## Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography |

Optics Express, Vol. 20, Issue 10, pp. 10724-10749 (2012)

http://dx.doi.org/10.1364/OE.20.010724

Acrobat PDF (2572 KB)

### Abstract

Differential X-ray phase-contrast tomography (DPCT) refers to a class of promising methods for reconstructing the X-ray refractive index distribution of materials that present weak X-ray absorption contrast. The tomographic projection data in DPCT, from which an estimate of the refractive index distribution is reconstructed, correspond to one-dimensional (1D) derivatives of the two-dimensional (2D) Radon transform of the refractive index distribution. There is an important need for the development of iterative image reconstruction methods for DPCT that can yield useful images from few-view projection data, thereby mitigating the long data-acquisition times and large radiation doses associated with use of analytic reconstruction methods. In this work, we analyze the numerical and statistical properties of two classes of discrete imaging models that form the basis for iterative image reconstruction in DPCT. We also investigate the use of one of the models with a modern image reconstruction algorithm for performing few-view image reconstruction of a tissue specimen.

© 2012 OSA

## 1. Introduction

1. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **13**, 6296–6304 (2005). [CrossRef] [PubMed]

5. J. Brankov, M. 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]

7. M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. **90**, 224101 (2007). [CrossRef]

8. 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]

9. D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. **42**, 2015–2025 (1997). [CrossRef] [PubMed]

10. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. **42**, 866–868 (2003). [CrossRef]

11. G. Faris and R. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. **27**, 5202–5212 (1988). [CrossRef] [PubMed]

12. J. Stricker, “Analysis of 3-d phase objects by moiré deflectometry,” Appl. Opt. **23**, 3657–3659 (1984). [CrossRef] [PubMed]

11. G. Faris and R. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. **27**, 5202–5212 (1988). [CrossRef] [PubMed]

13. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA **107**, 13576–13581 (2010). [CrossRef] [PubMed]

14. Z. Huang, K. Kang, L. Zhang, Z. Chen, F. Ding, Z. Wang, and Q. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A **79**, 013815 (2009). [CrossRef]

11. G. Faris and R. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. **27**, 5202–5212 (1988). [CrossRef] [PubMed]

15. T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. **38**, 4542–4545 (2011). [CrossRef] [PubMed]

*et al.*[15

15. T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. **38**, 4542–4545 (2011). [CrossRef] [PubMed]

## 2. Background: Imaging model for differential X-ray phase-contrast tomography

### 2.1. Data function and imaging model in continuous form

*δ*(

*x,y,z*) ≡ 1 −

*n*(

*x,y,z*) denote the compactly supported and bounded object function we seek to reconstruct, where

*n*(

*x,y,z*) is the real-valued refractive index distribution. We will employ the notation

*δ*(

**r**

_{2};

*z*) ≡

*δ*(

*x,y,z*), where

**r**

_{2}= (

*x, y*), as a convenient description of a transverse slice of the 3D object function.

1. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **13**, 6296–6304 (2005). [CrossRef] [PubMed]

8. 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]

10. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. **42**, 866–868 (2003). [CrossRef]

16. A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Meth. A **352**, 622–628 (1995). [CrossRef]

17. A. Momose, “Phase-sensitive imaging and phase tomography using x-ray interferometers,” Opt. Express **11**, 2303–2314 (2003). [CrossRef]

9. D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. **42**, 2015–2025 (1997). [CrossRef] [PubMed]

18. T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature **373**, 595–598 (1995). [CrossRef]

24. I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, “Biological imaging by x-ray phase tomography using diffraction-enhanced imaging,” Jpn. J. Appl. Phys. **44**, 8219–8221 (2005). [CrossRef]

*θ*is measured in the (

*x*

_{r}*, z*) plane located at

*y*

*=*

_{r}*d*and will be denoted by

*I*(

*x*

_{r}*, z,*

*θ*;

*K*). Here

*K*represents an integer-valued index that specifies the state of the imaging system. For example, in crystal analyzer-based systems, distinct values of

*K*would correspond to different orientations of the analyzer crystal. Alternatively, in grating interferometry when a phase-stepping procedure [1

1. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **13**, 6296–6304 (2005). [CrossRef] [PubMed]

8. 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]

*K*correspond to different translational positions of the grating that is being scanned.

*δ*(

**r**

_{2};

*z*) from knowledge of

*g*(

*x*

_{r}*,*

*θ*;

*z*). When

*g*(

*x*

_{r}*,*

*θ*;

*z*) is measured at a large number of view angles

*θ*, this can be accomplished by use of analytic image reconstruction algorithms [11

**27**, 5202–5212 (1988). [CrossRef] [PubMed]

28. M. A. Anastasio and X. Pan, “Region-of-interest imaging in differential phase-contrast tomography,” Opt. Lett. **32**, 3167–3169 (2007). [CrossRef] [PubMed]

### 2.2. General forms of discrete imaging models

*s*and

*h*are integer-valued detector element indices and

*t*is the tomographic view index. Here,

*L*×

*L*, and

*Q*denotes the number of samples measured in each dimension. The quantity Δ

*denotes the angular sampling interval between the uniformly distributed view angles. The reconstruction algorithms described below can be applied in the case of non-uniformly sampled measurement data as well. The general forms of the reconstruction algorithms would remain unchanged for the case of non-uniformly sampled measurement data; However, the explicit forms of the system matrices would be changed. Although not indicated in Eq. (2), the measured discrete data will also be degraded by the averaging effects of the sampling aperture. Additionally, the effects of finite temporal and spatial beam coherence will effectively blur the data function*

_{θ}*g*[

*s,t;h*]. These effects can limit the attainable spatial resolution in the reconstructed DPCT images. Because the reconstruction problem is inherently 2D, we will consider the problem of reconstructing a transverse slice of the object function located at

*z*=

*h*Δ

*. Let the vector*

_{d}**g**∈

*denote a lexicographically ordered representation of*

^{M}*g*[

*s,h,t*]. The dimension

*M*is defined by the product of the number of detector row elements and the number of view angles.

*N*-dimensional approximation of

*δ*(

**r**

_{2};

*z*=

*h*Δ

*) can be formed as where the subscript*

_{d}*a*indicates that

*δ*

*(*

_{a}**r**;

_{2}**z**) is an approximation of

*δ*(

**r**;

_{2}**z**), {

*ϕ*

*(*

_{n}**r**

_{2})} are a set of expansion functions, and {

*h*. Let the 2D function

*δ*

*(*

_{a}**r**

_{2};

*z*=

*h*Δ

*) be contained within a disk of radius*

_{d}*r*

_{0}. The discrete data function satisfies assuming that

**R**

*δ*

*(*

_{a}**r**

_{2};

*z*=

*h*Δ

*) is differentiable ∀*

_{d}*x*

*∈ (−*

_{r}*r*

_{0},

*r*

_{0}). For certain choices of the expansion functions, such as the pixels described below, this differentiability requirement will not be met. Moreover, when computing Eq. (4), as required by iterative image reconstruction algorithms, the operator

**S**is a smoothing operator that acts with respect to the

*x*

*coordinate and ensures that*

_{r}**SR**

*δ*

*(*

_{a}**r**

_{2};

*z*=

*h*Δ

*) is differentiable. The composite operator*

_{d}**R**

*ϕ*

*(*

_{n}**r**

_{2}) is differentiable ∀

*x*

*∈ (−*

_{r}*r*

_{0},

*r*

_{0}), as satisfied by the Kaiser-Bessel expansion functions investigated below, Eq. (4) can be expressed as In matrix form, each of Eqs. (4)–(6) can be expressed as where

**g**is a lexicographically ordered representation of the sampled data function,

**H**is an

*M*×

*N*system matrix, and

**b**is a

*N*× 1 vector of expansion coefficients that has an

*n*-th element given by

**b**from knowledge of

**g**and

**H**. From the estimated

**b**, the object function estimate - the sought after image - can be obtained by use of Eq. (3). In the special case in which the expansion functions are classical pixels, the estimates of

**b**and

*δ*

*(*

_{a}**r**

_{2};

*z*=

*h*Δ

*) coincide. Explicit forms for the system matrix*

_{d}**H**are found by specifying the expansion functions

*ϕ*

*(*

_{n}**r**

_{2}) and implementation of the operator

*z*=

*h*Δ

*and the discrete index*

_{d}*h*will be suppressed hereafter. For use with the pixel basis functions, three different discrete implementations of the operator

## 3. Construction of system matrices for iterative image reconstruction in DPCT

### 3.1. System matrix construction employing pixel basis functions

*x*) = 1 for

*x*

_{n}*, y*

*) specifies the coordinate of the*

_{n}*n*th lattice point on a uniform Cartesian lattice, and

*ɛ*is the spacing between those lattice points. A description of the system matrix construction for use with pixel expansion functions provided below. According to Eq. (5), this will require specifying methods for : (1) numerically approximating

**R**

*δ*

*(*

_{a}**r**;

_{2}**z**) and (2) computing a regularized discrete derivative operator

**R**

*δ*

*(*

_{a}**r**;

_{2}*z*) [29–31

31. S. Lo, “Strip and line path integrals with a square pixel matrix: A unified theory for computational CT projections,” IEEE Trans. Med. Imag. **7**, 355–363 (1988). [CrossRef]

*w*

*is the weighting factor that corresponds to the contribution of the*

_{stj}*j*-

*th*expansion function to the projection data recorded at detector location [

*s,t*], and

*b*

*is the*

_{j}*j*-

*th*component of

**b**. By defining

**p**∈

*to be a lexicographically ordered representation of*

^{M}*p*[

*s,t*], Eq. (8) can be expressed in matrix form as where in which S is the total number of discrete projection data for each view and the notation [

**H**

*]*

^{R}*denotes the element of*

_{m,n}**H**

*corresponding to the*

^{R}*m*-th row and

*n*-th column. In our numerical studies, we adopted a ’ray-driven’ method to establish

**H**

*[30*

^{R}30. R. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. **12**, 252–255 (1985). [CrossRef] [PubMed]

32. J. Monaghan, “Smoothed particle hydrodynamics,” Rep. Prog. Phys. **68**, 1703–1760 (2005). [CrossRef]

33. A. Chaniotis and D. Poulikakos, “High order interpolation and differentiation using b-splines,” J. Comput. Phys. **197**, 253–274 (2004). [CrossRef]

**p**′ ∈

**denote a 1D discrete derivative of**

^{M}**p**that approximates samples of

**p**′ as where

*p*′

*is the*

_{k}*k*-th element of

**p**′, K is the total number of neighbouring particles,

*p*

*and*

_{i}*p*

*are the*

_{k}*i*-th and

*k*-th elements of

**p**respectively, and

**W**(

*x*

*,*

_{r}*h*) is a kernel function with a smoothing length h. In our studies we employed three different kernel functions of the form: linear, quadratic spline, and cubic spline [32

32. J. Monaghan, “Smoothed particle hydrodynamics,” Rep. Prog. Phys. **68**, 1703–1760 (2005). [CrossRef]

33. A. Chaniotis and D. Poulikakos, “High order interpolation and differentiation using b-splines,” J. Comput. Phys. **197**, 253–274 (2004). [CrossRef]

*ρ*

*is defined as In matrix form, Eq. (11) is expressed as where explicit forms of*

_{k}**H**

*are provided in the appendix that correspond to different choices of*

^{D}**W**(

*x*

*,*

_{r}*h*).

### 3.2. System matrix construction employing generalized Kaiser-Bessel window functions

34. R. Lewitt, “Multidimensional digital image representations using generalized Kaiser-Bessel window functions,” J. Opt. Soc. Am. A **7**, 1834–1846 (1990). [CrossRef] [PubMed]

35. R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol. **37**, 705–716 (1992). [CrossRef] [PubMed]

*I*

*(·) is the*

_{m}*m*-

*th*order modified Bessel function,

**r**

*≡ |*

_{b}**r**

_{2}–

**r**

*| with*

_{n}**r**

*= (*

_{n}*x*

*,*

_{n}*y*

*) denoting the blob center, and*

_{n}*a*and

*α*determine the blob’s radius and specific shape.

*ξ*≡

*x*

*–*

_{r}*x*

*cos*

_{n}*θ*–

*y*

*sin*

_{n}*θ*. As demonstrated by Lewitt [34

34. R. Lewitt, “Multidimensional digital image representations using generalized Kaiser-Bessel window functions,” J. Opt. Soc. Am. A **7**, 1834–1846 (1990). [CrossRef] [PubMed]

*ξ*| ≤

*a*and zero otherwise. As derived in Appendix B, the 1D derivative of this quantity is given by

**H**is sparse because only a relatively few blobs contribute to each component of

^{blob}**g**.

*k*-th order spatial derivative of

*m*>

*k*[34

34. R. Lewitt, “Multidimensional digital image representations using generalized Kaiser-Bessel window functions,” J. Opt. Soc. Am. A **7**, 1834–1846 (1990). [CrossRef] [PubMed]

*m*= 2 was chosen. This ensured that the first-order derivatives of the blobs were continuous.

## 4. Comparison of numerical and statistical properties of system matrices

### 4.1. SVD analysis of the system matrices

36. M. Bertero, *Introduction to Inverse Problems in Imaging* (Taylor & Francis, 1998). [CrossRef]

**H**

*were constructed as described in Sec. 3.1 for the cases where linear, quadratic spline, and cubic spline kernel functions*

^{pixel}**W**(

*x*

*,*

_{r}*h*) were employed [32

32. J. Monaghan, “Smoothed particle hydrodynamics,” Rep. Prog. Phys. **68**, 1703–1760 (2005). [CrossRef]

33. A. Chaniotis and D. Poulikakos, “High order interpolation and differentiation using b-splines,” J. Comput. Phys. **197**, 253–274 (2004). [CrossRef]

*μ*m. The window size of h was chosen to be two times detector pixel pitch, three times detector pixel pitch, and four times detector pixel pitch for linear interpolation, quadratic spline and cubic spline kernel, respectively. The object was assumed to be contained within an area of dimension 6.4 mm × 6.4 mm. For the pixel-based studies, a 128 × 128 array of 50

*μ*m square pixels was employed to discretize the object. Accordingly, the system matrices

**H**

*were of dimension 46080 (256 × 180) by 16384 (128 × 128). For the case of blob expansion functions, the same scanning configuration was considered. Six system matrices*

^{pixel}**H**

*were constructed as described in Sec. 3.2 for the cases where the blob parameters were chosen as*

^{blob}*m*= 2, radius

*a*= 75

*μ*m (1.5 times sampling interval) or 100

*μ*m (2 times sampling interval), and

*α*= 2, 6, or 10.4. Hereafter, we will refer to the blob radius relative to the image grid spacing. For example, we use indicate a = 1.5 to represent a physical radius of 75

*μ*m and a = 2 to represent a physical radius of 100

*μ*m. The value of

*α*= 10.4 was chosen because it results in a quasi-bandlimited blob function that has been demonstrated to suppress artifacts in other tomographic image reconstruction applications [35

35. R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol. **37**, 705–716 (1992). [CrossRef] [PubMed]

39. S. Matej and R. Lewitt, “Practical considerations for 3-D image reconstruction using spherically symmetric volume elements,” IEEE Trans. Med. Imag. **15**, 68–78 (1996). [CrossRef]

40. T. Obi, S. Matej, R. Lewitt, and G. Herman, “2.5-D simultaneous multislice reconstruction by series expansion methods from fourier-rebinned pet data,” IEEE Trans. Med. Imag. **19**, 474–484 (2000). [CrossRef]

*μ*m. The dimension of

**H**

*is the same as that of*

^{blob}**H**

*. The spectrum of singular values was computed for all system matrices using the Matlab programming environment [41].*

^{pixel}**H**

*for the three different weighting kernels. The matrix constructed by use of the cubic spline kernel is the most ill-conditioned, while the system matrix constructed by use of the linear kernel is the least ill-conditioned. This behavior is expected since the cubic spline kernel imposes the most smoothness on the data, followed by the quadratic spline and linear kernels.*

^{pixel}*a*= 1.5 and varying shape parameter

*α*. These results indicate that the parameter

*α*will generally affect the stability of the system matrix. In this case,

*α*= 2.0 corresponds to the most poorly conditioned system matrix while

*α*= 10.4 corresponds to the best conditioned system matrix. The spectra for the case when the blob relative radius

*a*was increased to 2 (physical size 100

*μ*m) are displayed in Fig. 4. The parameter

*α*is again observed to have a significant effect on the stability of the system matrices. The system matrix corresponding to

*α*= 2 is the most ill-conditioned, while the one corresponding to

*α*= 10.4 is the least ill-conditioned. In order to gain insight into this behavior, one can examine the normalized differential projection profile of one blob as shown in Fig. 5. One observes that the profile is more localized when

*α*increases from 2 to 10.4, which results in a better conditioned system matrix.

**H**

*with*

^{blob}*α*= 10.4. and relative radius

*a*= 1.5 and

*a*= 2 and the third to

**H**

*employing the linear weighting kernel. The two blob-based spectra possess a slower rate of decay than the pixel-based spectra, indicating that that the blob-based system matrices will yield more stable reconstruction problems than will pixel-based ones.*

^{pixel}### 4.2. Investigation of image variance and spatial resolution

#### 4.2.1. Simulation data and image reconstruction algorithm

*δ*(

**r**

_{2};

*z*). The physical size of the phantom was 25.6 mm × 25.6 mm. The phantom was composed of nine uniform disks possessing different values and physical sizes, which were blurred with a Gaussian kernel of width 0.15 mm. From knowledge of the phantom, the elements of the differential projection data

**g**were computed analytically. The scanning geometry employed assumed 180 tomographic views that were uniformly spaced over a

*π*angular range. At each view, the detector was assumed to possess 1024 elements of pitch 25

*μ*m.

44. J. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imag. **13**, 290–300 (1994). [CrossRef]

**b̂**. The analytic solution of the PLS algorithm with

*L*

_{2}regularization can be written as a pseudo-inverse operator

**H**

^{+}acting on

**g**. The pseudo-inverse operator

**H**

^{+}can be decomposed as a linear combination of certain outer-product operators, whose coefficients are the reciprocals of the singular values of the operator

**H**[36

36. M. Bertero, *Introduction to Inverse Problems in Imaging* (Taylor & Francis, 1998). [CrossRef]

**b̂**represent approximate solutions of the optimization program where

*γ*is a regularization parameter, with the set

*𝒩*

*containing the index values of the four neighbour points of the*

_{n}*n*th value of

**b**. From knowledge of

**b̂**, estimates of the object function

*δ*

*(*

_{a}**r**

_{2};

*z*) were obtained by use of Eq. (3). For the cases where blob expansion functions were employed, the estimates of

*δ*

*(*

_{a}**r**

_{2};

*z*) were sampled by use of a 2D Dirac delta sampling function onto a Cartesian grid and the resulting values stored as a matrix for analysis and display.

**H**

*or*

^{pixel}**H**

*. For the pixel-based studies, the object was represented by a 512 × 512 pixel array with a 50*

^{blob}*μ*m pitch. Three different pixel-based matrices

**H**

*were constructed corresponding to the weighting kernel functions described in Sec. 3.1. For the blob-based studies, six different system matrices were employed that corresponded to blob parameters relative radius a = 1.5 (physical size 75*

^{pixel}*μ*m), relative radius a = 2 (physical size 100

*μ*m), and

*α*= 2, 6, or 10.4. In all cases, 512 × 512 blobs were employed to represent the object function and the distance (sampling interval) between the blobs was 50

*μ*m. For each system matrix, five sets of 100 noisy images were reconstructed for distinct values of the regularization parameter specified by

*γ*= 10, 200, 1000, 2000, or 5000.

**H**

*and*

^{pixel}**H**

*. Example images reconstructed from noisy data sets by use of*

^{blob}**H**

*and*

^{pixel}**H**

*are shown in Figs. 8(a) and 8(b). The system matrix*

^{blob}**H**

*utilized linear interpolation and*

^{pixel}**H**

*utilized blob parameters relative radius*

^{blob}*a*= 2,

*m*= 2, and

*α*= 10.4. The regularization parameter was set at

*γ*= 10 for both cases. Horizontal profiles through the centers of the images in Figs. 8(a) and 8(b) are shown in Fig. 9. The solid blue line (pixel-based result) appears to overshoot some of the boundaries and has more oscillations than the dashed red line (blob-based result). Note that the grey levels of the true object were recovered with good fidelity due to the fact that the object was contained within the field-of-view of the simulated imaging system and therefore there was no truncation of the data function with respect to the detector coordinate.

#### 4.2.2. Empirical determination of image statistics and resolution measures

46. M. A. Anastasio, M. Kupinski, and X. Pan, “Noise propagation in diffraction tomography: Comparison of conventional algorithms with a new reconstruction algorithm,” IEEE Trans. Nucl. Sci. **45**, 2216–2223 (1998). [CrossRef]

47. J. Zhang, M. A. Anastasio, P. La Rivière, and L. Wang, “Effects of different imaging models on least-squares image reconstruction accuracy in photoacoustic tomography,” IEEE Trans. Med. Imag. **28**, 1781–1790 (2009). [CrossRef]

*x*denotes the coordinate along the image profile,

*I*

_{1}and

*I*

_{2}indicate the image values on the two sides of the boundary with

*I*

_{1}<

*I*

_{2},

*μ*is the true boundary location, and erf(

*x*) is the error function, and

*σ*is the associated standard deviation. We adopted the full-width at half-maximum (FWHM) value of the fitted error function as a summary measure of spatial resolution [47

47. J. Zhang, M. A. Anastasio, P. La Rivière, and L. Wang, “Effects of different imaging models on least-squares image reconstruction accuracy in photoacoustic tomography,” IEEE Trans. Med. Imag. **28**, 1781–1790 (2009). [CrossRef]

*γ*produced a collection of (variance, FWHM) pairs for each system matrix, which were plotted to characterize the trade-offs between spatial resolution and noise levels in the reconstructed images.

*γ*= 10, while the right-most point on each curve corresponds to

*γ*= 5000. As expected, when the value of

*γ*increases, the image variance decreases at the cost of spatial resolution.

**H**

*produced images with smaller variances at any of the attained spatial resolution values than did the other two system matrices. These observations are consistent with the singular value spectra displayed in Fig. 2, where the linear and cubic spline-based system matrices were demonstrated to yield the best and worst, respectively, conditioned system matrices for the pixel-based studies.*

^{pixel}*α*= 10.4 were lower than those corresponding to the other

*α*values for both relative radius

*a*= 1.5 [Fig. 11(a)] and relative radius

*a*= 2 [Fig. 11(b)]. The curves corresponding to the shape parameter

*α*= 2.0 were higher than the others for both values of

*a*. These observation are consistent with the singular value spectra displayed in Figs. 3 and 4, where the system matrices

**H**

*corresponding to*

^{blob}*α*= 10.4 and

*α*= 2.0 were demonstrated to yield the best and worst, respectively, conditioned system matrices for the blob-based studies.

**H**

*with*

^{blob}*α*= 10.4 and relative radius

*a*= 1.5 and relative radius

*a*= 2 and the third to

**H**

*employing the linear weighting kernel. The two blob-based curves are everywhere lower than the pixel-based curve, indicating images produced by use of*

^{pixel}**H**

*can possess improved variance-resolution trade offs than those produced by use of*

^{blob}**H**

*. Below we demonstrate and investigate the use of*

^{pixel}**H**

*for reconstructing images of biological tissue from few-view experimental differential projection data.*

^{blob}## 5. Application to few-view image reconstruction

### 5.1. Experimental data and image reconstruction algorithm

4. S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. **16**, 562–572 (2009). [CrossRef] [PubMed]

*μ*m. In the studies described below, certain subsets of these data were employed for few-view image reconstruction. A phase-stepping procedure was employed, which utilized four steps, to compute the differential projection data at each tomographic view angle. We refer the reader to reference [4

4. S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. **16**, 562–572 (2009). [CrossRef] [PubMed]

48. E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory **52**, 489–509 (2006). [CrossRef]

**b**||

_{TV}represents the TV norm of the vector

**b**and

*ɛ*is the specified data tolerance. It has been demonstrated that this image reconstruction strategy can be highly effective at mitigating data-incompleteness for certain classes of objects [50–52

52. X. Han, J. Bian, D. Eaker, T. Kline, E. Y. Sidky, E. Ritman, and X. Pan, “Algorithm-enabled low-dose micro-CT imaging,” IEEE Trans. Med. Imag. **30** pp. 606–620 (2011). [CrossRef]

51. E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. **53**, 4777–4807 (2008). [CrossRef] [PubMed]

51. E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. **53**, 4777–4807 (2008). [CrossRef] [PubMed]

**H**

*with*

^{blob}*m*= 2, relative radius a = 2 (physical size 14

*μ*m, which is twice the sampling interval 7

*μ*m) and

*α*= 10.4 was constructed as described in Sec. 3.2. The values of the data tolerance

*ɛ*employed were 43.8 and 58.2 for the reconstruction problems involving 90 and 180 view angles, respectively. From knowledge of

**b̂**, estimates of

*δ*

*(*

_{a}**r**

_{2};

*z*) were obtained by use of Eq. (3) and were subsequently sampled by use of a 2D Dirac delta sampling function with a period of 7

*μ*m onto a Cartesian grid for display. Because it is commonly employed in current applications of DPCT, we also reconstructed images by use of a modified FBP algorithm that acts directly on the differential projection data [11

**27**, 5202–5212 (1988). [CrossRef] [PubMed]

### 5.2. Reconstructed images

*δ*(

**r**) only up to a constant. Because the true values of

*δ*(

**r**) were not available, we did not investigate this. All the images presented were normalized into the same scale.

## 6. Summary

## Appendix A: Explicit construction of the pixel-based system matrices

**H**

*employed in our numerical studies were constructed by use of Eq. (15). Specifically, because*

^{pixel}**H**

*is defined by Eq. (10) with the elements provided in reference [30*

^{R}30. R. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. **12**, 252–255 (1985). [CrossRef] [PubMed]

**H**

*for the three kernel functions*

^{D}**W**(

*x*

*,*

_{r}*d*) employed.

**H**

*can be expressed as follows where*

^{D}**H**

*(*

^{tt}*t*= 1, 2,··· ,

*T*) is a

*S*×

*S*matrix, T is the total number of projection views and S is the number of sampled projection data at each view. Explicit forms of

**H**

*are determined by different interpolation kernels*

^{tt}**W**(

*x*

*,*

_{r}*h*). Three types of

**H**

*corresponding to three different kernels*

^{tt}**W**(

*x*

*,*

_{r}*h*) adopted in the paper are provided as follows:

## Linear interpolation kernel

*n*

*is a normalization constant which is determined by the dimensionality and the smoothing length*

_{d}*h*. The value of

*h*was set to 2 times the projection sampling interval, and

*n*

*is equal to*

_{d}**H**

*corresponding to use of*

^{tt}**W**

_{1}(

*x*

*,*

_{r}*d*) can be expressed as where the the boundary condition elements are appropriately defined. In our studies, the projection data were not truncated and the object was embedded in uniform background medium. In this case, the boundary condition elements were set to zero.

## Quadratic spline

*n*

*is equal to*

_{d}**H**

*corresponding to use of*

^{tt}**W**

_{2}(

*x*

*,*

_{r}*d*) can be expressed as

## Cubic spline

*n*is equal to

_{d}**H**

*corresponding to use of*

^{tt}**W**

_{3}(

*x*

*,*

_{r}*d*) can be expressed as

## Appendix B: The derivation of the Eq. (18) in Sec.3.2

*ξ*≡

*x*

*–*

_{r}*x*

*cos*

_{n}*θ*–

*y*

*sin*

_{n}*θ*. As demonstrated by Lewitt [34

**7**, 1834–1846 (1990). [CrossRef] [PubMed]

*ξ*| ≤

*a*and zero otherwise. The gradient of the modified bessel function has the following relationship as [53] where

*z*is the distance to the center of the blob and

*m*is a real number. Let

## Acknowledgments

## References and links

1. | T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express |

2. | A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. |

3. | A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase–contrast X–ray computed tomography for observing biological soft tissues,” Nat. Med. |

4. | S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. |

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

6. | Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast: computed tomography using compressed sensing,” in “Proc. SPIE ,” |

7. | M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. |

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

9. | D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. |

10. | A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. |

11. | G. Faris and R. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. |

12. | J. Stricker, “Analysis of 3-d phase objects by moiré deflectometry,” Appl. Opt. |

13. | P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA |

14. | Z. Huang, K. Kang, L. Zhang, Z. Chen, F. Ding, Z. Wang, and Q. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A |

15. | T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. |

16. | A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Meth. A |

17. | A. Momose, “Phase-sensitive imaging and phase tomography using x-ray interferometers,” Opt. Express |

18. | T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature |

19. | M. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. Galatsanos, Y. Yang, J. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. |

20. | F. Dilmanian, Z. Zhong, B. Ren, X. Wu, L. Chapman, I. Orion, and W. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. |

21. | K. Pavlov, C. Kewish, J. Davis, and M. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys. |

22. | S. Fiedler, A. Bravin, J. Keyriläinen, M. Fernández, P. Suortti, W. Thomlinson, M. Tenhunen, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation dei-ct technique with clinical ct, mammography and histology,” Phys. Med. Biol. |

23. | A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett. |

24. | I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, “Biological imaging by x-ray phase tomography using diffraction-enhanced imaging,” Jpn. J. Appl. Phys. |

25. | K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. |

26. | Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys. |

27. | D. Paganin, |

28. | M. A. Anastasio and X. Pan, “Region-of-interest imaging in differential phase-contrast tomography,” Opt. Lett. |

29. | A. Kak and M. Slaney, |

30. | R. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. |

31. | S. Lo, “Strip and line path integrals with a square pixel matrix: A unified theory for computational CT projections,” IEEE Trans. Med. Imag. |

32. | J. Monaghan, “Smoothed particle hydrodynamics,” Rep. Prog. Phys. |

33. | A. Chaniotis and D. Poulikakos, “High order interpolation and differentiation using b-splines,” J. Comput. Phys. |

34. | R. Lewitt, “Multidimensional digital image representations using generalized Kaiser-Bessel window functions,” J. Opt. Soc. Am. A |

35. | R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol. |

36. | M. Bertero, |

37. | R. Fatehi, M. Fayazbakhsh, and M. Manzari, “On discretization of second-order derivatives in smoothed particle hydrodynamics,” Proceedings of World Academy of Science, Engineering and Technology . |

38. | S. Matej and R. Lewitt, “Image representation and tomographic reconstruction using spherically-symmetric volume elements,” in Nuclear Science Symposium and Medical Imaging Conference, 1992., Conference Record of the 1992 IEEE, (IEEE, 1992), pp. 1191–1193. |

39. | S. Matej and R. Lewitt, “Practical considerations for 3-D image reconstruction using spherically symmetric volume elements,” IEEE Trans. Med. Imag. |

40. | T. Obi, S. Matej, R. Lewitt, and G. Herman, “2.5-D simultaneous multislice reconstruction by series expansion methods from fourier-rebinned pet data,” IEEE Trans. Med. Imag. |

41. | D. Hanselman and B. Littlefield, |

42. | V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. |

43. | T. Köhler, K. Engel, and E. Roessl, “Noise properties of grating-based x-ray phase contrast computed tomography,” Med. Phys. |

44. | J. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imag. |

45. | H. Barrett, K. Myers, and S. Dhurjaty, |

46. | M. A. Anastasio, M. Kupinski, and X. Pan, “Noise propagation in diffraction tomography: Comparison of conventional algorithms with a new reconstruction algorithm,” IEEE Trans. Nucl. Sci. |

47. | J. Zhang, M. A. Anastasio, P. La Rivière, and L. Wang, “Effects of different imaging models on least-squares image reconstruction accuracy in photoacoustic tomography,” IEEE Trans. Med. Imag. |

48. | E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory |

49. | E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. |

50. | E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” Journal of X-Ray Science and Technology |

51. | E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. |

52. | X. Han, J. Bian, D. Eaker, T. Kline, E. Y. Sidky, E. Ritman, and X. Pan, “Algorithm-enabled low-dose micro-CT imaging,” IEEE Trans. Med. Imag. |

53. | M. Abramovitz and I. Stegun, |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 30, 2011

Revised Manuscript: January 3, 2012

Manuscript Accepted: January 9, 2012

Published: April 25, 2012

**Virtual Issues**

Vol. 7, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Qiaofeng Xu, Emil Y. Sidky, Xiaochuan Pan, Marco Stampanoni, Peter Modregger, and Mark A. Anastasio, "Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography," Opt. Express **20**, 10724-10749 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-10724

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### References

- T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express13, 6296–6304 (2005). [CrossRef] [PubMed]
- A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45, 5254–5262 (2006). [CrossRef]
- A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase–contrast X–ray computed tomography for observing biological soft tissues,” Nat. Med.2, 473–475 (1996). [CrossRef] [PubMed]
- S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat.16, 562–572 (2009). [CrossRef] [PubMed]
- J. Brankov, M. 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]
- Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast: computed tomography using compressed sensing,” in “Proc. SPIE,” 7258A1–A8 (2009)
- M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett.90, 224101 (2007). [CrossRef]
- 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]
- D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol.42, 2015–2025 (1997). [CrossRef] [PubMed]
- A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys.42, 866–868 (2003). [CrossRef]
- G. Faris and R. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt.27, 5202–5212 (1988). [CrossRef] [PubMed]
- J. Stricker, “Analysis of 3-d phase objects by moiré deflectometry,” Appl. Opt.23, 3657–3659 (1984). [CrossRef] [PubMed]
- P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA107, 13576–13581 (2010). [CrossRef] [PubMed]
- Z. Huang, K. Kang, L. Zhang, Z. Chen, F. Ding, Z. Wang, and Q. Fang, “Alternative method for differential phase-contrast imaging with weakly coherent hard x rays,” Phys. Rev. A79, 013815 (2009). [CrossRef]
- T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys.38, 4542–4545 (2011). [CrossRef] [PubMed]
- A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Meth. A352, 622–628 (1995). [CrossRef]
- A. Momose, “Phase-sensitive imaging and phase tomography using x-ray interferometers,” Opt. Express11, 2303–2314 (2003). [CrossRef]
- T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature373, 595–598 (1995). [CrossRef]
- M. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. Galatsanos, Y. Yang, J. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol.48, 3875–3895 (2003). [CrossRef]
- F. Dilmanian, Z. Zhong, B. Ren, X. Wu, L. Chapman, I. Orion, and W. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol.45, 933–946 (2000). [CrossRef] [PubMed]
- K. Pavlov, C. Kewish, J. Davis, and M. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys.33, 1596–1605 (2000). [CrossRef]
- S. Fiedler, A. Bravin, J. Keyriläinen, M. Fernández, P. Suortti, W. Thomlinson, M. Tenhunen, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation dei-ct technique with clinical ct, mammography and histology,” Phys. Med. Biol.49, 175–188 (2004). [CrossRef] [PubMed]
- A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett.86, 124105–124105-3 (2005). [CrossRef]
- I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, “Biological imaging by x-ray phase tomography using diffraction-enhanced imaging,” Jpn. J. Appl. Phys.44, 8219–8221 (2005). [CrossRef]
- K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt.26, 349–393 (1988). [CrossRef]
- Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Jpn. J. Appl. Phys.46, 89–91 (2007). [CrossRef]
- D. Paganin, Coherent X-ray Optics, (Oxford University Press, 2006). [CrossRef]
- M. A. Anastasio and X. Pan, “Region-of-interest imaging in differential phase-contrast tomography,” Opt. Lett.32, 3167–3169 (2007). [CrossRef] [PubMed]
- A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Service Center, Piscataway, NJ, 1988).
- R. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys.12, 252–255 (1985). [CrossRef] [PubMed]
- S. Lo, “Strip and line path integrals with a square pixel matrix: A unified theory for computational CT projections,” IEEE Trans. Med. Imag.7, 355–363 (1988). [CrossRef]
- J. Monaghan, “Smoothed particle hydrodynamics,” Rep. Prog. Phys.68, 1703–1760 (2005). [CrossRef]
- A. Chaniotis and D. Poulikakos, “High order interpolation and differentiation using b-splines,” J. Comput. Phys.197, 253–274 (2004). [CrossRef]
- R. Lewitt, “Multidimensional digital image representations using generalized Kaiser-Bessel window functions,” J. Opt. Soc. Am. A7, 1834–1846 (1990). [CrossRef] [PubMed]
- R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol.37, 705–716 (1992). [CrossRef] [PubMed]
- M. Bertero, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998). [CrossRef]
- R. Fatehi, M. Fayazbakhsh, and M. Manzari, “On discretization of second-order derivatives in smoothed particle hydrodynamics,” Proceedings of World Academy of Science, Engineering and Technology. 30 pp. 243–246 (2008).
- S. Matej and R. Lewitt, “Image representation and tomographic reconstruction using spherically-symmetric volume elements,” in Nuclear Science Symposium and Medical Imaging Conference, 1992., Conference Record of the 1992 IEEE, (IEEE, 1992), pp. 1191–1193.
- S. Matej and R. Lewitt, “Practical considerations for 3-D image reconstruction using spherically symmetric volume elements,” IEEE Trans. Med. Imag.15, 68–78 (1996). [CrossRef]
- T. Obi, S. Matej, R. Lewitt, and G. Herman, “2.5-D simultaneous multislice reconstruction by series expansion methods from fourier-rebinned pet data,” IEEE Trans. Med. Imag.19, 474–484 (2000). [CrossRef]
- D. Hanselman and B. Littlefield, Mastering Matlab 7 (Pearson Education, 2005).
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- J. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imag.13, 290–300 (1994). [CrossRef]
- H. Barrett, K. Myers, and S. Dhurjaty, Foundations of Image Science (Wiley-Interscience, 2003), 2nd ed.
- M. A. Anastasio, M. Kupinski, and X. Pan, “Noise propagation in diffraction tomography: Comparison of conventional algorithms with a new reconstruction algorithm,” IEEE Trans. Nucl. Sci.45, 2216–2223 (1998). [CrossRef]
- J. Zhang, M. A. Anastasio, P. La Rivière, and L. Wang, “Effects of different imaging models on least-squares image reconstruction accuracy in photoacoustic tomography,” IEEE Trans. Med. Imag.28, 1781–1790 (2009). [CrossRef]
- E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006). [CrossRef]
- E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math.59, 1207–1223 (2006). [CrossRef]
- E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” Journal of X-Ray Science and Technology14, 119–139 (2006).
- E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol.53, 4777–4807 (2008). [CrossRef] [PubMed]
- X. Han, J. Bian, D. Eaker, T. Kline, E. Y. Sidky, E. Ritman, and X. Pan, “Algorithm-enabled low-dose micro-CT imaging,” IEEE Trans. Med. Imag.30 pp. 606–620 (2011). [CrossRef]
- M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

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