Springer, 1993. — 461 p.Scale is a concept the antiquity of which can hardly be traced. Certainly the familiar phenomena that accompany scale changes in optical patterns are mentioned in the earliest written records. The most obvious topological changes such as the creation or annihilation of details have been a topic of fascination to philosophers, artists and later scientists. This appears to be the case for all cultures from which extensive written records exist. For instance, Chinese 17thc artist manuals remark that "distant faces have no eyes". The merging of details is also obvious to many authors, e.g., Lucretius mentions the fact that distant islands look like a single one. The one topological event that is (to the best of my knowledge) mentioned only late (by John Ruskin in his "Elements of drawing" of the mid 19thc) is the splitting of a blob on blurring. The change of images on a gradual increase of resolution has been a recurring theme in the arts (e.g., the poetic description of the distant armada in Calderon's The Constant Prince) and this "mystery" (as Ruskin calls it) is constantly exploited by painters. It was facts as these that induced me to attempt a mathematical study of scale, especially when my empirical researches in the psychophysics of spatial contrast in the peripheral visual field of human observers convinced me that the brain samples and represents the optic array at many scales simultaneously. Events accelerated around 1980. I remember that I met Andy Witkin at the 1st David Marr memorial workshop thrown by Whitman Richards at Cold Spring Harbor: I believe that there I heard the term "scale-space" for the F t time. By then I had set up the diffusion equation formulation that connects the scales formally though I didn't publish because the whole thing still had too much ad hockery to it. However, shortly thereafter the "causality constraint" occurred to me (triggered by a fascination with cartographic "generalization") and I saw the pieces dick together except for a few frayed ends. (Although the notion of "causality" in the scale domain remains indeed a corner stone of the theory in my opinion, its original formulation-though correct-was perhaps unfortunate because often misunderstood even to the point where people believe it to be false.) Since then much has happened and I feel that today "scale-space" can stand on its own. "Pure scale-space" can be constructed from a few basic assumptions that essentially express total a priori ignorance. Thus no scale, place or orientation is potentially special in any way. The resulting pristine structure has the beauty and fascination of something inevitable, a discovery rather than a mere construction. However, it tends to leave practical people unsatisfied because in any real application one knows many things that might advantageously be exploited in the analysis. This might be any form of prior knowledge, including rather abstract notions of what structure to expect, so that the actual data can be used to control the operations performed on themselves. It seems natural to try to use the freedom left in pure scale-space to attach such control handles and thereby turn the nymph into a handmaiden. It is perhaps fair to say that no one today knows how to proceed in an apparently necessary way here, thus the theory sprouts into a multitude of complementary and concurrent approaches. One thing that the reader will notice is the nature of the pictures that go to illustrate many of the contributions: They are quite distinct in character from the type of illustration one meets in the pure diffusion literature. The difference lies in the very sharpness of the "blurred" images. Here we meet with the same fascination one finds in e.g., Canaletto's figures in his paintings of Venice's squares: As the viewer looks at the painted figures farther and farther away in the perspective of the pavement details are lost though the figures are always made up of sharply delimited blobs of paint. In the near foreground a single blob may stand for the button of a waistcoat, in the far distance a whole head, yet everything appears mysteriously "sharp" at any distance. It is as with cartographic generalization where the general shape of a city may suddenly give way to a circular discoLinear Scale-Space I: Basic Theory. Linear Scale-Space II: Early Visual Operations. Anisotropic Diffusion. Vector-Valued Diffusion. Bayesian Rationale for the Variational Formulation. Variational Problems with a Free Discontinuity Set. Minimization of Energy Functional with Curve-Represented Edges. Approximation, Computation, and Distortion in the Variational Formulation. Coupled Geometry-Driven Diffusion Equations for Low-Level Vision. Morphological Approach to Multiscale Analysis: From Principles To Equations. Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach. On Optimal Control Methods in Computer Vision and Image Processing. Nonlinear Scale-Space. A Differential Geometric Approach to Anisotropic Diffusion. Numerical Analysis of Geometry-Driven Diffusion Equations.
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