Hui Wang, Hui Huang. Group Representation of Global Intrinsic Symmetries, Computer Graphics Forum (Proceedings of Pacific Graphics), 2017, 36:(7): 51-61. [Project Page]
Global intrinsic symmetry detection of 3D shapes has received considerable attentions in recent years. However, unlike extrinsic symmetry that can be represented compactly as a combination of an orthogonal matrix and a translation vector, representing the global intrinsic symmetry itself is still challenging. Most previous works based on point-to-point representations of global intrinsic symmetries can only find reflectional symmetries, and are inadequate for describing the structure of a global intrinsic symmetry group. In this paper, we propose a novel group representation of global intrinsic symmetries, which describes each global intrinsic symmetry as a linear transformation of functional space on shapes. If the eigenfunctions of the Laplace-Beltrami operator on shapes are chosen as the basis of functional space, the group representation has a block diagonal structure. We thus prove that the group representation of each symmetry can be uniquely determined from a small number of symmetric pairs of points under certain conditions, where the number of pairs is equal to the maximum multiplicity of eigenvalues of the Laplace-Beltrami operator. Based on solid theoretical analysis, we propose an efficient global intrinsic symmetry detection method, which is the first one able to detect all reflectional and rotational global intrinsic symmetries with a clear group structure description. Experimental results demonstrate the effectiveness of our approach.
Global intrinsic symmetry detection of 3D shapes has received considerable attentions in recent years. However, unlike extrinsic symmetry that can be represented compactly as a combination of an orthogonal matrix and a translation vector, representing the global intrinsic symmetry itself is still challenging. Most previous works based on point-to-point representations of global intrinsic symmetries can only find reflectional symmetries, and are inadequate for describing the structure of a global intrinsic symmetry group. In this paper, we propose a novel group representation of global intrinsic symmetries, which describes each global intrinsic symmetry as a linear transformation of functional space on shapes. If the eigenfunctions of the Laplace-Beltrami operator on shapes are chosen as the basis of functional space, the group representation has a block diagonal structure. We thus prove that the group representation of each symmetry can be uniquely determined from a small number of symmetric pairs of points under certain conditions, where the number of pairs is equal to the maximum multiplicity of eigenvalues of the Laplace-Beltrami operator. Based on solid theoretical analysis, we propose an efficient global intrinsic symmetry detection method, which is the first one able to detect all reflectional and rotational global intrinsic symmetries with a clear group structure description. Experimental results demonstrate the effectiveness of our approach.
Global intrinsic rotational (left) and reflectional (right) symmetries that we compute on an Octopus model, where symmetries are
represented by the colors, comparing them with the original colored shape in the middle.
represented by the colors, comparing them with the original colored shape in the middle.
Hui Wang, Junjie Cao, Xiuping Liu, Jianmin Wang, Tongrang Fan, Jianping Hu. Least-squares images for edge-preserving smoothing, Computational Visual Media, 2015, 1(1): 27-35. [PDF]
In this paper, we propose least-squares images (LS-images) as a basis for a novel edge-preserving image smoothing method. The LS-image requires the value of each pixel to be a convex linear combination of its neighbors, i.e., to have zero Laplacian, and to approximate the original image in a least-squares sense. The edge-preserving property inherits from the edge-aware weights for constructing the linear combination. Experimental results demonstrate that the proposed method achieves high quality results compared to previous state-of-the- art works. We also show diverse applications of LS-images, such as detail manipulation, edge enhancement, and clip-art JPEG artifact removal.
In this paper, we propose least-squares images (LS-images) as a basis for a novel edge-preserving image smoothing method. The LS-image requires the value of each pixel to be a convex linear combination of its neighbors, i.e., to have zero Laplacian, and to approximate the original image in a least-squares sense. The edge-preserving property inherits from the edge-aware weights for constructing the linear combination. Experimental results demonstrate that the proposed method achieves high quality results compared to previous state-of-the- art works. We also show diverse applications of LS-images, such as detail manipulation, edge enhancement, and clip-art JPEG artifact removal.
Hui Wang, Junjie Cao, Xiuping Liu, Tongrang Fan, Jianmin Wang. Mesh Detail Editing by Filtering Differential Edge Coordinates,Computer Aided Drafting, Design and Manufacturing, 2014, 24(4): 1-6.
In this paper, we propose a novel geometrical detail editing method for triangulated mesh models based on filtering robust differential edge coordinates. The introduced detail editing consists of not only feature-preserving denoising for removing scanner noises, but also interactive detail editing for weakening or enhancing some specific geometric details. Various detail editing results are obtained by reconstructing the mesh from new processed differential edge coordinates, which are filtered from the view of signal processing, in linear least square sense. Experimental results and comparisons with other methods demonstrate that our method is effective and robust.
In this paper, we propose a novel geometrical detail editing method for triangulated mesh models based on filtering robust differential edge coordinates. The introduced detail editing consists of not only feature-preserving denoising for removing scanner noises, but also interactive detail editing for weakening or enhancing some specific geometric details. Various detail editing results are obtained by reconstructing the mesh from new processed differential edge coordinates, which are filtered from the view of signal processing, in linear least square sense. Experimental results and comparisons with other methods demonstrate that our method is effective and robust.
Various detail editing results of a head model. (a) Original mesh; (b)~(e) are results of filtering the differential edge coordinates by band-stop, band-enhancement, low-pass, and high-enhancement filters respectively.
Xiuping Liu, Shuhua Li, Risheng Liu, Jun Wang, Hui Wang (corresponding author), Junjie Cao. Properly constrained orthonormal functional maps for intrinsic symmetries, Computer & Graphics (Special issue of SMI), 2015, 46: 198-208.
Intrinsic symmetry detection, phrased as finding intrinsic self-isometries, courts much attention in recent years. However, extracting dense global symmetry from the shape undergoing moderate non-isometric deformations is still a challenge to the state-of-the-art methods. To tackle this problem, we develop an automatic and robust global intrinsic symmetry detector based on functional maps. The main challenges of applying functional maps lie in how to amend the previous numerical solution scheme and construct reliable and enough constraints. We address the first challenge by formulating the symmetry detection problem as an objective function with descriptor, regional and orthogonality constraints and solving it directly. Compared with refining the functional map by a post-processing, our approach does not break existing constraints and generates more confident results without sacrificing efficiency. To conquer the second challenge, we extract a sparse and stable symmetry-invariant point set from shape extremities and establish symmetry electors based on the transformation, which is constrained by the symmetric point pairs from the set. These electors further cast votes on candidate point pairs to extract more symmetric point pairs. The final functional map is generated with regional constraints constructed from the above point pairs. Experimental results on TOSCA and SCAPE Benchmarks show that our method is superior to the state-of-the-art methods.
Intrinsic symmetry detection, phrased as finding intrinsic self-isometries, courts much attention in recent years. However, extracting dense global symmetry from the shape undergoing moderate non-isometric deformations is still a challenge to the state-of-the-art methods. To tackle this problem, we develop an automatic and robust global intrinsic symmetry detector based on functional maps. The main challenges of applying functional maps lie in how to amend the previous numerical solution scheme and construct reliable and enough constraints. We address the first challenge by formulating the symmetry detection problem as an objective function with descriptor, regional and orthogonality constraints and solving it directly. Compared with refining the functional map by a post-processing, our approach does not break existing constraints and generates more confident results without sacrificing efficiency. To conquer the second challenge, we extract a sparse and stable symmetry-invariant point set from shape extremities and establish symmetry electors based on the transformation, which is constrained by the symmetric point pairs from the set. These electors further cast votes on candidate point pairs to extract more symmetric point pairs. The final functional map is generated with regional constraints constructed from the above point pairs. Experimental results on TOSCA and SCAPE Benchmarks show that our method is superior to the state-of-the-art methods.
The results of our method for nearly self-isometric shapes(centaur, michael, victoria and gorilla).
Shengfa Wang, Junjie Cao, Hui Wang (corresponding author), Baochang Han, Zhixun Su. Primary Correspondences between Intrinsically Symmetrical Shapes, Journal of Information & Computational Science, 2014, 11(9): 2975-2982.
Finding structure-preserving correspondences between pairs of shapes is a common operation in many geometry processing and applications. Usually the ambiguity of primary correspondences and symmetrical correspondences arises in the matching problem between two intrinsically symmetrical shapes. As intrinsically symmetrical shapes are ubiquitous, it is essential to handle this problem. In this paper, we present an approach to establish automatic sparse symmetry-aware correspondences between two shapes using graph matching. We propose a directed-angle scheme to capture the global structure of meshes. Then the graph matching is generalized to high-order potentials, which contains intrinsic descriptors, intrinsic distances and directed-angles among arbitrary three points in deformation-invariant embedding space of the mesh. Comprehensive experiments demonstrate that our method works well.
Finding structure-preserving correspondences between pairs of shapes is a common operation in many geometry processing and applications. Usually the ambiguity of primary correspondences and symmetrical correspondences arises in the matching problem between two intrinsically symmetrical shapes. As intrinsically symmetrical shapes are ubiquitous, it is essential to handle this problem. In this paper, we present an approach to establish automatic sparse symmetry-aware correspondences between two shapes using graph matching. We propose a directed-angle scheme to capture the global structure of meshes. Then the graph matching is generalized to high-order potentials, which contains intrinsic descriptors, intrinsic distances and directed-angles among arbitrary three points in deformation-invariant embedding space of the mesh. Comprehensive experiments demonstrate that our method works well.
Hui Wang, Patricio Simari, Zhixun Su, Hao Zhang. Spectral Global Intrinsic Symmetry Invariant Functions, Graphics Interface 2014, pp. 209-215. [Project Page]
We introduce spectral Global Intrinsic Symmetry Invariant Functions (GISIFs), a class of GISIFs obtained via eigendecomposition of the Laplace-Beltrami operator on compact Riemannian manifolds, and provide associated theoretical analysis. We also discretize the spectral GISIFs for 2D manifolds approximated either by triangle meshes or point clouds. In contrast to GISIFs obtained from geodesic distances, our spectral GISIFs are robust to topological changes. Additionally, for symmetry analysis, our spectral GISIFs represent a more expressive and versatile class of functions than the classical Heat Kernel Signatures (HKSs) and Wave Kernel Signatures (WKSs). Finally, using our defined GISIFs on 2D manifolds, we propose a class of symmetry-factored embeddings and distances and apply them to the computation of symmetry orbits and symmetry-aware segmentations.
We introduce spectral Global Intrinsic Symmetry Invariant Functions (GISIFs), a class of GISIFs obtained via eigendecomposition of the Laplace-Beltrami operator on compact Riemannian manifolds, and provide associated theoretical analysis. We also discretize the spectral GISIFs for 2D manifolds approximated either by triangle meshes or point clouds. In contrast to GISIFs obtained from geodesic distances, our spectral GISIFs are robust to topological changes. Additionally, for symmetry analysis, our spectral GISIFs represent a more expressive and versatile class of functions than the classical Heat Kernel Signatures (HKSs) and Wave Kernel Signatures (WKSs). Finally, using our defined GISIFs on 2D manifolds, we propose a class of symmetry-factored embeddings and distances and apply them to the computation of symmetry orbits and symmetry-aware segmentations.
Spectral Global Intrinsic Symmetry Invariant Functions (GISIFs) computed on a five-point star with rotational symmetries; fi_j denotes a GISIF computed using eigenfunctions of the Laplace-Beltrami operator corresponding to repeated eigenvalues i through j (see Eq. 7).
Hui Wang, Zhixun Su, Junjie Cao, Ye Wang, Hao Zhang. Empirical mode decomposition on surfaces, Graphical Models (Special issue of GMP), 2012, 74(4): 173-183.
Empirical Mode Decomposition (EMD) is a powerful tool for analysing non-linear and nonstationary signals, and has drawn a great deal of attentions in various areas. In this paper,we generalize the classical EMD from Euclidean space to the setting of surfaces represented as triangular meshes. Inspired by the EMD, we also propose a feature-preserving smoothing method based on extremal envelopes. The core of our generalized EMD on surfaces is an envelope computation method that solves a bi-harmonic field with Dirichlet boundary conditions. Experimental results show that the proposed generalization of EMD on surfaces works well. We also demonstrate that the generalized EMD can be effectively utilized in filtering scalar functions defined over surfaces and surfaces themselves.
Empirical Mode Decomposition (EMD) is a powerful tool for analysing non-linear and nonstationary signals, and has drawn a great deal of attentions in various areas. In this paper,we generalize the classical EMD from Euclidean space to the setting of surfaces represented as triangular meshes. Inspired by the EMD, we also propose a feature-preserving smoothing method based on extremal envelopes. The core of our generalized EMD on surfaces is an envelope computation method that solves a bi-harmonic field with Dirichlet boundary conditions. Experimental results show that the proposed generalization of EMD on surfaces works well. We also demonstrate that the generalized EMD can be effectively utilized in filtering scalar functions defined over surfaces and surfaces themselves.
Hui Wang, Hongyin Chen, Zhixun Su, Junjie Cao, Fengshan Liu, Xiquan Shi. Versatile surface detail editing via Laplacian coordinates, The Visual Computer, 2011, 27(4): 401-411.
This paper presents a versatile detail editing approach for triangular meshes based on filtering the Laplacian coordinates. More specifically, we first compute the Laplacian coordinates of the mesh vertices, then filter the Laplacian coordinates, and finally reconstruct the mesh from the filtered Laplacian coordinates by solving a linear least square system. The proposed detail editing method includes not only feature preserving smoothing but also enhancing. Furthermore, the proposed approach allows interactive editing of some user-specified frequencies and regions. Experimental results demonstrate that our method is much more versatile and faster than the existing methods.
This paper presents a versatile detail editing approach for triangular meshes based on filtering the Laplacian coordinates. More specifically, we first compute the Laplacian coordinates of the mesh vertices, then filter the Laplacian coordinates, and finally reconstruct the mesh from the filtered Laplacian coordinates by solving a linear least square system. The proposed detail editing method includes not only feature preserving smoothing but also enhancing. Furthermore, the proposed approach allows interactive editing of some user-specified frequencies and regions. Experimental results demonstrate that our method is much more versatile and faster than the existing methods.
Zhixun Su, Hui Wang, Junjie Cao. Mesh Denoising based on Differential Coordinates, Shape Modeling International 2009, pp.1-6.
In this paper, we propose a novel triangle mesh denoising method based on the differential coordinates. The proposed approach consists of the application of the mean filter to differential coordinates of the mesh and the reconstruction of mesh vertices' Cartesian coordinates to make them fit to the modified differential coordinates. The presented method is simple, stable and able to effectively remove large noise. Experimental results demonstrate that the proposed Mesh Mean Filter does not cause surface shrinkage and shape distortion during the denoising process, and preserves geometric detail features to a certain extent.
In this paper, we propose a novel triangle mesh denoising method based on the differential coordinates. The proposed approach consists of the application of the mean filter to differential coordinates of the mesh and the reconstruction of mesh vertices' Cartesian coordinates to make them fit to the modified differential coordinates. The presented method is simple, stable and able to effectively remove large noise. Experimental results demonstrate that the proposed Mesh Mean Filter does not cause surface shrinkage and shape distortion during the denoising process, and preserves geometric detail features to a certain extent.