Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee

Dey, Tamal K.; Zhao, Wulue

doi:10.1007/3-540-45749-6_36

Cited by 33 publications

(41 citation statements)

References 14 publications

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“…The set Q 2 captures points going to be processed in the next algorithm iteration (detailed further below). Next, we sort the current set Q 1 on increasing ρ (line 15), allowing the second inner loop (lines [17][18][19] to process these in increasing order of their density values. This second loop performs the detection phase of the thinning algorithm.…”

Section: B Algorithmmentioning

confidence: 99%

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An Unified Multiscale Framework for Planar, Surface, and Curve Skeletonization

Jalba

Sobiecki

Telea

2016

IEEE Trans. Pattern Anal. Mach. Intell.

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Computing skeletons of 2D shapes, and medial surface and curve skeletons of 3D shapes, is a challenging task. In particular, there is no unified framework that detects all types of skeletons using a single model, and also produces a multiscale representation which allows to progressively simplify, or regularize, all skeleton types. In this paper, we present such a framework. We model skeleton detection and regularization by a conservative mass transport process from a shape's boundary to its surface skeleton, next to its curve skeleton, and finally to the shape center. The resulting density field can be thresholded to obtain a multiscale representation of progressively simplified surface, or curve, skeletons. We detail a numerical implementation of our framework which is demonstrably stable and has high computational efficiency. We demonstrate our framework on several complex 2D and 3D shapes.

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Section: B Algorithmmentioning

confidence: 99%

“…Mesh-based methods use a surface sampling of ∂ Ω, which allows processing higher-resolution shapes. Mesh methods include Voronoi diagrams to compute polygonal skeletons [19]. Amenta et al compute the Power Crust, an approximation of a surface and its medial axis by a subset of Voronoi points [2].…”

Section: Introductionmentioning

confidence: 99%

An Unified Multiscale Framework for Planar, Surface, and Curve Skeletonization

Jalba

Sobiecki

Telea

2016

IEEE Trans. Pattern Anal. Mach. Intell.

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“…Once a suitable molecular surface is extracted, we compute its medial axis using the algorithm in (Dey and Zhao, 2004). Medial axis is usually very noisy and often does not prove to be a very useful and robust descriptor of the shape under investigation.…”

Section: Secondary Structure Detectionmentioning

confidence: 99%

“…Variants of the topological structures of this function have been used earlier for shape modeling of three dimensional objects. In computational geometry, the distance function has been used to reconstruct piecewise triangulated surfaces from discrete point samples (Chaine, 2003; Giesen and John, 2003) and to compute the stable medial axis (Chazal and Lieutier, 2004; Siddiqi et al, 1998; Dey and Zhao, 2004). While a surface approximation and its medial axis serve as crude shape descriptors, deeper insights about the feature space of the shape can also be obtained using the critical points of the distance function.…”

Section: Introductionmentioning

confidence: 99%

Detection of secondary and supersecondary structures of proteins from cryo-electron microscopy

Bajaj

Goswami²,

Zhang

2012

Journal of Structural Biology

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Recent advances in three-dimensional electron microscopy (3D EM) have enabled the quantitative visualization of the structural building blocks of proteins at improved resolutions. We provide algorithms to detect the secondary structures (α-helices and β-sheets) from proteins for which the volumetric maps are reconstructed at 6–10Å resolution. Additionally, we show that when the resolution is coarser than 10Å, some of the super-secondary structures can be detected from 3D EM maps. For both these algorithms, we employ tools from computational geometry and differential topology, specifically the computation of stable/unstable manifolds of certain critical points of the distance function induced by the molecular surface. Our results connect mathematically well-defined constructions with bio-chemically induced structures observed in proteins.

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“…Several MA regularization techniques have been independently developed but rather share common ideas and mainly vary in how the MA elements are selected to prune. We briefly review them and for in-depth surveys, see [2,32,10,26,34].…”

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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Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee

Cited by 33 publications

References 14 publications

An Unified Multiscale Framework for Planar, Surface, and Curve Skeletonization

An Unified Multiscale Framework for Planar, Surface, and Curve Skeletonization

Detection of secondary and supersecondary structures of proteins from cryo-electron microscopy

Regularizing 3D medial axis using medial scaffold transforms

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