Journal of Mathematical Imaging and Vision
Journal of Mathematical Imaging and Vision
Using Catastrophe Theory to Derive Trees from Images
Journal of Mathematical Imaging and Vision
Gradient Structure of Image in Scale Space
Journal of Mathematical Imaging and Vision
Exploring and exploiting the structure of saddle points in Gaussian scale space
Computer Vision and Image Understanding
The Representation and Matching of Images Using Top Points
Journal of Mathematical Imaging and Vision
On manifolds in Gaussian scale space
Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
Content based image retrieval using multiscale top points a feasibility study
Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
Image reconstruction from multiscale critical points
Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
Transitions of multi-scale singularity trees
DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
A comparison of the deep structure of α-scale spaces
DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
Discrete representation of top points via scale space tessellation
Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
Dynamics of a mean-shift-like algorithm and its applications on clustering
Information Processing Letters
Uniqueness Results for Image Reconstruction from Features on Curves in α-Scale Spaces
Journal of Mathematical Imaging and Vision
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The main theorem we present is a version of a "Folklore Theorem" from scale-space theory for nonnegative compactly supported functions from Rn 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.