Medial axis approximation and unstable flow complex

Giesen, Joachim; Ramos, Edgar A.; Sadri, Bardia

doi:10.1145/1137856.1137905

Cited by 18 publications

(15 citation statements)

References 12 publications

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“…At a conceptual level, our advection-based importance measure is related to the flow complex by Giesen et al [46], [47]. We briefly outline next the similarities and differences.…”

Section: Comparison With Other Methodsmentioning

confidence: 99%

Computing Multiscale Curve and Surface Skeletons of Genus 0 Shapes Using a Global Importance Measure

Reniers

Wijk

Telea

2009

IEEE Trans. Comput.

108

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We present a practical algorithm for computing robust, multiscale curve and surface skeletons of 3D objects. Based on a model which follows an advection principle, we assign to each point on the skeleton a part of the object surface, called the collapse. The size of the collapse is used as a uniform importance measure for the curve and surface skeleton, so that both can be simplified by imposing a single threshold on this intuitive measure. The simplified skeletons are connected by default, without special precautions, due to the monotonicity of the importance measure. The skeletons possess additional desirable properties: They are centered, robust to noise, hierarchical, and provide a natural skeleton-to-boundary mapping. We present a voxel-based algorithm that is straightforward to implement and simple to use. We illustrate our method on several realistic 3D objects.

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“…At a conceptual level, our advection-based importance measure is related to the flow complex by Giesen et al [46], [47]. We briefly outline next the similarities and differences.…”

Section: Comparison With Other Methodsmentioning

confidence: 99%

Computing Multiscale Curve and Surface Skeletons of Genus 0 Shapes Using a Global Importance Measure

Reniers

Wijk

Telea

2009

IEEE Trans. Comput.

108

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“…Using the original separation result of [DGRS05], Theorem 4 can be generalized to allow adaptive samples, where the sampling density varies with the local feature size. The proof essentially remains unchanged modulo using the adaptive analogue of 3 proven in [GRS06]. For adaptive noisy samples, one can achieve a similar result using a corresponding separation theorem of [CL06].…”

Section: Discussionmentioning

confidence: 87%

“…The following lemma is an adaptation of a similar lemma from [GRS06] for uniforms samples. The proof is provided in the Appendix.…”

Section: Shallow Versus Deep Critical Pointsmentioning

confidence: 99%

“…between the union of stable manifolds of deep critical points and the complement of the manifold, is the core result of this paper and its raison d'etre. Standard distance flow arguments as those used in [Lie04, CL06, GRS06] fail in this case; see Section 5. We thus use a novel proof method that successively applies such flow arguments on a family of intermediate sets that are indexed by the indices of shallow critical points.…”

Section: Introductionmentioning

confidence: 99%

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Manifold Homotopy via the Flow Complex

Sadri

2009

Computer Graphics Forum

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It is known that the critical points of the distance function induced by a dense sample P of a submanifold Σ of R n are distributed into two groups, one lying close to Σ itself, called the shallow, and the other close to medial axis of Σ, called deep critical points. We prove that under (uniform) sampling assumption, the union of stable manifolds of the shallow critical points have the same homotopy type as Σ itself and the union of the stable manifolds of the deep critical points have the homotopy type of the complement of Σ. The separation of critical points under uniform sampling entails a separation in terms of distance of critical points to the sample. This means that if a given sample is dense enough with respect to two or more submanifolds of R n , the homotopy types of all such submanifolds together with those of their complements are captured as unions of stable manifolds of shallow versus those of deep critical points, in a filtration of the flow complex based on the distance of critical points to the sample. This results in an algorithm for homotopic manifold reconstruction when the target dimension is unknown.

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“…In [31], the m-Rep is extracted from an inner Voronoi skeleton via the pruning and merging of medial sheets while minimizing the change of the underlying shape according to two criteria similar to [33]: (i) the sheet area (number of vertices) and (ii) its corresponding shape volume. The recent flow complex (FC) [15] extracts the MA from the VD based a Morse analysis on the radius flow and is closely related to our approach. A possibly extendable core (set of 'unstable manifolds') is computed from the flow analysis on the VD to approximate the MA.…”

Section: Related Workmentioning

confidence: 99%

Regularizing 3D medial axis using medial scaffold transforms

Chang

Kimia

2008

2008 IEEE Conference on Computer Vision and Pattern Recognition

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This paper addresses a key bottleneck in the use of the 3D medial axis (MA) representation, namely, how the complex MA structure can be regularized so that similar, withincategory 3D shapes yield similar 3D MA that are distinct from the non-category shapes. We rely on previous work which (i) constructs a hierarchical MA hypergraph, the medial scaffold (MS), and (ii) the theoretical classification of the instabilities of this structure, or transitions (sudden topological changes due to a small perturbation). The shapes at transition point are degenerate. Our approach is to recognize the transitions which are close-by to a given shape and transform the shape to this transition point, and repeat until no close-by transitions exists. This move towards degeneracy is the basis of simplification of shape. We derive 11 transforms from 7 transitions and follow a greedy scheme in applying the transform. The results show that the simplified MA preserves with-in-category similarity, thus indicating its potential use in various applications including shape analysis, manipulation, and matching.

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Medial axis approximation and unstable flow complex

Cited by 18 publications

References 12 publications

Computing Multiscale Curve and Surface Skeletons of Genus 0 Shapes Using a Global Importance Measure

Computing Multiscale Curve and Surface Skeletons of Genus 0 Shapes Using a Global Importance Measure

Manifold Homotopy via the Flow Complex

Regularizing 3D medial axis using medial scaffold transforms

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