## Abstract

Phase-contrast x-ray computed tomography has a high potential to become clinically implemented because of its complementarity to conventional absorption-contrast.

In this study, we investigate noise-reducing but resolution-preserving analytical reconstruction methods to improve differential phase-contrast imaging. We apply the non-linear Perona-Malik filter on phase-contrast data prior or post filtered backprojected reconstruction. Secondly, the Hilbert kernel is replaced by regularized iterative integration followed by ramp filtered backprojection as used for absorption-contrast imaging. Combining the Perona-Malik filter with this integration algorithm allows to successfully reveal relevant sample features, quantitatively confirmed by significantly increased structural similarity indices and contrast-to-noise ratios. With this concept, phase-contrast imaging can be performed at considerably lower dose.

© 2014 Optical Society of America

## 1. Introduction

X-ray phase-contrast imaging has been developed as a powerful imaging method at synchrotron sources and is increasingly applied also with lab-based x-ray setups. Among various phase-contrast imaging techniques, differential phase-contrast (DPC) imaging has gained attraction [1], mainly as a complementary imaging modality with clinical prospects due to its high soft-tissue sensitivity and edge enhancing properties [2–4]. The phase information is contained in the complex index of refraction *n* = 1 − *δ* + i*β*, which describes the phase shift and the attenuation of the x-rays when interacting with matter, with the refractive decrement *δ* and the attenuation coefficient *β*, respectively. Phase contrast is obtained by determining the relative phase shift of the x-rays, ΔΦ, and is directly related to *δ* via

*r⃗*and wave vector

*k⃗*of the x-rays.

Several techniques have been developed to measure the phase information in addition to attenuation contrast (AC) [5]. In this study, grating-based x-ray computed tomography is used [6]. Its main component is a Talbot interferometer, which consists of a phase-grating and an analyzer grating and is followed by an x-ray detector, see fig. 1. By installing a source grating in front of a conventional x-ray tube, the coherence requirements of the subsequent Talbot interferometer are sufficiently fulfilled. With the so-called phase-stepping method [7], the relative phase-shift and change in average amplitude induced by the specimen is extracted. Therefrom, *β* and *δ* can be deduced [1, 8]. Furthermore, grating-based x-ray imaging permits access to so-called dark-field imaging, which records x-ray scattering on the subpixel-scale. Here, we will use it for image guidance only.

In order to reconstruct a computed tomography (CT) scan with the corresponding *δ*- and *β*-values for each voxel, filtered backprojection (FBP) is used. Despite the rapid development of iterative reconstruction methods (e.g. [9–11]), FBP remains a valuable tool for reconstruction due to its speed and maturity. Given the different noise structure in DPC compared to attenuation contrast, further optimization of FBP for DPC is still possible and due to the speed of FBP as well desired. For AC-CT, a ramp-filter, also known as Ram-Lak filter, is applied to the projection data before backprojection. However, in practice, this filter is most often modified in order to trade-off noise and resolution in the reconstructed image [12]. In DPC-CT, the integration of the differential data is included in the filtering step. This so-called Hilbert filter is equivalent to the Ram-Lak filter in AC-FBP [6]. In this study, we present an approach to improve FBP in DPC-CT in order to achieve a significantly lower noise level with minor loss in resolution.

Contrary to absorption images with dominant high-frequency noise, DPC images contain mainly low-frequency noise [13, 14]. Figure 2 shows the two-dimensional (2D) noise power spectra (NPS) of AC-CT and DPC-CT using the Ram-Lak and Hilbert filter, respectively.

Figure 2 demonstrates that noise in AC images is mainly present in the mid- to the high-frequency range whereas strong low-frequency noise can be observed in DPC-CT. As mathematically demonstrated by Raupach and Flohr [14], the integration of the differential projections changes the frequency (*ω*)-dependence of the noise power spectrum (NPS). The resulting NPS of DPC-CT exhibits a divergence for *ω* → 0 if an infinitely large detector is assumed. For a finite detector size, this low-frequency noise will, however, reach a maximum value. In the DPC image, the different noise-to-frequency dependence is expressed in a coarse noise structure, which may mask small details of the sample. Here, we investigate two different approaches to improve the image quality: 1. The Perona-Malik filter applied on DPC projections or reconstructed images permits anisotropic noise smoothing; 2. A regularized iterative integration of DPC projections couples the individual projection lines in vertical direction to reduce noise during an explicit integration process. The integrated projection data are subsequently reconstructed using the Ram-Lak filter. Finally, a combination of both methods will be presented.

## 2. Reconstruction and quantification methods

In this study, standard Hilbert FBP of DPC projections are obtained using a half-pixel shifted Hilbert kernel with −i sgn(*ω*)exp(i*πω*) (described e.g. in [15, 16]). Since in AC-CT, the high frequencies contain typically more noise than signal, this frequency range is often suppressed for noise reduction. The stronger high frequencies are restrained, the softer the edges become. The low frequencies, however, represent the ”flat structures” in an image. To reduce the dominant low-frequency noise in DPC-CT, high-pass filters can be applied [17] with the benefit of edge enhancement but at the expense of quantitativeness. Here, we analyze a different approach for DPC reconstruction.

*Perona-Malik filter*: Instead of modifying the reconstruction filter, the application of an edge-preserving smoothing filter might be advantageous. Here, the well-known and highly optimized Perona-Malik filter (PM filter) [18] is investigated. Its applicability has yet only been shown on AC images [19–21], which suggests to yield interesting results when applied in DPC-CT and allows for comparison with literature. Based on an anisotropic diffusion equation,

*f*(

*x*,

*y*,

*t*= 0) in dependence of the image’s gradient. The diffusivity

*γ*(

*f*(

*x*,

*y*,

*t*)) determines the velocity of the diffusion process and regulates the edge detection. For

*γ*= 1, i.e., if

*γ*is a constant function, the effect of this diffusion filter after a “time”

*t*

_{0}is equivalent to the application of a Gaussian filter with a standard deviation

*σ*= (2

_{PM}*t*

_{0})

^{1/2}. An implicit Euler’s method [22, 23] is applied to approximate the time derivative of the image. Thus, the partial derivative is discretized under the assumption of small ”time”-steps Δ

*t*[24] such that the image after one iteration step is given by:

*f*(

*x*,

*y*,

*t*) is abbreviated by

*f*(

*t*). For a given

*σ*, i.e.

_{PM}*t*

_{0}, the diffusion process runs

*N*times with

*t*

_{0}=

*N*· Δ

*t*, resulting in a final image

*f*(

*t*

_{0}). As diffusivity, we use

*ξ*to restrict the conservation of edges to those with a gradient much higher than

*ξ*. To simplify the parameter choice,

*ξ*is set to a fixed value of 0.08, showing a good performance for projections with a low signal-to-noise ratio. If images with a higher signal-to-noise ratio are treated, instead of modifying

*ξ*, a ratio of

*γ*is taken, accordingly. The Roberts cross operator [25] is used to compute the image’s gradient, ∇

*f*(

*x*,

*y*,

*t*), as well as the Laplace operation on

*f*over the spatial variables. Taking diagonal elements into account for derivation enhances the accuracy with longer computation time as a trade-off. Equations (2)–(4) can easily be extended to a three dimensional PM filter for a volume

*f*(

*x*,

*y*,

*z*,

*t*). Three parameters of the PM filter can be tuned: the strength of the Gaussian blurring for areas without edges

*σ*, the diffusivity

_{PM}*γ*, and the accuracy of solving the diffusion equation via the ”time”-step Δ

*t*. As only a restricted range of values for these three parameters yields meaningful results, the application of the PM filter stays on a relatively intuitive level compared to the complexity of, e.g., iterative reconstruction methods.

*Regularized iterative integration + Ram-Lak FBP*: A different approach for noise suppression is to integrate DPC projections prior to reconstruction. Multiplying the differential projections with the Hilbert kernel in Fourier space, as described above, includes the integration of the differential phase to determine the actual phase shift. Here, we separate the integration and the filtering step. As a simple cumulative sum of the uncoupled projection lines leads to strong streak artifacts due to line-wise noise summation, we use a regularized iterative integration (RII) algorithm adapted from Thüring et al. [26]. Thereby, the intensity values of neighboring pixels perpendicular to the integration direction are taken into account, which minimizes these stripe artifacts. The corresponding cost function *L* contains integration, regularization perpendicular to integration direction and offset determination (for sample sizes smaller than the field of view). Each pixel in the integration plane is indexed by (*i*, *j*). The cost function is minimized using the *ℓ*_{2}-norm. It consists of three terms:

*with the estimate of the integrated phase Φ̃*

_{i,j}*is evaluated in a least square sense for each neighboring pixel pair with respect to the standard deviation*

_{i,j}*σ*of the measured data.

_{i,j}*σ*is determined following [27], by matching the differentiated attenuation data to DPC data. The second term provides the coupling of neighboring projection lines in vertical direction using a tunable parameter

_{i,j}*λ*

_{1}to weight this sum with regard to the consistency term. The difference of neighboring pixel values is minimized in dependence of the local variance ${\left({\sigma}_{i,j}^{\mathit{loc}}\right)}^{2}$, which is evaluated within a Gaussian window of width Ω [px] around the respective pixel (

*i*,

*j*). The third term is added to the algorithm to mask the background such that side features do not influence the relevant signal if the sample does not cover the entire field of view. The signal suppression in an area determined by the mask

*m*is quantified by

*λ*

_{2}.

Three parameters of *L* can be varied: Ω(*σ _{loc}*),

*λ*

_{1}, and

*λ*

_{2}. For low-noise projections, the mask with suppression strength

*λ*

_{2}can be automatically chosen using the dark-field image as reference of the sample border. If noise dominates the scattering signal, the mask must be applied manually. In this study,

*λ*

_{2}is empirically set to a constant value of 10. The Gaussian blurring and the coupling strength should be set dependent on the noise level in the projections.

Applying the PM filter as well as the RII algorithm, the resulting projections and reconstructed images were evaluated concerning noise and resolution. Due to non-linear noise treatment using either of the two methods, the NPS requiring isotropic noise cannot be employed for noise quantification. Therefore, the contrast-to-noise ratio (CNR) was studied using the mean value of two contrast features
$\overline{{C}_{1}}$ and
$\overline{{C}_{2}}$ with respect to the root mean square of the corresponding standard deviations, *σ*_{C1} and *σ*_{C2}:

Similar as for the NPS, a comparison via the modulation transfer function obtained from one edge in the image does not reflect the overall edge sharpness after usage of a non-linear filter. To further quantify the quality of the reconstructed images, we chose the structural similarity (SSIM) index based on full-reference image quality assessment [28]. The SSIM quantifies the similarity of an image to a reference image on a scale from 0 (no similarity) to 1 (identical images) and is designed to match the visual perception of humans. This is implemented by an evaluation of structure and contrast on a local basis. The mean SSIM index is obtained from local areas using a 11 × 11 circular-symmetric Gaussian weighting function, which allows to handle non-linearly treated images. In this study, the generic MATLAB implementation of the algorithm described in [28] is used with the default parameter settings (”ssim” in Image Processing Toolbox Release 2014a, The Mathworks, Inc., Natick, Massachusetts, United States).

## 3. Experimental methods

For this study, we chose data sets of a formalin-fixated human heart specimen measured with varying exposure times *t _{exp}* = [0.025, 0.1, 0.225, 3.6] s per phase-step. Hence, we can investigate the reconstruction quality at different quantum noise levels. A rotating anode system was operated at 40 kV with a filament current of 70 mA (FR 591 by

*Enraf-Nonius*). For the tomographic scan, the specimen was mounted on a motorized rotation stage. To avoid readout noise and dark current, a photon counting detector was utilized (PILATUS II 100K by

*Dectris*). With 487×195 pixels of 172×172 μm

^{2}resolution, a field of 83.8×33.5 mm

^{2}was available offering a counting rate of 2 · 10

^{6}photons per second per pixel.

Source grating, phase-grating, and analyzer grating were produced by the *Karlsruhe Institute of Technology* and by the *Microworks GmbH*. The phase-stepping curve was obtained using 11 phase-steps of the analyzer grating taken at 1200 equidistant tomographic angles.

The heart specimen consists mainly of muscle and fat tissue crossed by an artery (cf. fig. 3).

It was placed in a falcon tube of 3 cm diameter together with a highly contrasting polymethylmethacrylat (PMMA) rod for calibration purposes.

## 4. Results

In this section, the standard Hilbert FBP, denoted by ’*Standard*’, is compared to the reconstruction methods described in section 2: ’*PM*’ designates the application of the Perona-Malik filter, either in 2D on projection data followed by standard Hilbert FBP, ’*PM*_{2D}’, or in three dimensions (3D) on reconstructed data, ’*PM*_{3D}’. For ’*RII*’, regularized iterative integration (RII) is performed prior to Ram-Lak FBP. ’*PM*_{2D} *+ RII*’ labels the concept of 2D PM filtering followed by the integration procedure. Reconstructed images indicated with ’*RII + PM*_{3D}’ are integrated, reconstructed, and finally PM filtered in the image domain.

To illustrate the performance of each reconstruction method, axial, coronal and sagittal slices of the sample are investigated. The reconstruction quality with regard to quantum noise is examined for a selection of three data sets acquired with *t _{exp}* = [0.025, 0.1, 0.225] s. Varying signal-to-noise ratios require an adjustment of the parameters for the PM filter and for the RII, which are given in table 1. For each case, the parameters are set to values yielding visually and quantitatively, i.e. by CNR determination, the best quality of the image. The optimization is performed for the axial slice as, in this plane, the weakest improvement can be expected due to the RII reducing stripe artifacts in coronal and sagittal projections.

For quantitative comparison, CNR values and SSIM indices are determined. Exemplarily, we present the CNR values of all axial and coronal slices, representative for the entire volume. The SSIM indices of the aforementioned axial and coronal slices are calculated with respect to the reference image taken with 3.6 s exposure time per phase-step (cf. fig. 3).

Figure 4 presents the five reconstruction concepts for varying exposure times in the axial slice.

The columns from left to right contain images obtained with *t _{exp}* = 0.225, 0.1, and 0.025 s, respectively. In the top row,

*Standard*reconstructed images are depicted. The shorter the exposure time of data acquisition, the stronger the correlated DPC noise, which masks relevant sample features and introduces artificial intensity variances.

In the *PM*_{2D} images, some of the noise-covered sample details are revealed as the projection lines become vertically coupled by the Gaussian smoothing. The non-linearity of the diffusion equation allows for edge preservation in dependence on the chosen *γ*-value. In the case of a low signal-to-noise ratio, a simultaneous smoothing of noise and edge features has to be expected, especially visible for *t _{exp}* = 0.025 s. However, the recovery of important sample information, like the boundary of fat and muscle tissue to the surrounding formalin, justifies the use of the PM filter.

Instead of reconstructing with the Hilbert filter, RII is performed prior to FBP in *RII*. The coarse noise structure is significantly reduced and refined. A strong noise level yet covers important sample details.

The parameters for the lowest exposure time of *PM*_{2D} *+ RII* are relaxed as the Gaussian smoothing by the 2D PM filter in addition to the regularization of the integration algorithm would lead to undesired edge blurring. In the resulting reconstructed images, residual noise is removed to a high extent.

By the combination of *RII + PM*_{3D}, the best results are obtained when compared to *Standard*, as here, the strongest noise reduction is observed while edges are not only preserved but even recovered. However, in particular for *t _{exp}* = 0.025 s, the DC level of the muscular tissue is substantially reduced. Hence, quantitativeness cannot be conserved for strong regularization but previously hidden features become recognizable.

The 3D performance of *RII + PM*_{3D} is demonstrated with axial, coronal, and sagittal slices for *t _{exp}* = 0.225, 0.1, and 0.025 s in fig. 5. For comparison, the images from a standard Hilbert FBP for

*t*= 0.025 s are depicted below.

_{exp}The removal of stripe artifacts can be well observed in the coronal and sagittal slices. This is accomplished by the RII - not explicitly shown here. Most importantly, the noise in all three dimensions is successfully reduced. Due to the strong regularization for *t _{exp}* = 0.025 s, the contrast in the corresponding images is diminished as the coupling of projections in vertical direction smooths signal and, especially, noise fluctuations prior to reconstruction. Slight staircase effects are visible in the second row images due to strong smoothing with the PM filter. However, a significant improvement of image quality in the entire volume can be stated.

To quantify these results, the CNR is determined from one axial and one coronal slice (as depicted in fig. 4 and fig. 5, respectively) for each of the five reconstruction methods shown. A rectangular, homogeneous area within the PMMA cylinder serves as *C*_{1} and a respective region of interest of the muscle tissue as *C*_{2}, cf. eq. (6). The resulting values are displayed in fig. 6.

All reconstruction methods with PM filter or RII yield higher CNR values than the standard Hilbert FBP. The CNR values of axial and coronal slices differ only slightly and a similar trend was observed for CNR values obtained from sagittal slices (not depicted) such that the following conclusions can be drawn for the entire volume. Interestingly, coarse low-frequency noise is visibly reduced in the case of *RII*, which replaces the Hilbert FBP (cf. fig. 5). The combination of PM filter and RII produces the highest CNR values, predominantly if the PM filter is applied post reconstruction in its 3D version. The *RII + PM*_{3D} method benefits from information of all three dimensions whereas in *PM*_{2D} *+ RII*, the angular dimension limits the PM filtering to 2D. For *t _{exp}* = 0.1, and 0.225 s, the CNR values are similar. However, in the corresponding images in fig. 5, edges are stronger blurred for

*t*= 0.1 s, which has to be included in the complete image analysis.

_{exp}To further prove these conclusions, image quality is measured using the SSIM. The standard Hilbert FBP reconstructed image obtained with *t _{exp}* = 3.6 s, see fig. 3, serves as a reference image to which the axial and coronal slices, partially presented in fig. 4 and fig. 5, are compared. Each calculated index is displayed as one bar in a histogram in fig. 7.

The SSIM analysis demonstrates an overall consistency with the CNR values discussed above. Compared to the CNR histogram, the difference of SSIM between *PM*_{2D} *+ RII* and *RII + PM*_{3D} is less distinct but follows the same trend. Additionally, the SSIM of the coronal slice confirms very well the findings for the axial view. Equivalent results were observed for sagittal slices (not displayed). As expected, the Hilbert FBP reconstructed image yields the lowest SSIM index, especially, in contrast to reconstructed images treated with the PM filter. Visually, the application of *RII* reduces low-frequency noise such that, in particular, previously hidden structures can be recovered. This is represented by an increased SSIM index compared to the standard Hilbert FBP. Image quality is most significantly improved using the combination of *RII* with 2D- and, in particular, 3D PM filtering.

Consequently, the qualitative and quantitative investigation allows to conclude that *RII + PM*_{3D} delivers the optimum performance of the discussed reconstruction methods with the limitation that for the lowest dose, RII introduces a bias leading to reduced contrast.

## 5. Discussion and conclusion

This study demonstrates that significant image improvement by application of the PM filter on differential projections and images is achievable. In contrast to the results shown for attenuation data in Xia et al. [19], the *PM*_{3D} on a reconstructed volume performs slightly better in CNR, SSIM index and visual noise perception than using the *PM*_{2D} on projections. We explain our result by the additional signal information introduced with the third dimension. However, the differences between PM filtering as a pre- or postprocessing step are very small and depend highly on the parameter choice. Currently, setting the parameters is user-dependent. For example, high *σ _{PM}* values can cause strong blurring of edges by which staircase effects might occur. This deteriorates the visual impression but does not reduce the CNR. Lowering

*σ*, however, results in a higher noise level, and consequently, in a lower CNR. In such cases, the determination of the SSIM index is relevant as it includes the importance of noise, structure and luminance according to the human visual system. In

_{PM}*RII*, if edge conservation is reduced to allow for stronger noise smoothing by increasing

*λ*

_{1}, the appearance of a grainy background pattern diminishes the image quality - similarly observed in [26]. Insignificant improvement of image quality by noise reduction and comparison of neighboring pixel intensities is achieved when too low values for

*λ*

_{1}are chosen. As both, the PM filter and the regularization of the RII algorithm, reduce noise up to a certain threshold, their performance worsens for volumes with low signal-to-noise ratios, when edges cannot be separated from noise spikes anymore. Knowing these limits, for instance by determination of CNR, SNR and edge response or by the SSIM index, the parameter setting can be automated to facilitate the applicability of the proposed reconstruction concept.

Visually, it can be perceived that with *RII*, the noise grains become finer while sample information is not only preserved but even revealed. In frequency space, this corresponds to a shift of the NPS towards higher frequencies, which yet is not an appropriate quantification tool due to the non-linear image treatment. Strong regularization can, however, lead to loss of quantitativeness and reduced contrast. Compared to high-pass filtering where relevant signal information is suppressed for noise reduction, with *RII* a superior method for DPC noise treatment in all three dimensions is presented.

Compared to advancing iterative reconstruction techniques, lower complexity of the proposed methods significantly facilitates their implementation. As a next step, a combination of the RII algorithm with, e.g. recently developed, computationally efficient bilateral filters [29] would be of interest to find the optimal image quality achievable with this concept.

All of the newly investigated reconstruction methods yield a better performance with regard to CNR, SSIM index and visual appearance than the standard Hilbert FBP. Using the PM filter only can already very efficiently improve the CNR and the SSIM index. An enhancement of these image quality measures is as well achieved when integration of the DPC projection is performed prior to reconstruction. Combining the two methods delivers the best result (visually and) in terms of CNR as well as SSIM index. With the *RII + PM*_{3D} the highest improvement was achieved. Hence, the qualitative observations made in the image study could be well confirmed by the quantitative analysis.

The successful treatment of DPC data by the combination of PM filter and RII allows to measure phase-contrast at significantly lower dose while revealing a high amount of relevant sample features. Thereby, this work contributes importantly to future clinical application of phase-contrast imaging.

## Acknowledgments

We acknowledge financial support through the DFG Cluster of Excellence Munich-Centre for Advanced Photonics (MAP), the DFG Gottfried Wilhelm Leibniz program and the European Research Council (ERC, FP7, StG 240142). This work was carried out with the support of the Karlsruhe Nano Micro Facility (KNMF, www.kit.edu/knmf), a Helmholtz Research Infrastructure at Karlsruhe Institute of Technology (KIT). The authors acknowledge the Department of Clinical Radiology, Ludwig-Maximilians-Universität München, Germany for providing the heart specimen.

## References

**1. **F. Pfeiffer, ”Milestones and basic principles of grating-based x-ray and neutron phase-contrast imaging,” in Proceedings of the AIP Conference1466(1) (American Institute of Physics, 2012), pp. 2–11.

**2. **M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, and N. Hauser, ”The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Invest. Rad. **46**(12), 801–806 (2011). [CrossRef]

**3. **J. Kiyohara, C. Makifuchi, K. Kido, S. Nagatsuka, J. Tanaka, M. Nagashima, and A. Momose, ”Development of the Talbot-Lau interferometry system available for clinical use,” in *International Workshop on XNPIG* (AIP Publishing, 2012), vol. 1466, pp. 97–102.

**4. **M. Bech, A. Tapfer, A. Velroyen, A. Yaroshenko, B. Pauwels, J. Hostens, and F. Pfeiffer, ”In-vivo dark-field and phase-contrast x-ray imaging,” Sci. Rep. **3**(3209), 1–3 (2013). [CrossRef]

**5. **A. Bravin, P. Coan, and P. Suortti, ”X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. **58**(1), R1–R35 (2013). [CrossRef]

**6. **F. Pfeiffer, C. Kottler, O. Bunk, and C. David, ”Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. **98**(10), 108105 (2007). [CrossRef] [PubMed]

**7. **F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, ”Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. **2**(4), 258–261 (2006). [CrossRef]

**8. **J. Herzen, T. Donath, F. Pfeiffer, O. Bunk, C. Padeste, F. Beckmann, A. Schreyer, and C. David, ”Quantitative phase-contrast tomography of a liquid phantom using a conventional x-ray tube source,” Opt. Express **17**(12), 10010–10018 (2009). [CrossRef] [PubMed]

**9. **T. Koehler, B. Brendel, and E. Roessl, ”Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. **38**(8), 4542–4545 (2011). [CrossRef]

**10. **M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, and M. Unser, ”Fast iterative reconstruction of differential phase contrast x-ray tomograms,” Opt. Express **21**(5), 5511–5528 (2013). [CrossRef] [PubMed]

**11. **T. Gaass, G. Potdevin, M. Bech, P. B. Noël, M. Willner, A. Tapfer, F. Pfeiffer, and A. Haase, ”Iterative reconstruction for few-view grating-based phase-contrast CT – an in vitro mouse model,” Europhys. Lett. **102**(4), 48001 (2013). [CrossRef]

**12. **A. C. Kak and M. Slaney, *Principles of Computerized Tomographic Imaging* (IEEE, 1988).

**13. **T. Koehler, K. J. Engel, and E. Roessl, ”Noise properties of grating-based x-ray phase contrast computed tomography,” Med. Phys. **38**(7), S106–S116 (2011). [CrossRef]

**14. **R. Raupach and T. G. Flohr, ”Analytical evaluation of the signal and noise propagation in x-ray differential phase-contrast computed tomography,” Phys. Med. Biol. **56**(7), 2219–2244 (2011). [CrossRef] [PubMed]

**15. **F. Noo, J. Pack, and D. Heuscher, ”Exact helical reconstruction using native cone-beam geometries,” Phys. Med. Biol. **48**(23), 3787–3818 (2003). [CrossRef]

**16. **A. Faridani, R. Hass, and D. C. Solmon, ”Numerical and theoretical explorations in helical and fan-beam tomography,” J. Phys. Conf. Ser. **124**(1), 012024 (2008). [CrossRef]

**17. **K. Li, N. Bevins, J. Zambelli, and G.-H. Chen, ”A new image reconstruction method to improve noise properties in x-ray differential phase contrast computed tomography,” Proc. SPIE **8313**, 83131V (2012). [CrossRef]

**18. **P. Perona and J. Malik, ”Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. **12**(7), 629–639 (1990). [CrossRef]

**19. **J. Q. Xia, J. Y. Lo, K. Yang, C. E. Floyd, and J. M. Boone, ”Dedicated breast computed tomography: volume image denoising via a partial-diffusion equation based technique,” Med. Phys. **35**(5), 1950 (2008). [CrossRef] [PubMed]

**20. **E. Michel-González, M. H. Cho, and S.Y. Lee, ”Geometric nonlinear diffusion filter and its application to x-ray imaging,” Biomed. Eng. Online **10**(47), 1–16 (2011). [CrossRef]

**21. **Z. Liao, ”Low-dosed x-ray computed tomography imaging by regularized fully spatial fractional-order Perona-Malik diffusion,” Adv. Math. Phys. **2013**, 371868 (2013). [CrossRef]

**22. **L. Euler, *Institutionum Calculi Integralis* (Academiae Imperialis Scientiarum, 1768).

**23. **T. Sauer, *Numerical Analysis*, II. ed. (Addison Wesley, 2012).

**24. **R. H. Landau, M. J. Páez Mejía, and C. C. Bordeianu, *Computational Physics: Problem Solving with Computers* (Wiley-VCH, 2007). [CrossRef]

**25. **L. G. Roberts, *Machine Perception of Three-Dimensional Solids*, Lincoln Laboratory Technical Report #315 (Massachusetts Institute of Technology, 1963).

**26. **T. Thüring, P. Modregger, B. R. Pinzer, Z. Wang, and M. Stampanoni, ”Non-linear regularized phase retrieval for unidirectional x-ray differential phase contrast radiography,” Opt. Express **19**(25), 25545–25548 (2011). [CrossRef]

**27. **D. Hahn, P. Thibault, M. Bech, M. Stockmar, S. Schleede, I. Zanette, and A. Rack, ”Numerical comparison of X-ray differential phase contrast and attenuation contrast,” Biomed. Opt. Express **3**(6), 1141–1148 (2012). [CrossRef] [PubMed]

**28. **Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, ”Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. **13**(4), 600–612 (2004). [CrossRef] [PubMed]

**29. **J. Herzen, M. S. Willner, A. A. Fingerle, P. B. Noël, T. Koehler, E. Drecoll, E. J. Rummeny, and F. Pfeiffer, ”Imaging liver lesions using grating-based phase-contrast computed tomography with bi-lateral filter post-processing,” PLOS ONE **9**(1), e83369 (2014). [CrossRef] [PubMed]