Fast Mean‐Curvature Flow via Finite‐Elements Tracking

Chuang, Ming; Kazhdan, Michael

doi:10.1111/j.1467-8659.2011.01899.x

Cited by 11 publications

(14 citation statements)

References 31 publications

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“…We would like to find a solution to this problem within the presented framework. Finally, we would also like to consider integrating our technique into the highly optimized curvature flow solver recently presented by Chuang and Kazhdan [CK11].…”

Section: Discussionmentioning

confidence: 99%

Mean Curvature Skeletons

Tagliasacchi

Alhashim

Olson

et al. 2012

Computer Graphics Forum

189

197

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Figure 1: Given a watertight surface (a), the well-known medial axis transform (b) often produces too complex of a structure to be of practical use. Our skeletonization algorithm can produce intermediate meso-skeletons (c), which contain medial sheets where needed and curves where appropriate, while converging to a medially centered curve skeleton output (d). AbstractInspired by recent developments in contraction-based curve skeleton extraction, we formulate the skeletonization problem via mean curvature flow (MCF). While the classical application of MCF is surface fairing, we take advantage of its area-minimizing characteristic to drive the curvature flow towards the extreme so as to collapse the input mesh geometry and obtain a skeletal structure. By analyzing the differential characteristics of the flow, we reveal that MCF locally increases shape anisotropy. This justifies the use of curvature motion for skeleton computation, and leads to the generation of what we call "mean curvature skeletons". To obtain a stable and efficient discretization, we regularize the surface mesh by performing local remeshing via edge splits and collapses. Simplifying mesh connectivity throughout the motion leads to more efficient computation and avoids numerical instability arising from degeneracies in the triangulation. In addition, the detection of collapsed geometry is facilitated by working with simplified mesh connectivity and monitoring potential non-manifold edge collapses. With topology simplified throughout the flow, minimal post-processing is required to convert the collapsed geometry to a curve. Formulating skeletonization via MCF allows us to incorporate external energy terms easily, resulting in a constrained flow. We define one such energy term using the Voronoi medial skeleton and obtain a medially centred curve skeleton. We call the intermediate results of our skeletonization motion meso-skeletons; these consist of a mixture of curves and surface sheets as appropriate to the local 3D geometry they capture.

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Section: Discussionmentioning

confidence: 99%

Mean Curvature Skeletons

Tagliasacchi

Alhashim

Olson

et al. 2012

Computer Graphics Forum

189

197

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“…18). The fundamental connection between contraction and differential geometry was first revealed by [CK11] and then formalized in [TAOZ12]. To avoid this energy from isotropically scaling the contracting shape down to a point, a constrained problem is solved where the locations of voxels close to the shrinking surface are required not to move too far from their initial location on S. A variant was later proposed in [ATC * 08], where the optimization is directly formulated on the shape surface S. Like in [WL08], constrains are added to avoid the trivial solution and also retain important surface features.…”

Section: Contraction Methodsmentioning

confidence: 99%

“…Mean curvature is also directly connected to the discrete Laplace-Beltrami operator, leading to the aforementioned computational efficiency. 5 by projecting its solution in the embedding space can be used [CK11], or alternatively the quality of the triangulation must be adapted during the motion [TAOZ12]. To fix this, advanced methods that discretize Eqn.…”

Section: Contraction Methodsmentioning

confidence: 99%

3D Skeletons: A State‐of‐the‐Art Report

Tagliasacchi

Delame

Spagnuolo

et al. 2016

Computer Graphics Forum

203

152

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International audienceGiven a shape, a skeleton is a thin centered structure which jointly describes the topology and the geometry of the shape. Skeletons provide an alternative to classical boundary or volumetric representations, which is especially effective for applications where one needs to reason about, and manipulate, the structure of a shape. These skeleton properties make them powerful tools for many types of shape analysis and processing tasks. For a given shape, several skeleton types can be defined, each having its own properties, advantages, and drawbacks. Similarly, a large number of methods exist to compute a given skeleton type, each having its own requirements, advantages, and limitations. While using skeletons for two-dimensional (2D) shapes is a relatively well covered area, developments in the skeletonization of three-dimensional (3D) shapes make these tasks challenging for both researchers and practitioners. This survey presents an overview of 3D shape skeletonization. We start by presenting the definition and properties of various types of 3D skeletons. We propose a taxonomy of 3D skeletons which allows us to further analyze and compare them with respect to their properties. We next overview methods and techniques used to compute all described 3D skeleton types, and discuss their assumptions, advantages, and limitations. Finally, we describe several applications of 3D skeletons, which illustrate their added value for different shape analysis and processing tasks

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“…Representative approaches include surface contraction via mean curvature flow [Au et al 2008;Chuang and Kazhdan 2011;Tagliasacchi et al 2012], coupled graph contraction and surface clustering [Jiang et al 2013], and the use of distance transforms [Dey and Sun 2006], centroidal Voronoi tessellation [Lu et al 2012], Reeb graph construction [Hilaga et al 2001], or mesh segmentation [Katz and Tal 2003]. These approaches all depend on mesh connectivity or surface-based measures to control the skeletonization process.…”

Section: Related Workmentioning

confidence: 99%

L ₁ -medial skeleton of point cloud

et al. 2013

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We introduce L1-medial skeleton as a curve skeleton representation for 3D point cloud data. The L1-median is well-known as a robust global center of an arbitrary set of points. We make the key observation that adapting L1-medians locally to a point set representing a 3D shape gives rise to a one-dimensional structure, which can be seen as a localized center of the shape. The primary advantage of our approach is that it does not place strong requirements on the quality of the input point cloud nor on the geometry or topology of the captured shape. We develop a L1-medial skeleton construction algorithm, which can be directly applied to an unoriented raw point scan with significant noise, outliers, and large areas of missing data. We demonstrate L1-medial skeletons extracted from raw scans of a variety of shapes, including those modeling high-genus 3D objects, plant-like structures, and curve networks.

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Fast Mean‐Curvature Flow via Finite‐Elements Tracking

Cited by 11 publications

References 31 publications

Mean Curvature Skeletons

Mean Curvature Skeletons

3D Skeletons: A State‐of‐the‐Art Report

L ₁ -medial skeleton of point cloud

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Fast Mean‐Curvature Flow via Finite‐Elements Tracking

Cited by 11 publications

References 31 publications

Mean Curvature Skeletons

Mean Curvature Skeletons

3D Skeletons: A State‐of‐the‐Art Report

L 1 -medial skeleton of point cloud

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Resources

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L ₁ -medial skeleton of point cloud