Edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them. A critical, intermediate goal of edge detection is the detection and characterization of significant intensity changes. This paper discusses this part of the edge detection problem. To characterize the types of intensity changes derivatives of different types, and possibly different scales, are needed. Thus, we consider this part of edge detection as a problem in numerical differentiation. We show that numerical differentiation of images is an ill-posed problem in the sense of Hadamard. Differentiation needs to be regularized by a regularizing filtering operation before differentiation. This shows that this part of edge detection consists of two steps, a filtering step and a differentiation step. Following this perspective, the paper discusses in detail the following theoretical aspects of edge detection. 1) The properties of different types of filters-with minimal uncertainty, with a bandpass spectrum, and with limited support-are derived. Minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties. 2) Relationships among several 2-D differential operators are established. In particular, we characterize the relation between the Laplacian and the second directional derivative along the gradient. Zero crossings of the Laplacian are not the only features computed in early vision. 3) Geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and Morse theory.

R. Abraham and J. Robbin, Transversal Mappings and Flows. X + 161 S. m. Fig. New York/Amsterdam 1967. W. A. Benjamin, Inc. Preis geb. $ 12,50

On edge detection

/pdf/sur-la-conjugaison-differentiable-des-diffeomorphismes-du-2klxn4vqsv.pdf

Sur la Conjugaison Différentiable des Difféomorphismes du Cercle a des Rotations

On Edge Detection

These lectures introduce the modern theory and practical numerical methods for continuation of solutions of nonlinear problems depending upon parameters. Bifurcations are one of the many types of singularities that occur along such solution paths and their computation and methods for switching branches are treated. Homotopy methods and degree theory are introduced as are global Newton methods, constructive determination of Brouwer fixed points, periodic solutions of O.D.E.s, Hopf bifurcations, etc. The treatment is elementary and basic. Advanced calculus and linear algebra are sufficient to understand almost all of the material. A chapter showing applications and numerical methods for nonlinear O.D.E.s is included.

Lectures on Numerical Methods in Bifurcation Problems

In this paper, we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real, axis, for semiclassical operators like −h 2 Δ+V(x), when h is small

R. Abraham and J. Robbin, Transversal Mappings and Flows. X + 161 S. m. Fig. New York/Amsterdam 1967. W. A. Benjamin, Inc. Preis geb. $ 12,50

Citations

On edge detection

Sur la Conjugaison Différentiable des Difféomorphismes du Cercle a des Rotations

On Edge Detection

Lectures on Numerical Methods in Bifurcation Problems

Geometric bounds on the density of resonances for semiclassical problems

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