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3D surface matching is fundamental for shape analysis. As a powerful method in geometric analysis, Ricci flow can flexibly design metrics by prescribed target curvature. In this paper we describe a novel approach for matching surfaces with complicated topologies based on hyperbolic Ricci flow. For surfaces with negative Euler characteristics, such...
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... this work, we applied hyperbolic conformal maps and hyperbolic hexagon decomposition for surface match- ing. Figure 1 gives an example, where a human face sur- face is decomposed to 2 parts by 3 geodesics between each two boundaries and each part is mapped to a hyperbolic hexagon. Hyperbolic conformal maps by hyperbolic Ricci flow method have no singularity, and can be efficiently used for matching surfaces with complicated topology, without any restriction on the number of holes. ...
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... It was developed by Richard Hamilton in the 1980's [1] as an approach to proving the Thruston geometrization conjecture, and therefore the Pioncaré conjecture, an endeavour that was eventually successful with the work of Grisha Pereleman in the early 2000's [2,3]. Since then, Ricci flow has remained an important tool for investigating the interplay between geometry and topology, and has also found a number of applications in image analysis, from facial recognition [4] to cancer detection [5], and in space-time physics [6,7,8]. Applications such as these require a robust numerical approach, and preliminary work has even shown the potential for numerical simulations to inform analytical Ricci flow research [9]. ...
Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh resolution is increased. The manifolds were chosen to have varying degrees of homogeneity, and include Nil and Gowdy manifolds, a three-torus initially embedded in Euclidean four-space, and a perturbation of a flat three-torus. The piecewise flat Ricci flow of the first two are shown to converge to analytic and numerical partial differential equation solutions respectively, with the remaining two flowing asymptotically to flat metrics.
... The conformal flow tool, such as Ricci flow, has already been used in the applications of brain analysis , shape analysis , network routing (Sarkar et al., 2009;Gao et al., 2016), shape modeling (Patanè et al., 2014), shape registration , face matching (Zeng et al., 2008), geometric structures (Jin et al., 2007b), and so on. In the future, we will explore the applicability of Calabi flow in these kinds of applications. ...
In this paper, we introduce discrete Calabi flow to the graphics research community and present a novel conformal mesh parameterization algorithm. Calabi energy has a succinct and explicit format. Its corresponding flow is conformal and convergent under certain conditions. Our method is based on the Calabi energy and Calabi flow with solid theoretical and mathematical base. We demonstrate our approach on dozens of models and compare it with other related flow based methods, such as the well-known Ricci flow and conformal equivalence of triangle meshes (CETM). Our experiments show that the performance of our algorithm is comparably the same with other methods. The discrete Calabi flow in our method provides another perspective on conformal flow and conformal parameterization.
... Discrete Surface Ricci Flow Discrete surface Ricci flow algorithm has been introduced in [Jin et al. 2008;Yang et al. 2009] and applied for shape analysis in [Zeng et al. 2010b], surface registration [Wang et al. 2007;Zeng et al. 2008], computational topology [Zeng et al. 2009a], geometric modeling ], image processing [Lui et al. 2010], virtual colonoscopy [Zeng et al. 2010a], and many other applications. In engineering fields, smooth surfaces are often approximated by simplicial complexes (triangle meshes). ...
Extending a shape-driven map to the interior of the input shape and to the surrounding volume is a difficult problem since it typically relies on the integration of shape-based and volumetric information, together with smoothness conditions, interpolating constraints, preservation of feature values at both a local and global level. This survey discusses the main volumetric approximation schemes for both 3D shapes and d-dimensional data, and provides a unified discussion on the integration of surface-based and volume-based shape information. Then, it describes the application of shape-based and volumetric techniques to shape modeling through volumetric parameterization and polycube splines; feature-driven approximation through kernels and radial basis functions. We also discuss the Hamilton's Ricci flow, which is a powerful tool to compute the conformal shape structure and to design Riemannian metrics of manifolds by prescribed curvatures. We conclude the presentation by discussing applications to shape analysis and medicine.
... Surfaces with boundaries are conformally deformed to constant curvature surfaces with circular holes. Discrete Surface Ricci Flow Discrete surface Ricci flow algorithm has been introduced in Yang et al. 2009] and applied for shape analysis in [Zeng et al. 2010b] , surface registra- tion Zeng et al. 2008] , computational topol- ogy [Zeng et al. 2009a], geometric modeling , image processing [Lui et al. 2010], virtual colonoscopy [Zeng et al. 2010a], and many other applications. In engineering fields, smooth surfaces are often approximated by simplicial complexes (triangle meshes). ...
Extending a shape-driven map to the interior of the input shape and to the surrounding volume is a difficult problem since it typically relies on the integration of shape-based and volumetric information, together with smoothness conditions, interpolating constraints, preservation of feature values at both a local and global level.
In this context, this course revises the main out-of-sample approximation schemes for both 3D shapes and d-dimensional data, and provides a unified discussion on the integration of surface- and volume-based shape information. Then, it describes the application of shape-based and volumetric techniques to shape modeling and analysis through the definition of volumetric shape descriptors; shape processing through volumetric parameterization and polycube splines; feature-driven approximation through kernels and radial basis functions.
We also discuss the Hamilton's Ricci flow, which is a powerful tool to compute the conformal structure of the shapes and to design Riemannian metrics of manifolds by prescribed curvatures and shape descriptors using conformal welding. We conclude the presentation by discussing applications to shape analysis and medicine, open problems, and future perspectives.
... Recently, conformal geometry has been introduced as a powerful surface shape analysis tool, such as surface matching, recognition and stitching [34] [9] surface registration [40], 3D face recognition [28] and 3D FER [19] [22]. Different from the previous works, in this paper, for the first time, we propose to use the unique surface conformal representation, i.e., conformal factor image (CFI) and mean curvature image (MCI), as expression features of the proposed 3D FER framework. ...
We propose a general and fully automatic framework for 3D facial expression recognition by modeling sparse representation of conformal images. According to Riemann Geometry theory, a 3D facial surface S embedded in ℝ3, which is a topological disk, can be conformally mapped to a 2D unit disk D through the discrete surface Ricci Flow algorithm. Such a conformal mapping induces a unique and intrinsic surface conformal representation denoted by a pair of functions defined on D, called conformal factor image (CFI) and mean curvature image (MCI). As facial expression features, CFI captures the local area distortion of S induced by the conformal mapping; MCI characterizes the geometry information of S. To model sparse representation of conformal images for expression classification, both CFI and MCI are further normalized by a Mobius transformation. This transformation is defined by the three main facial landmarks (i.e. nose tip, left and right inner eye corners) which can be detected automatically and precisely. Expression recognition is carried out by the minimal sparse expression-class-dependent reconstruction error over the conformal image based expression dictionary. Extensive experimental results on the BU-3DFER dataset demonstrate the effectiveness and generalization of the proposed framework.
... Euclidean Ricci flow has been applied to shape analysis in [37]. Hyperbolic Ricci flow has been applied to 3D face matching and registration in [38]. Prior Ricci flow works were based on the conventional circle packing metric. ...
Ricci flow is a powerful curvature flow method, which is invariant to rigid motion, scaling, isometric, and conformal deformations. We present the first application of surface Ricci flow in computer vision. Previous methods based on conformal geometry, which only handle 3D shapes with simple topology, are subsumed by the Ricci flow-based method, which handles surfaces with arbitrary topology. We present a general framework for the computation of Ricci flow, which can design any Riemannian metric by user-defined curvature. The solution to Ricci flow is unique and robust to noise. We provide implementation details for Ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry. Our Ricci flow-based method can convert all 3D problems into 2D domains and offers a general framework for 3D shape analysis. We demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems, such as 3D shape matching and registration, and shape indexing. Surfaces with large nonrigid anisotropic deformations can be registered using Ricci flow with constraints of feature points and curves. We show how conformal equivalence can be used to index shapes in a 3D surface shape space with the use of Teichmüller space coordinates. Experimental results are shown on 3D face data sets with large expression deformations and on dynamic heart data.
... The third category is based on holomorphic differentials, which induces different conformal mappings, by combining them, better results can be achieved for surface matching and registration [26]. The third rows in figure 1 and 4 analysis in [27], and hyperbolic Ricci flow has been used for face matching and registration in [28]. Recently, discrete surface Yamabe flow has been introduced by Luo in [32], it has been reintroduced in [33]. ...
Recently, various conformal geometric methods have been presented for non-rigid surface matching and registration. This work proposes to improve the robustness of conformal geometric methods to the boundaries by incorporating the symmetric information of the input surface. We presented two symmetric conformal mapping methods, which are based on solving Riemann-Cauchy equation and curvature flow respectively. Experimental results on geometric data acquired from real life demonstrate that the symmetric conformal mapping is insensitive to the boundary occlusions. The method outperforms all the others in terms of robustness. The method has the potential to be generalized to high genus surfaces using hyperbolic curvature flow.
... The conformal mapping preserves the symmetry in the following ways: first the image of the mapping is still symmetric; second, the area distortion factor on the image is symmetric as well.Figure 1 shows the symmetric conformal mappings, which are much more robust to the boundary occlusions and inconsistency. methods for the application of surface matching and registration, including Harmonic Maps [37], [30], [10], Riemann-Cauchy Equation (such as least square conformal maps (LSCMs) introduced in [15]) [29], [28], Holomorphic Differentials [36], and Ricci Flow [9], [35], [34]. Recently, discrete surface Yamabe flow has been introduced by Luo in [18], which has been reintroduced in [1]. ...
Recently, various conformal geometric methods have been presented for non-rigid surface matching and registration. This work proposes to improve the robustness of conformal geometric methods to the boundaries by incorporating the symmetric information of the input surface. We presented two symmetric conformal mapping methods, which are based on solving Riemann-Cauchy equation and curvature flow respectively. Experimental results on geometric data acquired from real life demonstrate that the symmetric conformal mapping is insensitive to the boundary occlusions. The method outperforms all the others in terms of robustness. The method has the potential to be generalized to high genus surfaces using hyperbolic curvature flow.
... In various situations of surface registration, different constraints like landmarkmatching and conformality-preserving are enforced to get desirable diffeomorphisms [16] [11] [14]. A good diffeomorphism usually preserves local geometry well. ...
In this work, we propose a novel idea of representing surface diffeomorphisms using Beltrami coefficients, which are complex-valued functions defined on the surface that describe lo-cal non-conformal distortions of the surface map. According to Quasiconformal Teichmuller theory, there is an one-to-one correspondence between the set of surface diffeomorphisms fixing three points and the set of Beltrami coefficients with L ∞ -norm strictly less than 1. Every surface diffeomorphism is associated with a unique Beltrami coefficient. Conversely, given every such coefficient, we can re-construct the associated diffeomorphism exactly using a flow, called the Beltrami holomorphic flow, which solves the Beltrami equation. The use of Beltrami coefficients to represent surface diffeomor-phisms is a powerful method because it captures the most essential features of surface maps, such as conformality distortions, rotational changes and dilations. By adjusting the Beltrami coefficient, we can adjust the surface diffeomorphism accordingly to obtain desired properties of the map. More-over, the Beltrami holomorphic flow guarantees to give a smooth sequence of surface diffeomorphisms. Therefore, a sequence of surface diffeomorphisms can be represented by a sequence of Beltrami coef-ficient and can be reconstructed by the Beltrami holomorphic flow. Using this approach, we propose several applications to properties adjustment of surface maps. It includes the accurate alignment of landmark curves, the reconstruction of surface diffeomorphisms, the construction of Riemann maps and the restoration of diffeomorphic and conformality properties. We apply our algorithms on differ-ent Riemann surfaces. Experimental results show that the Beltrami coefficient can effectively assist us to represent and adjust surface diffeomorphisms.
Surface registration plays a fundamental role in shape analysis and geometric processing. Generally, there are three criteria in evaluating a surface mapping result: diffeomorphism, small distortion, and feature alignment. To fulfill these requirements, this work proposes a novel model of the space of point landmark constrained diffeomorphisms. Based on Teichmüller theory, this mapping space is generated by the Beltrami coefficients, which are infinitesimally Teichmüller equivalent to 0. These Beltrami coefficients are the solutions to a linear equation group. By using this theoretic model, optimal registrations can be achieved by iterative optimization with linear constraints in the diffeomorphism space, such as harmonic maps and Teichmüller maps, which minimize different types of distortion. The theoretical model is rigorous and has practical value. Our experimental results demonstrate the efficiency and efficacy of the proposed method.
