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In Computed Tomography: Algorithms, Insight, and Just Enough
Theory, readers will learn about the fundamental computational
methods used for image reconstruction in computed tomography (CT).
Unique in its emphasis on the interplay of modeling, computing, and
algorithm development, the book presents underlying mathematical
models for motivating and deriving the basic principles of CT
reconstruction methods, along with insight into their advantages,
limitations, and computational aspects. Computed Tomography:
Algorithms, Insight, and Just Enough Theory: Develops the
mathematical and computational aspects of three main classes of
reconstruction methods. Emphasizes the link between CT and
numerical methods, which is rarely found in current literature.
Describes the effects of incomplete data using both microlocal
analysis and the singular value decomposition (SVD). Contains
computer exercises using MATLAB that allow readers to experiment
with the algorithms and make the book suitable for teaching and
self-study. This book is aimed at students, researchers, and
practitioners. As a textbook, it is appropriate for the following
courses: Advanced Numerical Analysis, Special Topics on Numerical
Analysis, Topics on Data Science, Topics on Numerical Optimization,
and Topics on Approximation Theory.
Chordal graphs play a central role in techniques for exploiting
sparsity in large semidefinite optimization problems, and in
related convex optimization problems involving sparse positive
semidefinite matrices. Chordal graph properties are also
fundamental to several classical results in combinatorial
optimization, linear algebra, statistics, signal processing,
machine learning, and nonlinear optimization. This book covers the
theory and applications of chordal graphs, with an emphasis on
algorithms developed in the literature on sparse Cholesky
factorization. These algorithms are formulated as recursions on
elimination trees, supernodal elimination trees, or clique trees
associated with the graph. The best known example is the
multifrontal Cholesky factorization algorithm but similar
algorithms can be formulated for a variety of related problems,
such as the computation of the partial inverse of a sparse positive
definite matrix, positive semidefinite and Euclidean distance
matrix completion problems, and the evaluation of gradients and
Hessians of logarithmic barriers for cones of sparse positive
semidefinite matrices and their dual cones. This monograph shows
how these techniques can be applied in algorithms for sparse
semidefinite optimization. It also points out the connections with
related topics outside semidefinite optimization, such as
probabilistic networks, matrix completion problems, and partial
separability in nonlinear optimization.
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