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. R
elevant parts of catastrophe theory in the context of the scale-space parad
igm are briefly reviewed, and subsequently rewritten into coordinate indepe
ndent form. This enables one to implement topological descriptors using a c
onveniently defined coordinate system.