Ricci Flow for 3D Shape Analysis

Gu, Xianfeng; Wang, Sen; Kim, Junho; Zeng, Yun; Wang, Yan; Qin, Hong; Samaras, Dimitris

doi:10.1109/iccv.2007.4409028

Cited by 41 publications

(27 citation statements)

References 24 publications

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“…On the other hand, the geometric flows [34,10,3,35,7,29,6] previously seemed a separate field which may be bridged by this paper. Here we merely attempt to show the inherent link between shape smoothing (scale space processing) and geometric flows.…”

Section: Prior Workmentioning

confidence: 83%

Intrinsic Geometric Scale Space by Shape Diffusion

Zou

Hua

Lai

et al. 2009

IEEE Trans. Visual. Comput. Graphics

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Abstract-This paper formalizes a novel, intrinsic geometric scale space (IGSS) of 3D surface shapes. The intrinsic geometry of a surface is diffused by means of the Ricci flow for the generation of a geometric scale space. We rigorously prove that this multiscale shape representation satisfies the axiomatic causality property. Within the theoretical framework, we further present a featurebased shape representation derived from IGSS processing, which is shown to be theoretically plausible and practically effective. By integrating the concept of scale-dependent saliency into the shape description, this representation is not only highly descriptive of the local structures, but also exhibits several desired characteristics of global shape representations, such as being compact, robust to noise and computationally efficient. We demonstrate the capabilities of our approach through salient geometric feature detection and highly discriminative matching of 3D scans.

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Section: Prior Workmentioning

confidence: 83%

Intrinsic Geometric Scale Space by Shape Diffusion

Zou

Hua

Lai

et al. 2009

IEEE Trans. Visual. Comput. Graphics

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“…A closely related approach utilizes inner metrics to describe shape deformations, which are prescribed directly on the surface [2]. Other groups study 3D shapes using level sets [23], curvature flows [12], the iterative closest point (ICP) algorithm [1] or the medial axis [3,10]. The representation of 3D objects by their boundaries, which form parameterized surfaces, also provides a natural framework for statistical shape analysis.…”

Section: Introductionmentioning

confidence: 99%

A Novel Riemannian Framework for Shape Analysis of Annotated Surfaces

Kurtek¹

2015

Procedings of the Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Sha

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We present a novel, parameterization-invariant method for shape analysis of annotated surfaces. While the method can handle various types of annotation including color and texture, in this paper we focus on soft landmark annotations. Landmark annotations are commonly provided in various applications including medical imaging where an expert marks points of interest on the objects. Most methods in current literature either study shapes using landmarks only or surfaces only. In either case, the analyst is forced to ignore a lot of useful information. We propose a novel representation of surfaces that can jointly incorporate shape and landmark annotation. Our framework properly removes all shape preserving transformations from the representation space including translation, scale, rotation and re-parameterization. We present results of comparing, averaging and classifying annotated shapes on toy and real data.

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“…If u : S → R is a function defined on the surface, we can define another metric G = e 2u G, that is conformal to the original metric, since the two metrics are proportional. The Gaussian curvaturek of the new metric changes by [2] …”

Section: Bounding the Conformal Factormentioning

confidence: 99%

Measuring Geodesic Distances via the Uniformization Theorem

Aflalo

Kimmel

2012

Lecture Notes in Computer Science

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Abstract. According to the Uniformization Theorem any surface can be conformally mapped into a flat domain, that is, a domain with zero Gaussian curvature. The conformal factor indicates the local scaling introduced by such a mapping. This process could be used to compute geometric quantities in a simplified flat domain. For example, the computation of geodesic distances on a curved surface can be mapped into solving an eikonal equation in a plane weighted by the conformal factor. Solving an eikonal equation on the weighted plane can then be done with regular sampling of the domain using, for example, the fast marching method. The connection between the conformal factor on the plane and the surface geometry can be justified analytically. Still, in order to construct consistent numerical solvers that exploit this relation one needs to prove that the conformal factor is bounded.In this paper we provide theoretical bounds over the conformal factor and introduce optimization formulations that control its behavior. It is demonstrated that without such a control the numerical results are unboundedly inaccurate. Putting all ingredients in the right order, we introduce a method for computing geodesic distances on a two dimensional manifold by using the fast marching algorithm on a weighed flat domain.

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Ricci Flow for 3D Shape Analysis

Cited by 41 publications

References 24 publications

Intrinsic Geometric Scale Space by Shape Diffusion

Intrinsic Geometric Scale Space by Shape Diffusion

A Novel Riemannian Framework for Shape Analysis of Annotated Surfaces

Measuring Geodesic Distances via the Uniformization Theorem

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