Numerical Experiments. Zy(k + 1) + 1 Ly(k) − xy(k + 1) , where Zy(k) = Z µˆy(k), Gˆy k , and we have used the Minkowski sum of zonotopes as per Sec. III-A. We leave alternative methods of approximating continuous time to future work. For example, one could compute the convex hull between timesteps with constrained zonotopes [45], [50], but the resulting numerical representations are typically large. One could also apply standard zonotope reachability methods [40], [43], though the resulting set representations may not be differentiable.
Numerical Experiments. To test the method, we generate a 2D image of the size 128 by 128 and assume that the object is made up of three simulated materials that arise in polyenergetic image reconstruction – adipose, air and bones. For bones, we use the main component, cal- cium, to represent it. One application of polyenergetic image reconstruction is breast imaging, which requires low dose radiation for patients. To realize this application, we generate an energy spectrum with potential 26 keV with the help of function “spek- trSpectrum” [53]. We also select a low radiation dose of 1e5 total photons for the x-ray energy spectrum. The corresponding spectrum is shown in Figure 3.1. From Figure 3.1, we can find that the photon flux density is above zero when the energy is between 3 kev and 28 kev. Based on this observation, the discrete energies for the simulated source x-xxx xxxx are chosen from 3 keV to 28 keV, with an interval of 1 keV. Photon flux density The plots of linear attenuation coefficients to materials adipose, air and calcium are shown in Figure 3.2. In Figure 3.2, the red, blue and black curves represent Adipose Air Calcium 105 Linear attenuation coefficient 100 Figure 3.2: The linear attenuation curves for adipose (red), air (blue) and calcium (black). adipose tissue, air and calcium, respectively, and the xxxx patch corresponds to the area of energy flux that is not equivalent to zero. From Figure 3.2, we can see that the curvatures of air and adipose are similar, while the curve of calcium has a K-edge [52]. The similarity of curvatures between adipose and air might cause the collinearity of linear attenuation coefficient matrix C and so as the ill-conditioning of Hessian, while the K-edge might result in difficulty for reconstruction. The simulations of the true object, shown in Figure 3.3, contain four distinct regions: 100% adipose, 0% air, 0% calcium; 0% adipose, 100% air, 0% calcium; 0% adipose, 0% air, 100% calcium; 50% adipose, 50% air, 0% calcium1. In Figure 3.3, the 1We actually tested many different combinations of mixed materials, for example, 20% adipose, 60% air and 20% calcium. The results are are very similar to the one case considered in this experiment, thus to conserve space, we omit the results. yellow color represents regions that contain 100% of the corresponding material, the turquoise color indicates regions that contain 50% of the adipose and air materials, the blue color indicates that the corresponding material does not exist in this ...
Numerical Experiments. To test the preconditioners and the optimization method, we generate a 2D image of the size 128 by 128 as the object. We also assume that this object is made up of two materials, plexiglass and polyvinyl chloride (PVC). Thus we can obtain 2 material maps corresponding to the weights of these two materials. The original material maps are shown in Figure 4.2. In Figure 4.2, the yellow color represents that it has the corresponding material in this area, while the blue color shows that it does not have the corresponding material in this area. Therefore, we can see that the object is a circle and both materials are distributed inside this circle. Inside this circle, plexiglass dominates most areas except three dots occupied by PVC. We can also 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 see that the images corresponding to these two materials compensate each other and they are completely separable. The goal of this numerical experiment is to reconstruct these two images such that different material maps present the corresponding material compositions.
Numerical Experiments. It is well known that the image deblurring problem requires regularization to stabilize the inversion process when there is noise in y and/or in A. Note that even if the data y has no noise (which is highly unlikely in any real problem), because we use only an approximation of the true boundary ele- ments (e.g, with AZ, AP , AR, AA, or AS), there is effectively noise in A. For the numerical results reported in this section we use standard Tikhonov regularization [25, 33, 36, 68], min }y AXx}2 α}x}2( , where AX is one of AZ, AP , AR, AA, or AS. Our implementation can be ob- tained from RestoreTools1 patched with synthetic boundary conditions mod- ification2 , or Python RestoreTools (PYRET)3. The following experiments are done with the function HyBR (hybrid bidiagonalization regularization) [13, 14], which implements a modified version of LSQR [56], in Restore- Tools. If the true image is known (as we do in our simulations) HyBR can easily compute Tikhonov solutions with optimal regularization parameters. RestoreTools also facilitates the implementation by providing functions to efficiently implement matrix-vector multiplications. In our first set of experiments, we use the “Xxxxxxx” image (Figure 2.3) as the main test image. The following 4 cases are considered: • Gaussian blur (Section 2.4.1) • diagonal motion blur (Section 2.4.2) • Gaussian blur with additive Gaussian noise (Section 2.4.3) • diagonal motion blur with additive Gaussian noise (Section 2.4.4) 1http://xxx.xxxxxx.xxxxx.xxx/~{}xxxx/RestoreTools 2http://xxx.xxxxxx.xxxxx.xxx/~{}yfan/SyntheticBC/SyntheticBcPatch.tgz 3http://xxx.xxxxxx.xxxxx.xxx/~{}yfan/PYRET • DCT based preconditioning with AR (Section 2.4.5)
