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29
Spectral geometry processing with manifold harmonics
 Computer Graphics Forum
, 2008
"... the geometry into frequency space by computing the Manifold Harmonic Transform (MHT). C: Apply the frequency space filter on the transformed geometry. D: Transform back into geometric space by computing the inverse MHT. We present a new method to convert the geometry of a mesh into frequency space. ..."
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Cited by 71 (1 self)
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the geometry into frequency space by computing the Manifold Harmonic Transform (MHT). C: Apply the frequency space filter on the transformed geometry. D: Transform back into geometric space by computing the inverse MHT. We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the LaplaceBeltrami operator are used to define Fourierlike function basis and transform. Since this generalizes the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even just a few eigenvectors is currently impossible for meshes with more than a few thousand vertices. In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometryaware and orthogonal. On the practical side, we propose a bandbyband spectrum computation algorithm and an outofcore implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design.
Randomized Cuts for 3D Mesh Analysis
"... The goal of this paper is to investigate a new shape analysis method based on randomized cuts of 3D surface meshes. The general strategy is to generate a random set of mesh segmentations and then to measure how often each edge of the mesh lies on a segmentation boundary in the randomized set. The re ..."
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Cited by 60 (2 self)
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The goal of this paper is to investigate a new shape analysis method based on randomized cuts of 3D surface meshes. The general strategy is to generate a random set of mesh segmentations and then to measure how often each edge of the mesh lies on a segmentation boundary in the randomized set. The resulting “partition function” defined on edges provides a continuous measure of where natural part boundaries occur in a mesh, and the set of “most consistent cuts ” provides a stable list of global shape features. The paper describes methods for generating random distributions of mesh segmentations, studies sensitivity of the resulting partition functions to noise, tessellation, pose, and intraclass shape variations, and investigates applications in mesh visualization, segmentation, deformation, and registration.
Sparse points matching by combining 3D mesh saliency with statistical descriptors
, 2008
"... This paper proposes new methodology for the detection and matching of salient points over several views of an object. The process is composed by three main phases. In the first step, detection is carried out by adopting a new perceptuallyinspired 3D saliency measure. Such measure allows the detecti ..."
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Cited by 49 (4 self)
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This paper proposes new methodology for the detection and matching of salient points over several views of an object. The process is composed by three main phases. In the first step, detection is carried out by adopting a new perceptuallyinspired 3D saliency measure. Such measure allows the detection of few sparse salient points that characterize distinctive portions of the surface. In the second step, a statistical learning approach is considered to describe salient points across different views. Each salient point is modelled by a Hidden Markov Model (HMM), which is trained in an unsupervised way by using contextual 3D neighborhood information, thus providing a robust and invariant point signature. Finally, in the third step, matching among points of different views is performed by evaluating a pairwise similarity measure among HMMs. An extensive and comparative experimental session has been carried out, considering real objects acquired by a 3D scanner from different points of view, where objects come from standard 3D databases. Results are promising, as the detection of salient points is reliable, and the matching is robust and accurate.
Shape Analysis Using the Auto Diffusion Function
 Comp. Graph. Forum
"... Scalar functions defined on manifold triangle meshes is a starting point for many geometry processing algorithms such as mesh parametrization, skeletonization, and segmentation. In this paper, we propose the Auto Diffusion Function (ADF) which is a linear combination of the eigenfunctions of the Lap ..."
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Cited by 41 (0 self)
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Scalar functions defined on manifold triangle meshes is a starting point for many geometry processing algorithms such as mesh parametrization, skeletonization, and segmentation. In this paper, we propose the Auto Diffusion Function (ADF) which is a linear combination of the eigenfunctions of the LaplaceBeltrami operator in a way that has a simple physical interpretation. The ADF of a given 3D object has a number of further desirable properties: Its extrema are generally at the tips of features of a given object, its gradients and level sets follow or encircle features, respectively, it is controlled by a single parameter which can be interpreted as feature scale, and, finally, the ADF is invariant to rigid and isometric deformations. We describe the ADF and its properties in detail and compare it to other choices of scalar functions on manifolds. As an example of an application, we present a pose invariant, hierarchical skeletonization and segmentation algorithm which makes direct use of the ADF.
SpectralDriven IsometryInvariant Matching of 3D Shapes
, 2009
"... This paper presents a matching method for 3D shapes, which comprises a new technique for surface sampling and two algorithms for matching 3D shapes based on pointbased statistical shape descriptors. Our sampling technique is based on critical points of the eigenfunctions related to the smaller eige ..."
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Cited by 23 (1 self)
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This paper presents a matching method for 3D shapes, which comprises a new technique for surface sampling and two algorithms for matching 3D shapes based on pointbased statistical shape descriptors. Our sampling technique is based on critical points of the eigenfunctions related to the smaller eigenvalues of the LaplaceBeltrami operator. These critical points are invariant to isometries and are used as anchor points of a sampling technique, which extends the farthest point sampling by using statistical criteria for controlling the density and number of reference points. Once a set of reference points has been computed, for each of them we construct a pointbased statistical descriptor (PSSD, for short) of the input surface. This descriptor incorporates an approximation of the geodesic shape distribution and other geometric information describing the surface at that point. Then, the dissimilarity between two surfaces is computed by comparing the corresponding sets of PSSDs with bipartite graph matching or measuring the L1distance between the reordered feature vectors of a proximity graph. Here, the reordering is given by the Fiedler vector of a Laplacian matrix
Inexact Matching of Large and Sparse Graphs Using Laplacian Eigenvectors
"... Abstract. In this paper we propose an inexact spectral matching algorithm that embeds large graphs on a lowdimensional isometric space spanned by a set of eigenvectors of the graph Laplacian. Given two sets of eigenvectors that correspond to the smallest nonnull eigenvalues of the Laplacian matric ..."
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Cited by 12 (3 self)
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Abstract. In this paper we propose an inexact spectral matching algorithm that embeds large graphs on a lowdimensional isometric space spanned by a set of eigenvectors of the graph Laplacian. Given two sets of eigenvectors that correspond to the smallest nonnull eigenvalues of the Laplacian matrices of two graphs, we project each graph onto its eigenenvectors. We estimate the histograms of these onedimensional graph projections (eigenvector histograms) and we show that these histograms are well suited for selecting a subset of significant eigenvectors, for ordering them, for solving the signambiguity of eigenvector computation, and for aligning two embeddings. This results in an inexact graph matching solution that can be improved using a rigid point registration algorithm. We apply the proposed methodology to match surfaces represented by meshes. 1
A Comprehensive Survey on ThreeDimensional Mesh Watermarking
, 2008
"... Threedimensional meshes have been used more and more in industrial, medical and entertainment applications during the last decade. Many researchers, from both the academic and the industrial sectors, have become aware of their intellectual property protection and authentication problems arising wit ..."
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Cited by 12 (6 self)
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Threedimensional meshes have been used more and more in industrial, medical and entertainment applications during the last decade. Many researchers, from both the academic and the industrial sectors, have become aware of their intellectual property protection and authentication problems arising with their increasing use. This paper gives a comprehensive survey on 3D mesh watermarking, which is considered an effective solution to the above two emerging problems. Our survey covers an introduction to the relevant state of the art, an attackcentric investigation, and a list of existing problems and potential solutions. First, the particular difficulties encountered while applying watermarking on 3D meshes are discussed, followed by a presentation and an analysis of the existing algorithms, distinguishing them between fragile techniques and robust techniques. Since the attacks play an important role in the design of 3D mesh watermarking algorithms, we also provide an attackcentric viewpoint of this state of the art. Finally, some future working directions are pointed out especially on the ways of devising robust and blind algorithms and on some new probably promising watermarking feature spaces.
A FAST ITERATIVE METHOD FOR SOLVING THE EIKONAL EQUATION ON TRIANGULATED SURFACES ∗
"... Abstract. This paper presents an efficient, finegrained parallel algorithm for solving the Eikonal equation on triangular meshes. The Eikonal equation, and the broader class of Hamilton–Jacobi equations to which it belongs, have a wide range of applications from geometric optics and seismology to b ..."
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Cited by 8 (4 self)
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Abstract. This paper presents an efficient, finegrained parallel algorithm for solving the Eikonal equation on triangular meshes. The Eikonal equation, and the broader class of Hamilton–Jacobi equations to which it belongs, have a wide range of applications from geometric optics and seismology to biological modeling and analysis of geometry and images. The ability to solve such equations accurately and efficiently provides new capabilities for exploring and visualizing parameter spaces and for solving inverse problems that rely on such equations in the forward model. Efficient solvers on stateoftheart, parallel architectures require new algorithms that are not, in many cases, optimal, but are better suited to synchronous updates of the solution. In previous work [W. K. Jeong and R. T. Whitaker, SIAM J. Sci. Comput., 30 (2008), pp. 2512–2534], the authors proposed the fast iterative method (FIM) to efficiently solve the Eikonal equation on regular grids. In this paper we extend the fast iterative method to solve Eikonal equations efficiently on triangulated domains on the CPU and on parallel architectures, including graphics processors. We propose a new local update scheme that provides solutions of firstorder accuracy for both architectures. We also propose a novel trianglebased update scheme and its corresponding data structure for efficient irregular data mapping to parallel singleinstruction multipledata (SIMD) processors. We provide detailed descriptions of the implementations on a single CPU, a multicore CPU with shared memory, and SIMD architectures with comparative results against stateoftheart Eikonal solvers. Key words. Hamilton–Jacobi equation, Eikonal equation, triangular mesh, parallel algorithm, shared memory multipleprocessor computer system, graphics processing unit
A robust spectral approach for blind watermarking of manifold surfaces
 Proceedings of the 10th ACM Workshop on Multimedia and Security
"... This paper proposes a robust, blind, and imperceptible spectral watermarking approach for manifold surfaces represented as triangle meshes. The basic idea is to transform the original mesh into frequency domain using the FourierLike Manifold Harmonics Transform. The manifold harmonics basis defined ..."
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Cited by 8 (1 self)
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This paper proposes a robust, blind, and imperceptible spectral watermarking approach for manifold surfaces represented as triangle meshes. The basic idea is to transform the original mesh into frequency domain using the FourierLike Manifold Harmonics Transform. The manifold harmonics basis defined on arbitrary topology surfaces is an intrinsic property of the manifold surfaces, i.e., it is only determined by the surface metric and independent of their resolution and embedding. This property makes our watermarking scheme immune to uniform affine attack (rotation, scaling, and translation) and robust against noiseaddition and mesh simplification attacks. The global manifold harmonics are computed using the finite element method combined with a bandbyband algorithm that can compute thousands of eigenvectors for large meshes with up to a million triangles. The watermark data is embedded by modifying the manifold harmonics descriptors magnitude in an imperceptible way. By using global spectral analysis, the detection of such watermarks does not require mesh registration or resampling, and analysis of the statistics of the manifold harmonics descriptors is exploited to devise an optimal blind detector. Experimental results show the imperceptibility of the watermark with low distortions, and its robustness against the most common attacks including the uniform affine transformations, random noise addition, mesh simplification, etc.
Embedding Retrieval of Articulated Geometry Models
"... Abstract—Due to the popularity of computer games and animation, research on 3D articulated geometry model retrieval is attracting a lot of attention in recent years. However, most existing works extract high dimensional features to represent models and suffer from practical limitations. First, misal ..."
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Abstract—Due to the popularity of computer games and animation, research on 3D articulated geometry model retrieval is attracting a lot of attention in recent years. However, most existing works extract high dimensional features to represent models and suffer from practical limitations. First, misalignment in high dimensional features may produce unreliable Euclidean distances and affect retrieval accuracy. Second, the curse of dimensionality also degrades efficiency. In this paper, we propose an embedding retrieval framework to improve the practicability of these methods. It is based on a manifold learning technique, the Diffusion Map (DM). We project all pairwise distances onto a low dimensional space. This improves retrieval accuracy because intercluster distances are exaggerated. Then we adapt the DensityWeighted Nyström extension and further propose a novel step to locally align the Nyström embedding to the eigensolver embedding so as to reduce extension error and preserve retrieval accuracy. Finally, we propose a heuristic to handle disconnected manifolds by augmenting the kernel matrix with multiple similarity measures and shortcut edges, and further discuss the choice of DM parameters. We have incorporated two existing matching algorithms for testing. Our experimental results show improvement in precision at high recalls and in speed. Our work provides a robust retrieval framework for the matching of multimedia data that lie on manifolds. Index Terms—geometry retrieval, articulated model retrieval, geometry analysis, geometry recognition. 1