Journal of Mathematical Imaging and Vision
Topological Numbers and Singularities in Scalar Images: Scale-Space Evolution Properties
Journal of Mathematical Imaging and Vision
Generic structure of two-dimensional dimages under Gaussian blurring
SIAM Journal on Applied Mathematics
Relative critical sets in RR(N) and applications to image analysis
Relative critical sets in RR(N) and applications to image analysis
Scale-Space Theory in Computer Vision
Scale-Space Theory in Computer Vision
Generic transitions of relative critical sets in parametrized families with applications to image analysis
The Relevance of Non-Generic Events in Scale Space Models
International Journal of Computer Vision
A Trained Spin-Glass Model for Grouping of Image Primitives
IEEE Transactions on Pattern Analysis and Machine Intelligence
Journal of Signal Processing Systems
3D winding number: theory and application to medical imaging
Journal of Biomedical Imaging - Special issue on modern mathematics in biomedical imaging
Multi-scale midline extraction using creaseness
ICAPR'05 Proceedings of the Third international conference on Pattern Recognition and Image Analysis - Volume Part II
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We propose a new computational method for segmenting topological subdimensional point-sets in scalar images of arbitrary spatial dimensions. The technique is based on calculating the homotopy class defined by the gradient vector in a subdimensional neighborhood around every image point. This neighborhood is defined as the linear envelope spawned over a given subdimensional vector frame. In the simplest case where the rank of this frame is maximal, we obtain a technique for localizing the critical points, i.e., extrema and saddle points. We consider, in particular, the important case of frames formed by an arbitrary number of the first largest by absolute value principal directions of the Hessian. The method then segments positive and and negative ridges as well as other types of critical surfaces of different dimensionalities. The signs of the eigenvalues associated to the principal directions provide a natural labeling of the critical subsets. The result, in general, is a constructive definition of a hierarchy of point-sets of different dimensionalities linked by inclusion relations. Because of its explicit computational nature, the method gives a fast way to segment height ridges or edges in different applications. The defined topological point-sets are connected manifolds and, therefore, our method provides a tool for geometrical grouping using only local measurements. We have demonstrated the grouping properties of our construction by presenting two different cases where an extra image coordinate is introduced. In one of the examples, we considered the image analysis in the framework of the linear scale-space concept, where the topological properties are gradually simplified through the scale parameter. This scale parameter can be taken as an additional coordinate. In the second example, a local orientation parameter was used for grouping and segmenting elongated structures.