Manifold splines with single extraordinary point

Gu, Xianfeng; He, Ying; Jin, Miao; Luo, Feng; Qin, Hong; Yau, Shing–Tung

doi:10.1145/1236246.1236258

Cited by 10 publications

(7 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here they establish the theoretical foundation for discrete Ricci flow by proving the existence and convergence of the discrete Ricci flow [17]. This facilitates the computational and numerical application of surface Ricci flow to problems such as surface parameterization, shape analysis and geometric graphics [20,14,19,49,50,38,39]. These applications all share the feature of representing surfaces as piecewise linear triangle meshes and using Ricci flow to deform the edge dissimilarities using their discretely estimated curvature.…”

Section: Chow and Luo Have Studied The Intrinsic Relations Between Thmentioning

confidence: 95%

“…However, they are based on different geometric models. For example, both Jin et al [20] and Gu et al [14] extend Luo and Chow's work on the combinatorial structures of triangular meshes by improving the gradient descent Ricci flow algorithm using Newton's method, for a hyperbolic space and a Euclidean space respectively. Zeng has investigated Ricci flow in hyperbolic geometry for characterizing 3D shapes [49,50].…”

Section: Chow and Luo Have Studied The Intrinsic Relations Between Thmentioning

confidence: 99%

See 1 more Smart Citation

Ricci flow embedding for rectifying non-Euclidean dissimilarity data

Hancock

Wilson

2014

Pattern Recognition

View full text Add to dashboard Cite

Pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition. One of the problems encountered in the analysis of such data, is that the dissimilarities are rarely Euclidean, while statistical learning algorithms often rely on Euclidean dissimilarities. Such non-Euclidean dissimilarities are often corrected or a consistent Euclidean geometry imposed on them via embedding. This paper commences by reviewing the available algorithms for analysing non-Euclidean dissimilarity data. The novel contribution is to show how Ricci flow can be used to embed and rectify non-Euclidean dissimilarity data. According to our representation, the data is distributed over a manifold consisting of patches. Each patch has a locally uniform curvature, and this curvature is iteratively modified by the Ricci flow. The raw dissimilarities are the geodesic distances on the manifold. Rectified Euclidean dissimilarities are obtained using Ricci flow to flatten the curved manifold by modifying the individual patch curvatures. We use two algorithmic components to implement this idea. First, we apply the Ricci flow independently to a set of surface patches that cover the manifold. Second, we use curvature regularisation to impose consistency on the curvatures of the arrangement of different surface patches. We perform experiments on three real world data sets, and use these to determine the importance of the different algorithmic components, i.e. Ricci flow and curvature regularisation. We conclude that curvature regularisation is an essential step needed to control the stability of the piecewise arrangement of patches under Ricci flow.

show abstract

Section: Chow and Luo Have Studied The Intrinsic Relations Between Thmentioning

confidence: 95%

Section: Chow and Luo Have Studied The Intrinsic Relations Between Thmentioning

confidence: 99%

Ricci flow embedding for rectifying non-Euclidean dissimilarity data

Hancock

Wilson

2014

Pattern Recognition

View full text Add to dashboard Cite

show abstract

“…Some more discrete Ricci flow algorithms regarding efficiency and adaptivity improvements are reported in Refs. [79,80]. For further conformal mapping algorithms, the readers are referred to Ref.…”

Section: Concurrent Designing Of An Npr Structure and Its Infill Matementioning

confidence: 99%

Concurrent optimization of structural topology and infill properties with a CBF-based level set method

et al. 2019

View full text Add to dashboard Cite

In this paper, a parametric level-set-based topology optimization framework is proposed to concurrently optimize the structural topology at the macroscale and the effective infill properties at the micro/meso scale. The concurrent optimization is achieved by a computational framework combining a new parametric level set approach with mathematical programming. Within the proposed framework, both the structural boundary evolution and the effective infill property optimization can be driven by mathematical programming, which is more advantageous compared with the conventional partial differential equation-driven level set approach. Moreover, the proposed approach will be more efficient in handling nonlinear problems with multiple constraints. Instead of using radial basis functions (RBF), in this paper, we propose to construct a new type of cardinal basis functions (CBF) for the level set function parameterization. The proposed CBF parameterization ensures an explicit impose of the lower and upper bounds of the design variables. This overcomes the intrinsic disadvantage of the conventional RBF-based parametric level set method, where the lower and upper bounds of the design variables oftentimes have to be set by trial and error. A variational distance regularization method is utilized in this research to regularize the level set function to be a desired distanceregularized shape. With the distance information embedded in the level set model, the wrapping boundary layer and the interior infill region can be naturally defined. The isotropic infill achieved via the mesoscale topology optimization is conformally fit into the wrapping boundary layer using the shape-preserving conformal mapping method, which leads to a hierarchical physical structure with optimized overall topology and effective infill properties. The proposed method is expected to provide a timely solution to the increasing demand for multiscale and multifunctional structure design.

show abstract

“…There are different ways to define discrete conformal metric deformation. Details can be found in and [Luo et al 2007]. …”

Section: Surface Ricci Flowmentioning

confidence: 99%

Surface- and volume-based techniques for shape modeling and analysis

Patanè

2013

SIGGRAPH Asia 2013 Courses

View full text Add to dashboard Cite

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. Course descriptionShape modeling typically handles a 3D shape as a two-dimensional surface, which describes the shape boundary and is represented as a triangular mesh or a point cloud. However, in several applications a volumetric surface representation is more suited to handle the complexity of the input shape. For instance, volumetric representations are used to accurately modeling the behavior of non-rigid deformations and volume constraints are imposed to avoid deformation artifacts. In shape matching, volumetric descriptors, such as Laplacian eigenfunctions, heat kernels, and diffusion distances, are defined starting from their surface-based counterparts.In the aforementioned applications, the underlying problem generally requires the prolongation of the surface-based information, which is typically represented as a shape-driven map, to the interior of the input shape or, more generally, to the surrounding volume. Extending a surface-based scalar function to a volumetric representation 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. Besides the underlying complexity and degrees of freedom in the definition of volumetric approximations of surface-based maps, out-of-sample approximations (i.e., the extension of a surface-based scalar function to a volume-based approximation) are essential to address a wide range of problems. For instance, volumetric Laplacian eigenfunctions are suited to define volumetric shape descriptors, which are consistent with their surface-based counterparts. In a similar way, harmonic volumetric functions have been used for the volumetric parameterization and the definition of (volumetric...

show abstract

Manifold splines with single extraordinary point

Cited by 10 publications

References 16 publications

Ricci flow embedding for rectifying non-Euclidean dissimilarity data

Ricci flow embedding for rectifying non-Euclidean dissimilarity data

Concurrent optimization of structural topology and infill properties with a CBF-based level set method

Surface- and volume-based techniques for shape modeling and analysis

Contact Info

Product

Resources

About