On the Behavior of Spatial Critical Points under Gaussian Blurring. A Folklore Theorem and Scale-Space Constraints

The main theorem we present is a version of a “Folklore Theorem” from scale-space theory for nonnegative compactly supported functions from R to R. The theorem states that, if we take the scale in scale-space sufficiently large, the Gaussian-blurred function has only one spatial critical extremum, a maximum, and no other critical points. Two other interesting results concerning nonnegative compactly supported functions, we obtain are: 1. a sharp estimate, in terms of the radius of the support, of the scale after which the set of critical points consists of a single maximum; 2. all critical points reside in the convex closure of the support of the function. These results show, for example, that all catastrophes take place within a certain compact domain determined by the support of the initial function and the estimate mentioned in 1. To illustrate that the restriction of nonnegativity and compact support cannot be dropped, we give some examples of functions that fail to satisfy the theorem, when at least one assumption is dropped.