided detection of masses in mammography using subregion Hotelling observers,” Med. Phys. **30(7)**, 1781–1787 (2003). URL http://link.aip.org/link/?MPH/30/1781/1. [CrossRef]

**K**that was easily inverted and used as a surrogate for

_{g}**K**

_{g}^{-1}in Eq. (35).

*T*(

**g**) to a decision threshold value. When applied to a collection of images, the true positive fraction (TPF) and false positive fraction (FPF) can be estimated (e.g., see Chapter 13 in [23]). In a binary detection task, the TPF and FPF correspond to the probability of detecting the sought-after signal correctly and the false alarm rate, respectively. By varying the decision threshold, a receiver operating characteristic (ROC) curve is obtained [23]. The ROC curve depicts the relationship between the TPF and FPF and completely summarizes the detection performance of the observer.

## 6. Numerical studies

### 6.1. Computer-simulation studies

*z*-axis and irradiate an object. A mathematical phantom consisting of uniform ellipsoids that were assigned different values of

*n*(

*r*⃗) was employed to represent the object. In the detectability study, the object contained only two ellipsoids; a large uniform ellipsoid that represented the object background and a smaller one that represented the signal of interest. The real and imaginary components of the refractive indices were assigned as

*δ*=8.6421×10

^{-7}and

*β*=6.7344×10

^{-10}for the background, and

*δ*=9.6385×10

^{-7}and

*β*=1.4236×10

^{-9}for the signal. These correspond to values of tumor and fat tissues in human breast. The longest semi-axes for the larger and smaller ellipsoids were 320

*µ*m and 99.2

*µ*m, respectively. Five measurement geometries were considered. In all cases, the first detectorplane was located at

*z*

_{1}=9 mm. The second detector-plane (see Fig. 1) was located at

*z*

_{2}=37 mm,

*z*

_{3}=78 mm,

*z*

_{4}=115 mm,

*z*

_{5}=176 mm, and

*z*

_{6}=250 mm. The detector that contained 128×128 elements of dimension of 5

*µ*m.

*ϕ*̃

*[*

_{m,n}*p,q*] and

*A*̃

*[*

_{m,n}*p,q*] were computed by use of discretized versions of Eqs. (8) and (9) with the 2D FT replaced by the 2D DFT. From these Fourier data, noisy estimates of

*ϕ*[

*r, s*] and

*A*[

*r, s*] were computed by use of the 2D inverse IFFT algorithm. Subsequently, empirical estimates [39

39. M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. **45**, 2216–2223 (1998). [CrossRef]

*ϕ*

*[*

_{m,n}*r,s*],

*ϕ*[

_{m,n}*r*′,

*s*′]} and Cov{

*A*[

_{m,n}*r, s*],

*A*[

_{m,n}*r*′,

*s*′]} were computed and compared against the theoretical predictions given by Eq. (31) or (33).

*ϕ*[

_{m,n}*r, s*] and

*A*[

_{m,n}*r, s*] were computed as described above. Half of the images corresponded to signal present and half to signal absent. The subregion Hotelling observer described in Section 5.1 was separately applied to the collections of

*ϕ*[

_{m,n}*r, s*] and

*A*[

_{m,n}*r, s*] images and the TPF and FPF values were computed at 200 different decision threshold values. From the (TPF,FPF) pairs ROC curves were plotted for each imaging geometry.

### 6.2. Numerical results

*z*=28, 69, 106, 167, and 241 mm are denoted by solid, dashed, dashed-dotted, dotted and thin solid curves, respectively. As predicted by the factor in Eqs. (12) and (16), the magnitude of the autocovariance image scales inversely with detector spacing. This explains why the conspicuity of the image noise in Figs. 2(a)–2(d) decreases as the detector spacing is increased.

**Signal detection study:**The ROC curves corresponding to the phase and absorption images for different imaging geometries are shown in Fig. 6. Figure 6(a) indicates an enhancement in signal detectability in the phase images with increasing values of detector spacing Δ

*z*. This is consistent with the noisy appearance of the phase images displayed in Figs. 2(a)–2(d), and the fact that the strength of the noise correlations diminishes with increasing Δ

*z*. Note that the ROC curves corresponding to Δ

*z*≥106mmnearly overlap with the axes of the figure, which indicates nearly perfect detection performance (due to the relatively low noise level in this example). The case of a higher noise level is considered in Fig. 7, which more clearly demonstrates monotonically increasing detection performance with detector spacing. For the absorption images, the ROC curves in Fig. 6(b) are not strongly affected by the imaging geometry due to the absence of poles in the reconstruction formula and signal-independent noise model. The relatively poor ROC curves are explained by the fact that the absorption contrast of the object is very small. However, a slight enhancement in the object detectability is observed due to the reduction of the image variance with the increased values of the detector spacings.

## 7. Summary and conclusions

24. H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A **7**, 1266–1278 (1990). [CrossRef] [PubMed]

## Acknowledgment

## References and links

1. | D. M. Paganin, |

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

3. | 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) |

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

5. | A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. |

6. | 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). |

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

8. | 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. |

9. | 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. |

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

11. | C.-Y. Chou, Y. Huang, D. Shi, and M. A. Anastasio, “Image reconstruction in quantitative X-ray phase-contrast imaging employing multiple measurements,” Opt. Express |

12. | M. Langer, P. Cloetens, J. P Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. |

13. | T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A |

14. | A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. |

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

16. | 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. |

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

18. | 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. |

19. | B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. |

20. | X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Meth. B |

21. | Y. Nesterets, S. Wilkins, T. Gureyev, A. Pogany, and A. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum.76(9) (2005). |

22. | T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express |

23. | H. Barrett and K. Myers, |

24. | H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A |

25. | H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. |

26. | K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, and G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A |

27. | C. K. Abbey and M. P. Eckstein, “Classification images for simple detection and discrimination tasks in correlated noise,” J. Opt. Soc. Am. A |

28. | T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays,” Phys. Rev. Lett. |

29. | D. Shi and M. A. Anastasio, “Intensity diffraction tomography with fixed detector plane,” Opt. Eng. |

30. | T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. Mayo, and S. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. |

31. | A. Papoulis and S. U. Pillai, |

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

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

34. | A. R. Pineda and H. H. Barrett, “What does DQE say about lesion detectability in digital radiography?” in Proc. of SPIE , vol. |

35. | R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. |

36. | W. E. Smith and H. H. Barrett, “Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A |

37. | M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express |

38. | A. H. Baydush, D. M. Catarious, C. K. Abbey, and C. E. Floyd, “Computer aided detection of masses in mammography using subregion Hotelling observers,” Med. Phys. |

39. | M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(110.4280) Imaging systems : Noise in imaging systems

(110.7440) Imaging systems : X-ray imaging

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: May 21, 2009

Revised Manuscript: July 23, 2009

Manuscript Accepted: July 24, 2009

Published: August 3, 2009

**Virtual Issues**

Vol. 4, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Cheng-Ying Chou and Mark A. Anastasio, "Influence of imaging geometry on noise texture in quantitative in-line X-ray
phase-contrast imaging," Opt. Express **17**, 14466-14480 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14466

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### References

- D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006). [CrossRef]
- 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]
- 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]
- 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]
- 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).
- R. A. Lewis, "Medical phase contrast x-ray imaging: current status and future prospects," Phys. Med. Biol. 49(16), 3573-3583 (2004). URL http://stacks.iop.org/0031-9155/49/3573. [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(10), 2845-2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845. [CrossRef]
- 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]
- 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]
- C.-Y. Chou, Y. Huang, D. Shi, and M. A. Anastasio, "Image reconstruction in quantitative X-ray phasecontrast imaging employing multiple measurements," Opt. Express 15(16), 10,002-10,025 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-16-10002.
- M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, "Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography," Med. Phys. 35(10), 4556-66 (2008). [CrossRef]
- T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13(8), 1670-1682 (1996). URL http://josaa.osa.org/abstract.cfm?URI=josaa-13-8-1670. [CrossRef]
- A. Barty, K. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175(4), 329-336 (2000). [CrossRef]
- J.-P. Guigay, "Fourier transform analysis of Fresnel diffraction patterns and in-line holograms," Optik 49, 121-125 (1977).
- 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, 29,132 (1999). [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]
- 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]
- B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, "Phase contrast radiography: Image modelling and optimization," Rev. Sci. Instrum. 75, 5271-5276 (2004). [CrossRef]
- X. Wu, H. Liu, and A. Yan, "Optimization of X-ray phase-contrast imaging based on in-line holography," Nucl. Instrum. Meth. B 234, 563-572 (2005). [CrossRef]
- Y. Nesterets, S. Wilkins, T. Gureyev, A. Pogany, and A. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76(9) (2005).
- T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, "Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging," Opt. Express 16(5), 3223-3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223. [CrossRef]
- H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).
- H. H. Barrett, "Objective assessment of image quality: effects of quantum noise and object variability," J. Opt. Soc. Am. A 7, 1266-1278 (1990). [CrossRef] [PubMed]
- H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, "Model observers for assessment of image quality," Proc. Natl. Acad. Sci. 90, 9758-9765 (1993). [CrossRef] [PubMed]
- K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, and G. W. Seeley, "Effect of noise correlation on detectability of disk signals in medical imaging," J. Opt. Soc. Am. A 2, 1752-1759 (1985). [CrossRef] [PubMed]
- C. K. Abbey and M. P. Eckstein, "Classification images for simple detection and discrimination tasks in correlated noise," J. Opt. Soc. Am. A 24, B110-B124 (2007). [CrossRef]
- T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays," Phys. Rev. Lett. 86, 5827-5830 (2001). [CrossRef] [PubMed]
- D. Shi and M. A. Anastasio, "Intensity diffraction tomography with fixed detector plane," Opt. Eng. 46, 107003 (2007). [CrossRef]
- T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. Mayo, and S. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93(6), 068103 (2004).
- A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw Hill, 2002).
- 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).
- A. R. Pineda and H. H. Barrett, "What does DQE say about lesion detectability in digital radiography?" in Proc. of SPIE, vol. 4320, pp. 561-569 (2001). [CrossRef]
- R. F. Wagner and D. G. Brown, "Unified SNR analysis of medical imaging systems," Phys. Med. Biol. 30(6), 489-518 (1985). URL http://stacks.iop.org/0031-9155/30/489. [CrossRef]
- W. E. Smith and H. H. Barrett, "Hotelling trace criterion as a figure of merit for the optimization of imaging systems," J. Opt. Soc. Am. A 3(5), 717-725 (1986). URL http://josaa.osa.org/abstract.cfm?URI=josaa-3-5-717. [CrossRef]
- M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, "Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks," Opt. Express 11(5), 460-475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460. [CrossRef]
- A. H. Baydush, D. M. Catarious, C. K. Abbey, and C. E. Floyd, "Computer aided detection of masses in mammography using subregion Hotelling observers," Med. Phys. 30(7), 1781-1787 (2003). URL http://link.aip.org/link/?MPH/30/1781/1. [CrossRef]
- M. A. Anastasio, M. Kupinski, and X. Pan, "Noise properties of reconstructed images in ultrasound diffraction tomography," IEEE Trans. Nucl. Sci. 45, 2216-2223 (1998). [CrossRef]

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