6.5.1
Performance metrics
We use the structural similarity measure (SSIM) [116, 117] and signal-to-noise ratio (SNR) for measuring the quality of the reconstructed image. SSIM is a similarity measure proposed by Z. Wang et al. which compares the luminance, contrast, and structure of images. SSIM is computed for a window of size(R× R)
around each image pixel. The SSIM measure for two images x andbx for the specified window is
SSIM(x,bx) = (2µxµxˆ+C1)(2σxxˆ+C2) (µ2
x+ µx2ˆ+C1)(σx2+ σx2ˆ+C2)
, (6.29)
where C1and C2are small constant values to avoid instability. µxand µxˆdenote the empirical mean of the
images x andbx in the specified window, respectively. The empirical variance of the corresponding images are σx and σxˆ. The covariance of two images is denoted by σxˆx for the corresponding window. In our
experiments, we choose C1= C2= (.001∗ L)2where L is the dynamic range of the image pixel values.
SSIM for the total image is obtained as the average of SSIM over all pixels. It takes values between 0 and 1 with 1 corresponding to the highest similarity.
Our other quality measure is the SNR. If x is the oracle andbx is the reconstructed image we have
SNR(x,bx) = max
a,b∈R20 log
||x||2
||x − abx+ b||2
. (6.30)
Higher values of the SNR correspond to a better match between the oracle and the reconstructed image.
6.5.2
Experimental result
To validate our reconstruction method, we conducted experiments with real data acquired using the TOM- CAT beam line of the Swiss Light Source at the Paul Scherrer Institut in Villigen, Switzerland. The syn- chrotron light is delivered by a 2.9 T super-bending magnet. The energy of the X-ray beam is 25keV [118]. We used nine phase steps over two periods to measure the displacement of the diffraction pattern described in Section 2. For each step a complete tomogram was acquired around 180 degrees; we used 721 uniformly distributed projection angles. Image acquisition was performed with a CCD camera whose pixel size was 7.4µm.
For our experiments we used a rat brain sample. The sample is embedded in liquid paraffin at room temperature. This is necessary to match the refractive index of the sample with its environment, so that the small-refraction-angle approximation holds. Finally the projections were post-processed, including flat-field and dark-field corrections, for the extraction of the phase gradient.
Figure 6.4 contains a comparison of the performance of the proposed algorithm against FISTA. For comparison, the convergence in [35] requires at least 65 iterations to converge. This allows for a rough yet informative comparison of computational costs, in terms of number of evaluations of the forward model or its adjoint (which are the most expensive operations in the schemes discussed here). In [35], it is required to compute the forward operator twice per iteration. Therefore, the cost estimate is 65× 2 = 130 evaluations of the forward operator. Meanwhile, our reconstruction scheme converges after 5 outer iterations with 2 conjugate-gradient inner iterations. Each conjugate gradient step requires one application of the forward op- erator and one application of its adjoint. Since the number of viewing angles is typically less than the size of the object, the cost of the adjoint operator is less than the computation of the forward operator. If we neglect this fact, we need(2× 2 × 5) = 20 evaluations of the forward operator. Based on these considerations, we
expect our algorithm to be substantially faster. It demonstrates the benefits of using a warm initialization as well as a problem-specific preconditioner for the linear optimization step.
Figure 6.4: Speed of convergence of different iterative techniques for solving our regularized problem. Significant gains over the standard FISTA algorithm can be obtained using our ADMM- based scheme. Observe that the number of inner iterations in the ADMM-PCG method does not significantly influences its convergence.
We further investigated the performance of the direct filtered back-projection and of the proposed iterative reconstruction techniques on a coronal section of the rat brain. The reconstructed images with 721 angles using GFBP and those produced by our method are shown in Figure 6.5. In terms of quality, for real data, our results are more less equivalent to using the Kaiser Bessel Window functions proposed in [35].
The GFBP reconstruction contains artifacts at the boundary of the image and at specific anatomical features. For example, the bottom-right sub images in the reconstructions of Figure 6.5 show the mammal-thalamic tract in this coronal section. One can clearly see oscillatory artifacts in the GFBP reconstruction. This could confuse the biologist or automated diagnostic systems for determining the nucleus and immoneurins in that region. The middle-right and top-right images show a part of the thalamus and the region between
the thalamus and the hippocampus, respectively. To reduce the artifacts in the GFBP technique, we also
implemented a smoothed version of the GFBP algorithm. Specifically, we modified the filter in the third step of Algorithm 3 as b q(ωy) = 1 2π 1 |ωy|× h k(ω y) , (6.31)
where h(ωy) is a lowpass filter and k is an exponent that acts as a smoothing parameter. We chose h(ω)
to be the standard Hamming window. The reconstructions are shown in the bottom row of Figure 6.5. They suggest that with GFBP there is a tradeoff between artifacts and image contrast. Note that for these
SSIM = .96 SNR = 35 dB SSIM = .78 SNR = 28 dB SSIM = .91 SNR = 36.9 dB SSIM = .15 SNR = 9.2 dB SSIM = .96 SNR = 28 dB SSIM = .91 SNR = 27.3 dB SSIM = .94 SNR = 36.4 dB SSIM = .33 SNR = 12.89 dB SSIM = .51 SNR = 21.4 dB SSIM = .60 SNR = 22.2 dB SSIM = .43 SNR = 26.1 dB SSIM = .15 SNR = 7.1 dB SSIM = .16 SNR = 10.69 dB SSIM = .22 SNR = 10.20 dB SSIM = .78 SNR = 25.92 dB SSIM = .41 SNR = 27.13 dB SSIM = .49 SNR = 29.1 dB SSIM = .96 SNR = 24.55 dB SSIM = .95 SNR = 35.8 dB SSIM = .89 SNR = 37.0 dB
(a)
(b)
(c)
(d)
(e)
(f)
Figure 6.5: Comparison of the reconstruction results for 721 viewing angles (first column) and 181 viewing angles (second column). (a,d) GFBP, (b,e) GFBP with smoothing kernel, (c,f) the iterative ADMM. The sub-images correspond to the region between the thalamus and the hip- pocampus (top), a part of the talamus (middle), and the Fornix (bottom). Notice the oscillatory artifacts produced by GFBP. Applying a smoothing kernel reduces the artifacts but also blurs the reconstruction.
experiments the parameter k was optimized so as to achieve the best SNR. The figure of merits (SNR and SSIM) are indicated below each image in Figure 6.5. Visually the reconstructed image with 721 angles using our proposed technique is more faithful in comparison with the GFBP approach. Therefore, we consider it as our gold standard for investigating the dependence of our algorithm on the number of views as shown in Figure 6.6. The SNR and SSIM values are computed for the main region of the sample which includes
the brain. they suggest that we can reduce the number of views with our method at least fourfold while essentially maintaining the quality of the standard reconstruction method (FBP with a complete set of views).
Chapter 7
Improved Reconstruction Scheme for
X-ray Grating Interferometry
1 In Chapter 6, we applied our basic reconstruction framework (ADMM-PCG) in the context of GI. Our
experimental results show the feasibility of our framework. They suggest that the proposed technique out- performs the-state-of-the-art methods. The drawback of ADMM-PCG is that there is not a clear way to impose the convex constraints such as the positivity of the refractive index on the solution. Moreover, some line artifacts were visible on the reconstruction. In this chapter, we consider a GI problem with the same physical model as discussed in Chapter 6. We first use the constrained regularized weighted norm (CRWN) to improve the reconstruction. In the context of differential phase contrast imaging, we then investigate the situations in which the measurements are wrapped. We develop a reconstruction framework in order to simultaneously unwrap and reconstruct the object of interest.
One advantage of GI is that it provides simultaneously an absorption-contrast and a phase-contrast in- formation. In this regard, we develop a reconstruction framework to simultaneously retrieve the real and the imaginary part of the refractive index (complex refractive index reconstruction).