A Note on Local Morse Theory in Scale Space and Gaussian Deformations

In this note we study the local behavior of singularities occurring in scale space under Gaussian blurring. Based on ideas from singularity theory for vector fields this is done by considering deformations or unfoldings. To deal with the special nature of the problem the concept of Gaussian deformation is introduced. Using singularity theory the stability of these deformations is considered. New concepts of one-sided stability and one-sided equivalence are introduced. This way a classification of stable singularities is obtained which agrees with those known in literature.