Discrete Tomography: Foundations, Algorithms, and by Gabor T. Herman

By Gabor T. Herman

The visualization, building, and reconstruction of multidimensional photos are of extreme curiosity in technology and engineering this present day, and discrete tomography—which offers with the unique case during which the item to be reconstructed has a small variety of attainable values—offers a few major new analytical and computational tools.

Discrete Tomography: Foundations, Algorithms, and purposes provides a serious survey of recent tools, algorithms, and choose functions which are the rules of multidimensional picture development and reconstruction. The survey chapters, written through major foreign gurus, are self-contained adn current the newest study and leads to the sphere. The e-book covers 3 major parts: vital theoretical effects, to be had algorithms to make use of for reconstruction, and key functions the place new effects are indicative of better application. Following an intensive ancient review of the sector, the publication presents a trip throughout the quite a few mathematical and computational difficulties of discrete tomography. this is often by way of a piece on various algorithmic suggestions that may be used to accomplish genuine reconstructions from picture projections.

Topics and Features:

* historic review and precis chapter

* strong point and complexity in discrete tomography

* probabilistic modeling of discrete images

* binary tomography utilizing Gibb priors

* discrete tomography at the three-D torus and crystals

* binary steering

* 3D tomographic reconstruction from sparse radiographic data

* symbolic projections

The e-book is an important source for the newest advancements and instruments in discrete tomography. execs, researchers, and practitioners in arithmetic, laptop imaging, medical visualization, computing device engineering, and multidimensional snapshot processing will locate the e-book an authoritative consultant and connection with present study, tools, and applications.

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Additional info for Discrete Tomography: Foundations, Algorithms, and Applications

Sample text

Let R = (rl,"" r m ) and S = (SI,"" Sn) be a pair of compatible vectors. 1. n n j=1 j=1 L sj ~ L Sj, for 2::; l ::; n. (R, S) contains a binary matrix A. (R, S') contains a binary matrix A' constructed from A by a suitable permuta- tions of the columns. A can be obtained from A' (if they are different at all) by shifting 1's to the left in the rows of A'. Therefore we have (1. 7). 7) is true for the vectors Rand S. We are going to construct a binary matrix A by the following algorithm, whose output is illustrated in Fig.

Rosenfeld and A. Kak, "Digital Picture Processing," (Academic Press, New York) 1976. [60] G. T. Herman, "Geometry of Digital Spaces," (Birkhiiuser, Boston) 1998. [61] A. Del Lungo, M. Nivat and R. Pinzani, "The number of convex polyominoes reconstructible from their orthogonal projections," Discrete Math. 157,65-78 (1996). [62] E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, "Reconstructing convex polyominoes from horizontal and vertical projections," Theor. Comput. Sci. 155, 321-347 (1996).

9. Let A E 2(Q(R,S). 23) (k 2: 2). The corresponding switching (operation) is defined as changing the O's and the 1 's at all positions in the switching chain. It is clear that switching in a matrix A does not change the row and column sums of A. As an example, see the following matrices generated from each other by switching (the prescribed elements are denoted by x). 24) lOx The following theorems can be proven in a similar way as for the class 2(R, S) (see [49]). 8. A binary matrix with prescribed values is nonunique if and only if it has a switching chain.

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