By Charles L Byrne
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Additional resources for Applied iterative methods
In the latter case, the attenuation function is assumed to be constant over small pixels or voxels; these constants are the object of interest now. The problem has been reduced to solving a large system of linear equations, possibly subject to nonnegativity or other constraints. If the physical nature of the radiation is described using a statistical model, then the pixel values can be viewed as parameters to be estimated. The well-known maximum likelihood parameter estimation method can then be employed to obtain these pixel values.
We say that d ≥ 0 is the distance from the point x to the set B if, for every > 0, there is b in B, with ||x − b ||2 < d + , and no b in B with ||x − b||2 < d. It follows easily that, if B is closed and d = 0, then x is in B. The following proposition is fundamental in the study of convexity and can be found in most books on the subject; see, for example, the text by Goebel and Reich . 6. Given any nonempty closed convex set C and an arbitrary vector x in X , there is a unique member of C closest to x, denoted PC x, the orthogonal (or metric) projection of x onto C.
7. Show that the 1-norm is a norm. 8. Show that the ∞-norm is a norm. 9. Show that the 2-norm is a norm. Hint: For the triangle inequality, use Cauchy’s Inequality. 10. Show that the Q-weighted 2-norm is a norm. 11. Show that ρ(S 2 ) = ρ(S)2 . ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 48 4. 12. Show that, if S is Hermitian, then every eigenvalue of S is real. Hint: Suppose that Sx = λx. Then consider x† Sx. 13. Use the SVD of A to obtain the eigenvalue/eigenvector decompositions of B and C: N λi ui (ui )† , B= i=1 and N λi v i (v i )† .