Topological Numbers and Singularities in Scalar Images: Scale-Space Evolution Properties

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Abstract

Singular points of scalar images in any dimensions are classified by a topological number. This number takes integer values and can efficiently be computed as a surface integral on any closed hypersurface surrounding a given point. A nonzero value of the topological number indicates that in the corresponding point the gradient field vanishes, so the point is singular. The value of the topological number classifies the singularity and extends the notion of local minima and maxima in one-dimensional signals to the higher dimensional scalar images. Topological numbers are preserved along the drift of nondegenerate singular points induced by any smooth image deformation. When interactions such as annihilations, creations or scatter of singular points occurs upon a smooth image deformation, the total topological number remains the same.

Our analysis based on an integral method and thus is a nonperturbative extension of the order-by-order approach using sets of differential invariants for studying singular points.

Examples of typical singularities in one- and two-dimensional images are presented and their evolution induced by isotropic linear diffusion of the image is studied.

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