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We consider a region Ω 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” S E which measure how U bends along M. These are not traditional(More)
Blum's medial axes have great strengths, in principle, in intuitively describing object shape in terms of a quasi-hierarchy of figures. But it is well known that, derived from a boundary, they are damagingly sensitive to detail in that boundary. The development of notions of spatial scale has led to some definitions of multiscale medial axes different from(More)
We consider the average outward flux through a Jordan curve of the gradient vector field of the Euclidean distance function to the boundary of a 2D shape. Using an alternate form of the divergence theorem, we show that in the limit as the area of the region enclosed by such a curve shrinks to zero, this measure has very different behaviours at medial points(More)
Pizer and Eberly introduced the “core” as the analogue of the medial axis for greyscale images. For two-dimensional images, it is obtained as the “ridge” of a “medial function” defined on 2 + 1-dimensional scale space. The medial function is defined using Gaussian blurring and measures the extent to which a point is in the center of the object measured at a(More)
For contractible regions ωin ℝ3 with generic smooth boundary, we determine the global structure of the Blum medial axis M. We give an algorithm for decomposing M into “irreducible components” which are attached to each other along “fin curves”. The attaching cannot be described by a tree structure as in the 2D case. However, a simplified but topologically(More)
A new approach is presented for computing the interior medial axes of generic regions in R 3 bounded by C (4)-smooth parametric B-spline surfaces. The generic structure of the 3D medial axis is a set of smooth surfaces along with a singular set consisting of edge curves, branch curves, fin points and six junction points. In this work, the medial axis(More)
In deformable model segmentation, the geometric training process plays a crucial role in providing shape statistical priors and appearance statistics that are used as likelihoods. Also, the geometric training process plays a crucial role in providing shape probability distributions in methods finding significant differences between classes. The quality of(More)
The discrete m-rep, a medial representation of anatomical objects made from one or more meshes of medial atoms, has many attractive properties for biomedical image analysis. The nonlinear nature of the m-rep parameters captures nonlinear deformations in anatomical objects such as twisting and bending. Most uses of m-reps require extending them to one or(More)
We seek a form of object model that exactly and completely captures the interior of most non-branching anatomic objects and simultaneously is well suited for probabilistic analysis on populations of such objects. We show that certain nearly medial, skeletal models satisfy these requirements. These models are first mathematically defined in continuous(More)
In non-Euclidean data spaces represented by manifolds (or more generally stratified spaces), analogs of principal component analysis can be more easily developed using a backwards approach. There has been a gradual evolution in the application of this idea from using increasing geodesic subspaces of submanifolds in analogy with PCA to using a “backward(More)