Geometric Methods and Applications

Abstract

PrefaceThis book is an introduction to fundamental geometric concepts and tools needed for solving problems of a geometric nature with a computer. Our main goal is to present a collection of tools that can be used to solve problems in computer vision, robotics, machine learning, computer graphics, and geometric modeling.During the ten years following the publication of the first edition of this book, optimization techniques have made a huge comeback, especially in the fields of computer vision and machine learning. In particular, convex optimization and its special incarnation, semidefinite programming (SDP), are now widely used techniques in computer vision and machine learning, as one may verify by looking at the proceedings of any conference in these fields. Therefore, we felt that it would be useful to include some material (especially on convex geometry) to prepare the reader for more comprehensive expositions of convex optimization, such as Boyd and Vandenberghe [2], a masterly and encyclopedic account of the subject. In particular, we added Chapter 7, which covers separating and supporting hyperplanes.We also realized that the importance of the SVD (singular value decomposition) and of the pseudo-inverse had not been sufficiently stressed in the first edition of this book, and we rectified this situation in the second edition. In particular, we added sections on PCA (principal component analysis) and on best affine approximations and showed how they are efficienlty computed using SVD. We also added a section on quadratic optimization and a section on the Schur complement, showing the usefulness of the pseudo-inverse.In this second edition, many typos and small mistakes have been corrected, some proofs have been shortened, some problems have been added, and some references have been added. Here is a list containing brief descriptions of the chapters that have been modified or added.• Chapter 3, on the basic properties of convex sets, has been expanded. In particular, we state a version of Carathéodory's theorem for convex cones (Theorem 3.2), a version of Radon's theorem for pointed cones (Theorem 3.6), and Tverberg's theorem (Theorem 3.7), and we define centerpoints and prove their existence (Theorem 3.9). 3), and we prove rigorously how SVD yields PCA (Theorem 14.3), using the Rayleigh-Ritz ratio (Lemma 14.2). In Section 14.4, it is shown how to best approximate a set of data with an affine subspace in the least squares sense. Again, SVD can used to find solutions.• Chapter 15 is new, except for Section 15.1, which reproduces Section 13.2 from the first edition of this book. We added the definition of the positive semidefinite cone ordering, , on symmetric matrices, since it is extensively used in convex optimization. In Section 15.2, we find a necessary and sufficient condition (Proposition 15.2) for the quadratic function f (x) = 1 2 x Ax + x b to have a minimum in terms of the pseudo-inverse of A (where A is a symmetric matrix). We also show how to accommodate linear constraints of the form C x = 0 or...

Nonorthogonality constraints in open quantum and wave systems

“…is a so-called Hermitian form (or symmetric sesquilinear form) [46] because (i) it is linear in one argument (here the second, according to the standard physics notation for the scalar product) and (ii) symmetric under exchange of arguments and complex conjugation, i.e., (φ, ψ) = (ψ, φ) * .…”

“…[20,45] is not justified in general as a decaying quantum system can have non-decaying subspaces. For a positive semi-definite Hermitian form the Cauchy-Schwarz inequality holds, see, e.g., [46],…”

Real-Time Tool for Affine Transformations of Two Dimensional IFS Fractal

“…The set of the three points , , is said to be affine basis of the affine space of points if the set , is a vector base of , observed as a vector space [13,14]. We say that the point has barycentric coordinates , , relative to the basis , , , where 1, and we write (1) if and only if one of the following three equivalent conditions holds:…”

The exponential map for the group of similarity transformations and applications to motion interpolation

“…Since α > 0, we let λ = log α. Since R = I, it is known (for example, see Gallier [6] Theorem 18.1) that the exponential map exp : so(n) → SO(n) is surjective, and by the reasoning in the proof of Proposition 18.3 in Gallier [6], we may assume that the nonzero eigenvalues iθ j of the skew symmetric matrix Ω such that e Ω = R are such that θ j = k2π for all k ∈ Z. In fact, we may assume that 0 < θ j < π.…”