This book deals with vision as a computational problem. It presents visual reconstruction from the mechanical viewpoint, which is more natural for representation of a priori knowledge about visible surfaces or about distributions of visual quantities such as intensity, reflectance, optic flow, and curve orientation. An important class of reconstruction processes is presented. Two new concepts are introduced, analyzed, and illustrated: the weak continuity constraint in vision and the graduated nonconvexity algorithm for fitting piecewise continuous functions to visual data.
The chapters of the book are as follows:
(1) Modeling Piecewise Continuity
(2) Applications of Piecewise Continuous Reconstructions
(3) Introduction to Weak Continuity Constraints
(4) Properties of the Weak String and Membrane
(5) Properties of the Weak Rod and Plates
(6) The Discrete Problem
(7) The Graduated Non Convexity Algorithm and
(8) Conclusion
The first chapter presents visual reconstruction as a process of reducing visual data to stable descriptions. Problems discussed in chapter 2 are the applications of weak continuity constraints in visual reconstruction and include edge detection, stereoscopic vision, passive range finding, and describing curves. The next two chapters present the weak string as a one-dimensional discontinuity detecting filter and the weak membrane as its two-dimensional equivalent. Variational analysis, applied to certain special cases, leads to the notions of scale, contrast threshold, and gradient limit. But they cannot detect crease discontinuities. In chapter 5 it is shown that the rod and the plate can detect steps and creases simultaneously. Chapter 6 deals with energy minimization as a variational problem. For computational purposes, it is made discrete by using finite elements, together with line variables to handle discontinuities. Also, the various methods available for solving nonconvex problems are reviewed. The graduated nonconvexity algorithm--the method proposed in this book--is described in chapter 7. Convergence properties of the algorithm are also studied.
The basic purpose of this book is to present an original approach to the treatment of continuity in vision and to avoid obstructing the text with undue mathematical detail. The mathematics of weak continuity and the graduated nonconvexity algorithm are carefully developed in appendices. The mentioned aims are thoroughly fulfilled. The new methods of modeling continuity open new vistas on computer and biological vision, too.
To best understand the themes of the book, the reader should be familiar with the mathematics of the variational analysis. The references are new and good, and the index is adequate. Exercises are not included, and this book is not intended for a classroom textbook.
