Direct Reconstruction. The first experimental framework we explore is a direct image reconstruction using the baseline approximation of a blur operator. We first explain the setup that we will reuse throughout this work, then show the results for the baseline method. Using the same image deconvolution framework discussed in Section 2.1, we tested each SVD approximation method from Chapters 3-6 on three different blur operators. The first example we refer to as the Satellite Example, the second as the Grain Example, and the third as the Motion Example. The PSFs for each example are shown in Figure 3.1. For each example, we show the true image, the restoration with negative values set to zero (which typically hides noise in the restoration), the restoration with negative values kept, and the blurred image. We also give the peak signal-to-noise ratio (PSNR, see [1, 2]) compared to the true satellite image shown in Figure 3.2; a higher PSNR indicates a better reconstruction. Note that the images are displayed using MATLAB’s imagesc command, so the resulting visualizations have deep blue as the minimum value and yellow as the largest value in the image, regardless of the actual intensities. In all cases, the image is a 64 × 64 satellite image available from the RestoreTools package [39], and a full r = 64 is used. These are the constants among all examples. “Satellite” PSF “Grain” PSF “Motion” PSF Figure 3.1: PSFs for restorations. The three PSFs used for our image deconvolution examples. We vary a few aspects of the restorations between examples. These changing parameters are summarized in Table 3.1. Firstly, the PSFs are varied as shown in
Direct Reconstruction. We look at the same same reconstruction examples as we did in Subsection 3.4.1, starting with the Satellite Example. In this example, the baseline method gave a good approximation, and this continues to be the case with the truncation method. For this method, we split the truncation index k = 600 into A = 25 and m = 24. The result of the reconstruction is shown in Figure 4.1 below. The PSNR for this example is 46.9 dB, which matches the PSNR for the baseline method on this example. True x Restored x > 0 Restored x Blurred d Figure 4.1: Truncation Satellite Example Restoration. The truncation method yields a good reconstruction of the blurred image. In particular, the reconstruction with negative values truncated, the second from the left, is highly accurate, only missing the fine details of the original. As with the baseline method, the truncation method performs well for this case. There is a small amount of noise, discernible in the image that includes negative values in what should be the darkest regions of the image, but the object is still clearly visible. Most of the examples here use a sufficiently high truncation index to get reasonable reconstruction results. What happens if the truncation index is dropped? We will look at this question for the Satellite Example in this and the next chapter to contrast these two proposed methods, and understand how the truncation method performs with different truncation indices. If we restrict the truncation index to k = 256 with A = 16 = m, we get the reconstruction shown in Figure 4.2, which includes negative values to show the noise pattern more clearly.
Direct Reconstruction. The first test of the reordering method’s performance is its ability to reconstruct images when used as a direct method. It performs well, within the limitation that computing more singular values and vectors incurs larger cost. With a limited set of singular values and vectors, the reconstructions vary from good to borderline. In the Satellite Example, the reconstruction is about as good as the reconstruc- tions created by the baseline method, with slightly less fine detail due to inclusion of different singular vectors. This can be seen in Figure 5.1 below. This result is unsur- prising, as we expect the reordering method to work well for this example. The PSNR for this example is 45.1 dB, which is just under that of the baseline and truncation methods which both have a PSNR of 46.9 dB. True x Restored x > 0 Restored x Blurred d Figure 5.1: Reordering Satellite Example Restoration. The reordering method yields a good reconstruction for this example. Before proceeding to the Grain Example, we revisit the effect of restricting the Satellite Example to a smaller k = 256 terms. In Figure 5.2, we show enlarged versions of the reconstructions created using the truncation and reordering method. Different singular vectors are used in each restoration. The truncation method has a notable ringing effect, whereas the reordering method has different, more uniform noise pattern. The truncated method reconstructs a slimmer set of solar panels, whereas the reordering method gives wider, more uniformly accurate reconstructions of the solar panels. The truncation method has hints of the thin rod; the reordering method does not. Overall, the truncation method restoration is likely preferable. Truncated True Reordered Figure 5.2: Truncated versus Reordered Method for Restricted Satellite Example. This figure shows the truncated method restoration, true image, and reordering method restoration for the Grain Example. The truncation method has more fine details, but is less accurate on coarser details. The noise pattern differs between the restorations as well. The upshot is that it can be difficult to know ahead of time which method will give better performance for a specific application. The truncation method naturally includes singular vectors with lower singular values, so it reconstructs more fine de- tail. But, this can come at the expense of some of the broader data in the image. The restoration results depend on the characteristics of the blur operator and PSF (part...
