By Charles L. Byrne
Iterative Optimization in Inverse Problems brings jointly a few very important iterative algorithms for clinical imaging, optimization, and statistical estimation. It comprises fresh paintings that has now not seemed in different books and attracts at the author’s massive learn within the box, together with his lately built classification of SUMMA algorithms. relating to sequential unconstrained minimization equipment, the SUMMA classification features a wide variety of iterative algorithms renowned to researchers in a number of components, equivalent to facts and photo processing.
Organizing the themes from normal to extra particular, the publication first provides an outline of sequential optimization, the subclasses of auxiliary-function tools, and the SUMMA algorithms. the subsequent 3 chapters current specific examples in additional element, together with barrier- and penalty-function tools, proximal minimization, and forward-backward splitting. the writer additionally specializes in fixed-point algorithms for operators on Euclidean area after which extends the dialogue to incorporate distance measures except the standard Euclidean distance. within the ultimate chapters, particular difficulties illustrate using iterative equipment formerly mentioned. such a lot chapters include routines that introduce new rules and make the booklet compatible for self-study.
Unifying numerous likely disparate algorithms, the publication indicates the best way to derive new homes of algorithms via evaluating recognized homes of alternative algorithms. This unifying process additionally is helping researchers—from statisticians engaged on parameter estimation to snapshot scientists processing scanning information to mathematicians fascinated with theoretical and utilized optimization—discover precious similar algorithms in parts outdoors in their expertise.
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Extra info for Iterative Optimization in Inverse Problems
32) with KL(a, 0) = +∞, and KL(0, b) = b. Extend to nonnegative vectors coordinate-wise, so that J KL(x, z) = KL(xj , zj ). 33) j=1 Then KL(x, z) ≥ 0 and KL(x, z) = 0 if and only if x = z. Unlike the Euclidean distance, the KL distance is not symmetric; KL(Ax, b) and KL(b, Ax) are distinct, and we can obtain diﬀerent approximate solutions of Ax = b by minimizing these two distances with respect to nonnegative x. We discuss this point further in Chapter 11. 4 Convergence of MART In the consistent case, by which we mean that Ax = b has nonnegative solutions, we have the following convergence theorem for MART.
1 Barrier-Function Methods . . . . . . . . . . . . . . . . 2 Penalty-Function Methods . . . . . . . . . . . . . . . Auxiliary-Function Methods . . . . . . . . . . . . . . . . . . . 1 General AF Methods . . . . . . . . . . . . . . . . . . 2 AF Requirements . . . . . . . . . . . . . . . . . . . . 3 Majorization Minimization . . . . . . . . . . . . . . . 4 The Method of Auslander and Teboulle .
It is common in signal processing to speak of the wavelength of a sinusoidal signal; the wavelength associated with a given ω and c is λ= 2πc . 11) Therefore, we can measure cn for |n| not greater than 2L λ , which is the length of the interval [−L, L], measured in units of wavelength λ. We get more Fourier coeﬃcients when the product Lω is larger; this means that when L is small, we want ω to be large, so that λ is small and we can measure more Fourier coeﬃcients. As we saw previously, using these ﬁnitely many Fourier coeﬃcients to calculate the DFT reconstruction of f (x) can lead to a poor estimate of f (x), particularly when we don’t have many Fourier coeﬃcients.