Approximated Singular Values. As with the baseline method, we ask: how well does the truncation method (4.2) approximate the singular values of the matrix K? There is a pattern evident in both the Singular Value Atmospheric Blur Example and the Singular Value Motion Example: the truncation method consistently underestimates the smallest singular values. Recall that the truncation method keeps singular values of K1 that are less significant than some of the singular values it discards. The expected result is an under-estimate of the smallest singular values kept. We see this clearly in the Singular Value Atmospheric Blur Example, shown in Figure 4.7. The error starts out smaller than the baseline method produces, but then rises to be higher than the baseline for the smallest singular values. To see a comparison of all methods’ singular value approximations, see Appendix B. 0.015 Truncated singular values Calculated True
Approximated Singular Values. The second test of the reordering method (5.1) is its ability to approximate singular values. Its performance is problem-dependent as seen with the baseline and truncation methods. The first test is the Singular Value Atmospheric Blur Example. For this problem, the reordering method gives a close approximation of the singular values for most of the values, but the approximation worsens near the tail. Indeed, the relative error in the computed singular values increases as the singular value index increases, as seen in Figure 5.7 below. Notably, compared to the truncation method, the singular values start to significantly under-estimate the true singular values at a later point. The inflection here is around index 80, whereas for the truncation method this in- flection occurs around index 60. Otherwise, their performance is similar, making the reordering method more accurate overall. Compared to the baseline method, the reordering method has lower error for the highest singular values, and higher error for the smallest singular values. Appendix B compares all methods simultaneously.
