Handbook of Mathematical Methods in Imaging
Language: English
Pages: 2178
ISBN: 1493907891
Format: PDF / Kindle (mobi) / ePub
The Handbook of Mathematical Methods in Imaging provides a comprehensive treatment of the mathematical techniques used in imaging science. The material is grouped into two central themes, namely, Inverse Problems (Algorithmic Reconstruction) and Signal and Image Processing. Each section within the themes covers applications (modeling), mathematics, numerical methods (using a case example) and open questions. Written by experts in the area, the presentation is mathematically rigorous.
This expanded and revised second edition contains updates to existing chapters and 16 additional entries on important mathematical methods such as graph cuts, morphology, discrete geometry, PDEs, conformal methods, to name a few. The entries are cross-referenced for easy navigation through connected topics. Available in both print and electronic forms, the handbook is enhanced by more than 200 illustrations and an extended bibliography.
It will benefit students, scientists and researchers in applied mathematics. Engineers and computer scientists working in imaging will also find this handbook useful.
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! 1: while d dt  1 sin n2 t n à 2 D 2 The reverse diffusion process is also governed by an unbounded operator. As seen in (5), the operator L, which maps the temperature distribution g.x/ D u.x; T / to the initial temperature distribution f .x/ D u.x; 0/, is defined on the subspace ( 1 X D.L/ D g 2 L Œ0; W 2 ) e 2k n2 T 2 jbn j < 1 ; nD1 where bn D 2 Z g.s/ sin ns ds: 0 L is unbounded because the functions 'm .s/ D sin ms reside in D.L/ and satisfy k'm k2 D =2, but by the
(1983) 18. Deutsch, F.: Best Approximation in Inner Product Spaces. Springer, New York (2001) 19. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996) 20. Epstein, C.L.: Introduction to the Mathematics of Medical Imaging. Pearson Education, Upper Saddle River (2003) 21. Galilei, G.: Sidereus Nuncius (Trans.: van Helden, A.). University of Chicago Press, Chicago, 1989 (1610) Linear Inverse Problems 45 22. Gates, E.: Einstein’s Telescope. W.W.
are obtained by applying bounded linear operators R˛ W Y ! X to the data y ı for regularization parameters ˛ > 0, such explicit approach fails if either 94 J. Cheng and B. Hofmann in (1) (a) the forward operator F is nonlinear, (b) the domain D.F / is not a linear subspace of X , or (c) the mapping y ı 7! x˛ı is continuous but nonlinear for all ˛ > 0 even if F is linear. All three sources of nonlinearity make it necessary to define the regularized solutions in an implicit manner. The
logarithmic source conditions or on the method of approximate source conditions (cf. [58]) are not applicable since qualified nonlinearity conditions like (36) cannot be proven according to current knowledge. For a function x W R ! R with support on Œ0; 1, the autoconvolution x x is a function with support on Œ0; 2. Hence, the strength of ill-posedness for the deautoconvolution problem according to the forward operator (58) can be reduced if observations of ŒF .x/.s/ for all 0 Ä s Ä 2 are
discussed from quite a general point of view in Chapter Regularization Methods for Ill-Posed Problems in this handbook. • Variational methods are related to PDE restoration methods and are naturally developed for signals and images defined on a continuous subset Rd , d D 1; 2; : : : I for images d DR 2. Originally, the data-fidelity term is of the form (5) for A D Id and ˆ.u/ D .kDuk2 /dx, where is a convex function as those given in Table 1 (top). Since the beginning of the 1990s, a remarkable
