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Scale-based geometry for problems in computer imaging

Monday 27th November 2000 - 17:00 to 18:00
INI Seminar Room 1
For a number of different problems in computer imaging, we wish to be able to analyze the geometry of the objects in the image. Typical questions might concern: segmentation, which is the decomposition of an image into distinct objects and regions; registration, which is the alignment of different images of the same object; identification and description of features of the distinct objects in an image; etc. These goals must be accomplished in the presence of noise and various distortions. Hence, geometric structures which we associate to objects in images must not only capture important geometric features; but importantly for "most objects", they must be robust, i.e. they are stable under sufficiently small perturbations, and provide coherent structures with predictable properties.

We describe some of these problems and how methods of geometry and topology can be combined with methods of analysis to begin to provide answers to such questions. We recall the Blum medial axis and Canny edges as prototypes for more general structures. These have inherent limitations for "real world" problems, so we describe Witkin's approach and work of Koenderink which introduced "scale" as a basic parameter in images.

Scale can be introduced either via solutions to partial differential equations or by applying appropriate filters. We explain a framework based on methods of singularity theory which ensures the stability and genericity of scale-based geometric properties. Within this scale-based framework robust geometric methods can be applied to nondifferentiable objects. We illustrate this with explicit medial and edge based geometric structures which may be applied to both real world objects and the mathematical entities modeling them.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons