The topological structure of scale-space images

We investigate the deep structure of a scale-space image. The emphasis is on topology, i.e. we concentrate on critical points—points with vanishing gradient—and top-points—critical points with degenerate Hessian—and monitor their displacements, respectively generic morsifications in scale-space. Relevant parts of catastrophe theory in the context of the scale-space paradigm are briefly reviewed, and subsequently rewritten into coordinate independent form. This enables one to implement topological descriptors using a conveniently defined coordinate system.