Ridges, crests and sub-parabolic lines of evolving surfaces
International Journal of Computer Vision
Zoom-invariant vision of figural shape: the mathematics of cores
Computer Vision and Image Understanding
Generic structure of two-dimensional dimages under Gaussian blurring
SIAM Journal on Applied Mathematics
Properties of Ridges and Cores for Two-Dimensional Images
Journal of Mathematical Imaging and Vision
Relative critical sets in RR(N) and applications to image analysis
Relative critical sets in RR(N) and applications to image analysis
International Journal of Computer Vision
Symmetry Sets and Medial Axes in Two and Three Dimensions
Proceedings of the 9th IMA Conference on the Mathematics of Surfaces
Three-dimensional shape description using the symmetric axis transform
Three-dimensional shape description using the symmetric axis transform
Generic transitions of relative critical sets in parametrized families with applications to image analysis
Deformable M-Reps for 3D Medical Image Segmentation
International Journal of Computer Vision - Special Issue on Research at the University of North Carolina Medical Image Display Analysis Group (MIDAG)
Multiscale Medial Loci and Their Properties
International Journal of Computer Vision - Special Issue on Research at the University of North Carolina Medical Image Display Analysis Group (MIDAG)
Efficient Shape Modeling: ⋮-Entropy, Adaptive Coding, and Boundary Curves -vs- Blum's Medial Axis
International Journal of Computer Vision
Tree Structure for Contractible Regions in R3
International Journal of Computer Vision
Swept regions and surfaces: Modeling and volumetric properties
Theoretical Computer Science
Distance functions and skeletal representations of rigid and non-rigid planar shapes
Computer-Aided Design
Sampled medial loci for 3D shape representation
Computer Vision and Image Understanding
IPMI'05 Proceedings of the 19th international conference on Information Processing in Medical Imaging
A family of skeletons for motion planning and geometric reasoning applications
Artificial Intelligence for Engineering Design, Analysis and Manufacturing - Representing and Reasoning About Three-Dimensional Space
Estimating the statistics of multi-object anatomic geometry using inter-object relationships
DSSCV'05 Proceedings of the First international conference on Deep Structure, Singularities, and Computer Vision
Medial zones: Formulation and applications
Computer-Aided Design
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
Medial axis transform of a planar domain with infinite curvature boundary points
Computer Aided Geometric Design
Spaces and manifolds of shapes in computer vision: An overview
Image and Vision Computing
Research on 3D medial axis transform via the saddle point programming method
Computer-Aided Design
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We consider a region 驴 in R2 or R3 with generic smooth boundary B and Blum medial axis M, on which is defined a multivalued "radial vector field" U from points x on M to the points of tangency of the sphere at x with B. We introduce a "radial shape operator" Srad and an "edge shape operator" SE which measure how U bends along M. These are not traditional differential geometric shape operators, nonetheless we derive all local differential geometric invariants of B from these operators.This allows us to define from (M, U) a "geometric medial map" on M which corresponds, via a "radial map" from M to B, to the differential geometric properties of B. The geometric medial map also includes a description of the relative geometry of B. This is defined using the "relative critical set" of the radius function r on M. This set consists of a network of curves on M which describe where B is thickest and thinnest. It is computed using the covariant derivative of the tangential component of the unit radial vector field.We further determine how these invariants are related to the differential geometric invariants of M and how these invariants change under deforming diffeomorphisms of M.