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Shape Analysis Using the Auto Diffusion Function
Computer Graphics Forum
Abstract
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 Laplace-Beltrami 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.
... Gebal et al. [61] proposed the self-diffusion function (ADF) for shape analysis. ADF is considered to be a linear combination of the squared LBO eigenfunctions. ...
... In their research, they extracted multiple Reeb skeletons by the ADFt method with different t values and then combined them. Gebal proposed a method that extracted two joint regions from both ends of the marginal area of each Reeb graph [61], where U denotes the marginal area of the Reeb graph. When as the step-length. ...
... Vieira [19] adopted the concept of silhouette curvature to measure the complexity of a silhouette. As a modification of the previous contour curvature measurement methods, Secord [61] introduced the extreme value of contour curvature to emphasize high curvature areas of the contour. According to Secord [61], the extreme values of contour curvature can be used to obtain the corresponding optimal viewpoint. ...
Research on viewpoint in 3D scenes has been a cutting-edge subject in computer vision and 3D scene understanding. Calculating a high quality 3D viewpoint in virtual scenes is conducive to a more comprehensive analysis of 3D graphic structure, which can contribute to an overall understanding of virtual environments. 3D viewpoint research can also be applied to the analysis of structural relationships in virtual scenes and to exploit hidden hierarchical structures, thus it plays a role in fields such as scientific visualization computing or image-based scene modeling. Viewpoint research combined with relevant research results of human visual perception or visual psychology can be used to analyze the visual interest and focus areas of 3D objects in human vision, thus effectively improving the quality of mesh simplification and rendering efficiency, as well as providing an optimization basis for highly complex scenes. In fields such as 3D computer games, virtual reality, and landscape animation, research on high-quality viewpoints can be adopted to optimize global lighting and illumination. Additionally, practical applications can be realized through user attention analysis and aesthetic improvement strategies based on high-quality viewpoints, such as intelligent modeling and scene optimization calculations. In common fields such as 3D model retrieval, practical applications can be realized for saliency research, scientific visualization computing, and medical 3D imaging. In-depth research has been performed and extensive applications have been developed that make use of viewpoint analysis. In this paper, we introduce an optimization strategy to calculate high-quality virtual viewpoints for aesthetic images. A novel framework is proposed for modifying and optimizing the viewpoint calculation model by combining a multibranch CNN and a viewpoint correction method, thus realizing rendered images with a higher aesthetic quality. By attempting to fully integrate the visual perception with the geometric information calculation, we expect this method to achieve more comprehensive applications in many practice areas in the future, such as the realization of aesthetically based virtual camera path planning and analysis of the aesthetic characteristics of virtual environments.
... In addition, these points should not be altered by scaling, rotation, additive noise, articulation, and deformation. We propose here 3D salient point detector founded on the Auto Diffusion Function (ADF) introduced by Gbal et al. [9]. This method has been deployed successfully in a previous work [23] in the context of 3D shape retrieval. ...
... The main advantage of the ADF method is that it does not require the setting of any boundary conditions on feature points. In fact, its extremes prove to be the natural feature point in the extremities of the object which exhibit more capacity in capturing salient points as perceived by humans [9]. We note here that other methods have been developed in the literature to extract interest points from the mesh such as 3D Harris detector [27]. ...
... To supervise the number and the capacity of the extracted features, we just vary the parameter t. To make the function scale invariant, the exponential is divided by the second eigenvalue [9]. So the Auto Diffusion Function is defined as: ...
In this paper, we propose a robust and blind watermarking algorithm for 3D mesh models. Firstly, we extract salient feature points using a robust salient point detector based on Auto Diffusion Function (ADF). Afterwards, the mesh is segmented into different regions according to the detected salient points. Finally, we embed the watermark statistically into each region. During watermark embedding, the vertex norms ρ are decomposed into normalized bins. Then, the watermark is embedded by modifying the amplitude ρ depending on the watermark bit and the mean of each bin. In the extraction process, we extract the signature from each re-segmented region. Experiments conducted with a variety of mesh models evidenced the competitive performance of our watermarking scheme, in terms of the robustness and invisibility, when compared to other state of the art methods.
... Ovsjanikov et al. [37] used HKS to achieve a single point correspondence between the models. Gȩbal et al. [38] used the Laplace-Beltrami operator to construct an ADF (Auto Diffusion Function) similar to HKS to realize the shape analysis of 3D point cloud model. Lovato et al. [20] used ADF to achieve the feature points analysis of the 3D human point cloud model upon the method of [38]. ...
... Gȩbal et al. [38] used the Laplace-Beltrami operator to construct an ADF (Auto Diffusion Function) similar to HKS to realize the shape analysis of 3D point cloud model. Lovato et al. [20] used ADF to achieve the feature points analysis of the 3D human point cloud model upon the method of [38]. In the research field of 3D point cloud model segmentation, Benjamin et al. [39] calculate the continuous heat kernel value of 3D model. ...
... The ADF [38] method was used to extract the feature points of human model in literature [20]. The ADF method is developed by the heat kernel (HKS). ...
A novel approach of 3D human model segmentation is proposed, which is based on heat kernel signature and geodesic distance. Through calculating the heat kernel signature of the point clouds of human body model, the local maxima of thermal energy distribution of the model is found, and the set of feature points of the model is obtained. Heat kernel signature has affine invariability which can be used to extract the correct feature points of the human model in different postures. We adopt the method of geodesic distance to realize the hierarchical segmentation of human model after obtaining the semantic feature points of human model. The experimental results show that the method can overcome the defect of geodesic distance feature extraction. The human body models with different postures can be obtained with the model segmentation results of human semantic characteristics.
... In this light, several strategies have been introduced to extract local information in an efficient and theoretically sound fashion. For instance, a popular approach is based on the concept of diffusion geometry (Coifman and Lafon 2006) for the description of 3D shapes (Sun et al. 2009;Gebal et al. 2009;Aubry et al. 2011). The main idea consists of characterizing the neighborhood of a given point through an evolution process that measures how information propagates on the manifold. ...
... Our approach closely resembles methods based on diffusion geometry (Coifman and Lafon 2006;Sun et al. 2009;Gebal et al. 2009;Aubry et al. 2011), especially in its use of an infinite number of paths to characterize points and their relations. Nevertheless, by basing our descriptor directly on discrete time evolution and geodesic distances rather than on the differential operator, such as the Laplace-Beltrami operator, we are able to provide complementary information with respect to existing diffusion-based signatures. ...
... Diffusion-Based Methods. Another trend in shape analysis consists of exploiting diffusion (e.g., heat diffusion) properties on geometric shapes (Coifman and Lafon 2006;Sun et al. 2009;Gebal et al. 2009;Aubry et al. 2011). The general idea is to measure the propagation of information on 3D objects that, in some cases, can be interpreted as a random walk among surface points (Coifman and Lafon 2006;. ...
In shape analysis and matching, it is often important to encode information about the relation between a given point and other points on a shape, namely, its context. To this aim, we propose a theoretically sound and efficient approach for the simulation of a discrete time evolution process that runs through all possible paths between pairs of points on a surface represented as a triangle mesh in the discrete setting. We demonstrate how this construction can be used to efficiently construct a multiscale point descriptor, called the Discrete Time Evolution Process Descriptor, which robustly encodes the structure of neighborhoods of a point across multiple scales. Our work is similar in spirit to the methods based on diffusion geometry, and derived signatures such as the HKS or the WKS, but provides information that is complementary to these descriptors and can be computed without solving an eigenvalue problem. We demonstrate through extensive experimental evaluation that our descriptor can be used to obtain accurate results in shape matching in different scenarios. Our approach outperforms similar methods and is especially robust in the presence of large nonisometric deformations, including missing parts.
... In geometry processing and shape analysis, several applications have been addressed through the properties of the spectral kernels and distances, such as commute-time, biharmonic, diffusion, and wave distances. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and include random walks [Fouss et al. 2005;Ramani and Sinha 2013], heat diffusion [Bronstein et al. 2010a;Coifman and Lafon 2006;Gebal et al. 2009;Lafon et al. 2006;Luo et al. 2009], biharmonic [Lipman et al. 2010;Rustamov 2011b], and wave kernel [Bronstein and Bronstein 2011b;Aubry et al. 2011] distances. Laplacian spectral distances have been applied to shape segmentation [de Goes et al. 2008] and comparison Gebal et al. 2009;Memoli 2009;Ovsjanikov et al. 2010;Sun et al. 2009] with multi-scale and isometry-invariant signatures [Dey et al. 2010b;Lafon et al. 2006;Mèmoli and Sapiro 2005;Memoli 2011;Raviv et al. 2010;Rustamov 2007;Mahmoudi and Sapiro 2009]. ...
... Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and include random walks [Fouss et al. 2005;Ramani and Sinha 2013], heat diffusion [Bronstein et al. 2010a;Coifman and Lafon 2006;Gebal et al. 2009;Lafon et al. 2006;Luo et al. 2009], biharmonic [Lipman et al. 2010;Rustamov 2011b], and wave kernel [Bronstein and Bronstein 2011b;Aubry et al. 2011] distances. Laplacian spectral distances have been applied to shape segmentation [de Goes et al. 2008] and comparison Gebal et al. 2009;Memoli 2009;Ovsjanikov et al. 2010;Sun et al. 2009] with multi-scale and isometry-invariant signatures [Dey et al. 2010b;Lafon et al. 2006;Mèmoli and Sapiro 2005;Memoli 2011;Raviv et al. 2010;Rustamov 2007;Mahmoudi and Sapiro 2009]. In fact, they are intrinsic to the input shape, invariant to isometries, multi-scale, and robust to noise and tessellation. ...
... In fact, they are intrinsic to the input shape, invariant to isometries, multi-scale, and robust to noise and tessellation. Biharmonic [Lipman et al. 2010;Rustamov 2011b] and diffusion [Bronstein et al. 2010a;Coifman and Lafon 2006;Gebal et al. 2009;Lafon et al. 2006;Luo et al. 2009; Patanè and Spagnuolo 2013b] distances provide a tradeoff between a nearly geodesic behavior for small distances and the encoding of global surface properties for large distances, thus guaranteeing an intrinsic and multi-scale characterization of the input shape. The heat kernel [Berard et al. 1994] is also central in diffusion geometry [Belkin and Niyogi 2003;Coifman and Lafon 2006;Gine and Koltchinskii 2006;Singer 2006], dimensionality reduction with spectral embeddings [Belkin and Niyogi 2003;Xiao et al. 2010], and data classification [Smola and Kondor 2003]. ...
In geometry processing and shape analysis, several applications have been addressed through the properties of the spectral kernels and distances, such as commute-time, biharmonic, diffusion, and wave distances. Our survey is intended to provide a background on the properties, discretization, computation, and main applications of the Laplace-Beltrami operator, the associated differential equations (e.g., harmonic equation, Laplacian eigenproblem, diffusion and wave equations), Laplacian spectral kernels and distances (e.g., commute-time, biharmonic, wave, diffusion distances). While previous work has been focused mainly on specific applications of the aforementioned topics on surface meshes, we propose a general approach that allows us to review Laplacian kernels and distances on surfaces and volumes, and for any choice of the Laplacian weights. All the reviewed numerical schemes for the computation of the Laplacian spectral kernels and distances are discussed in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate method with respect to shape representation, computational resources, and target application.
... In geometry processing and shape analysis, several applications have been addressed through the properties of the spectral kernels and distances, such as commute-time, biharmonic, diffusion, and wave distances. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and include random walks [Fouss et al. 2005;Ramani and Sinha 2013], heat diffusion [Bronstein et al. 2010a;Coifman and Lafon 2006;Gebal et al. 2009;Lafon et al. 2006;Luo et al. 2009], biharmonic [Lipman et al. 2010;Rustamov 2011b], and wave kernel [Bronstein and Bronstein 2011b;Aubry et al. 2011] distances. Laplacian spectral distances have been applied to shape segmentation [de Goes et al. 2008] and comparison Gebal et al. 2009;Memoli 2009;Ovsjanikov et al. 2010;Sun et al. 2009] with multi-scale and isometry-invariant signatures [Dey et al. 2010b;Lafon et al. 2006;Mèmoli and Sapiro 2005;Memoli 2011;Raviv et al. 2010;Rustamov 2007;Mahmoudi and Sapiro 2009]. ...
... Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and include random walks [Fouss et al. 2005;Ramani and Sinha 2013], heat diffusion [Bronstein et al. 2010a;Coifman and Lafon 2006;Gebal et al. 2009;Lafon et al. 2006;Luo et al. 2009], biharmonic [Lipman et al. 2010;Rustamov 2011b], and wave kernel [Bronstein and Bronstein 2011b;Aubry et al. 2011] distances. Laplacian spectral distances have been applied to shape segmentation [de Goes et al. 2008] and comparison Gebal et al. 2009;Memoli 2009;Ovsjanikov et al. 2010;Sun et al. 2009] with multi-scale and isometry-invariant signatures [Dey et al. 2010b;Lafon et al. 2006;Mèmoli and Sapiro 2005;Memoli 2011;Raviv et al. 2010;Rustamov 2007;Mahmoudi and Sapiro 2009]. In fact, they are intrinsic to the input shape, invariant to isometries, multi-scale, and robust to noise and tessellation. ...
... In fact, they are intrinsic to the input shape, invariant to isometries, multi-scale, and robust to noise and tessellation. Biharmonic [Lipman et al. 2010;Rustamov 2011b] and diffusion [Bronstein et al. 2010a;Coifman and Lafon 2006;Gebal et al. 2009;Lafon et al. 2006;Luo et al. 2009; Patanè and Spagnuolo 2013b] distances provide a tradeoff between a nearly geodesic behavior for small distances and the encoding of global surface properties for large distances, thus guaranteeing an intrinsic and multi-scale characterization of the input shape. The heat kernel [Berard et al. 1994] is also central in diffusion geometry [Belkin and Niyogi 2003;Coifman and Lafon 2006;Gine and Koltchinskii 2006;Singer 2006], dimensionality reduction with spectral embeddings [Belkin and Niyogi 2003;Xiao et al. 2010], and data classification [Smola and Kondor 2003]. ...
... Rustamov [9] uses d2 distributions to characterize the Global Point Signature values across a mesh for model-tomodel comparison, showing a discernible difference between a variety of different models, while the same model with major and minor deformations would cluster together under the metric. Sun et al. [10] (and, simultaneously, Gȩ bal et al. [11]) address multiscale matching with the Heat Kernel Signature, side stepping the issue of rapid convergence at large time values t, by uniformly sampling values over the logarithmic scaled temporal domain of the signature, taking the L 2 -norm of the resultant vectors as a dissimilarity metric. Ovsjanikov et al. [12] follow-up on this work by describing a technique for point correspondence propagation and symmetry detection by identifying and registering a small number local maxima points of the Heat Kernel Signature within and across whole and partial models. ...
... The method we propose can be applied to any per-vertex signature with a single scalar parameter S(x, t), where x is a point on the shape and t is the scalar parameter. We experimented with signatures whose scalar parameter corresponds to scale: the Heat Kernel Signature [10,11], the Wave Kernel Signature [14], and a new signature we call the Smoothed Shape Diameter Function 3.1, based on smoothing the Shape Diameter Function [5]. ...
Shape similarity is a fundamental problem in geometry processing, enabling applications such as surface correspondence, segmentation, and edit propagation. For example, a user may paint a stroke on one finger of a model and desire the edit to propagate to all fingers. Automatic approaches have difficulty matching user expectations, either due to an algorithm’s inability to guess the scale at which the user is intending to edit or due to underlying deficiencies in the similarity metric (e.g., semantic information not present in the geometry).
We propose an approach to interactively design self-similarity maps. We investigate two primitive operations, useful in a variety of scenarios: region and curve similarity. Users select example similar and dissimilar regions. Starting with an automatically generated multi-scale shape signature, our approach solves for a scale parameter and thresholds that group the example regions as specified. We propose a new Smooth Shape Diameter Signature (SSDS) as a more efficient alternative to the Heat or Wave Kernel Signature. If no such parameters can be found, our approach modifies the shape signature itself. Given a curve drawn on the surface, we perform hybrid discrete/continuous optimization to find similar curves elsewhere.
We apply our approach for interactive editing scenarios: propagating mesh geometry, patterns duplication, and segmentation.
... Spectral kernels and distances are easily defined through a filtering of the Laplacian eigenpairs and include random walks [41], biharmonic [28,45], heat diffusion [10, 11, Preprint submitted to Elsevier arXiv:1906.03900v1 [cs.GR] 10 Jun 2019 17,25], wave [12] kernels and distances. Laplacian spectral distances have been applied to shape segmentation [19] and comparison [11,17,35,50] with multi-scale and isometry-invariant signatures [33,32,31]. ...
... [cs.GR] 10 Jun 2019 17,25], wave [12] kernels and distances. Laplacian spectral distances have been applied to shape segmentation [19] and comparison [11,17,35,50] with multi-scale and isometry-invariant signatures [33,32,31]. Main applications of the heat kernel include dimensionality reduction [7,53,16,4], diffusion geometry [6,47], graphs' embeddings [30] and analysis [18], shape comparison [9,13,43], data representation [54] and classification [34,46,49]. ...
Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion, wave kernel distances) are easily defined through a filtering of the Laplacian eigenpairs. They play a central role in several applications, such as dimensionality reduction with spectral embeddings, diffusion geometry, image smoothing, geometric characterisations and embeddings of graphs. Extending the results recently derived in the discrete setting~\citep{PATANE-STAR2016,PATANE-CGF2017} to the continuous case, we propose a novel definition of the Laplacian spectral kernels and distances, whose approximation requires the solution of a set of inhomogeneous Laplace equations. Their discrete counterparts are equivalent to a set of sparse, symmetric, and well-conditioned linear systems, which are efficiently solved with iterative methods. Finally, we discuss the optimality of the Laplacian spectrum for the approximation of the spectral kernels, the relation between the spectral and Green kernels, and the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues.
... Spectral kernels and distances are easily defined through a filtering of the Laplacian eigenpairs and include random walks [41], biharmonic [28,45], heat diffusion [10,11,17,25], wave [12] kernels and distances. Laplacian spectral distances have been ap-plied to shape segmentation [19] and comparison [11,17,35,50] with multi-scale and isometry-invariant signatures [33,32,31]. ...
... Spectral kernels and distances are easily defined through a filtering of the Laplacian eigenpairs and include random walks [41], biharmonic [28,45], heat diffusion [10,11,17,25], wave [12] kernels and distances. Laplacian spectral distances have been ap-plied to shape segmentation [19] and comparison [11,17,35,50] with multi-scale and isometry-invariant signatures [33,32,31]. Main applications of the heat kernel include dimensionality reduction [7,53,16,4], diffusion geometry [6,47], graphs' embeddings [30] and analysis [18], shape comparison [9,13,43], data representation [54] and classification [34,46,49]. ...
Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion, wave kernel distances) are easily defined through a filtering of the Laplacian eigenpairs. They play a central role in several applications, such as dimensionality reduction with spectral embeddings, diffusion geometry, image smoothing, geometric characterisations and embeddings of graphs. Extending the results recently derived in the discrete setting [38,39] to the continuous case, we propose a novel definition of the Laplacian spectral kernels and distances, whose approximation requires the solution of a set of inhomogeneous Laplace equations. Their discrete counterparts are equivalent to a set of sparse, symmetric, and well-conditioned linear systems, which are efficiently solved with iterative methods. Finally, we discuss the optimality of the Laplacian spectrum for the approximation of the spectral kernels, the relation between the spectral and Green kernels, and the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues.
... Many works represent shapes as graphs of segments. Reeb graphs are sometimes used as a skeletonization of the shape [GBAL09,BB13]. However, Reeb graphs are sensitive to small features in the shape and are therefore not ideal for representing the structure of a shape in a robust and consistent way (see also Figure 11 below). ...
... For the descriptor function, we use the heat kernel signature or HKS [SOG09,GBAL09] for a given range of time steps. In our experiments, we used 15 time steps between t = 0.03 and t = 0.25 (the shapes are normalized to have unit areas). ...
We present a robust method to find region‐level correspondences between shapes, which are invariant to changes in geometry and applicable across multiple shape representations. We generate simplified shape graphs by jointly decomposing the shapes, and devise an adapted graph‐matching technique, from which we infer correspondences between shape regions. The simplified shape graphs are designed to primarily capture the overall structure of the shapes, without reflecting precise information about the geometry of each region, which enables us to find correspondences between shapes that might have significant geometric differences. Moreover, due to the special care we take to ensure the robustness of each part of our pipeline, our method can find correspondences between shapes with different representations, such as triangular meshes and point clouds. We demonstrate that the region‐wise matching that we obtain can be used to find correspondences between feature points, reveal the intrinsic self‐similarities of each shape and even construct point‐to‐point maps across shapes. Our method is both time and space efficient, leading to a pipeline that is significantly faster than comparable approaches. We demonstrate the performance of our approach through an extensive quantitative and qualitative evaluation on several benchmarks where we achieve comparable or superior performance to existing methods.
... Intrinsic spectral geometry processing has thus yielded numerous useful techniques for vision and graphics, often due to its isometry invariance. This includes semi-localized, articulation invariant feature extraction, such as the heat (Gebal et al., 2009;Sun et al., 2009) and wave (Aubry et al., 2011) kernel signatures, later extended to learned generalizations (Boscaini et al., 2015b). Similar techniques can Visual explanation of the use of spectral geometry in characterizing intrinsic versus extrinsic shape. ...
A complete representation of 3D objects requires characterizing the space of deformations in an interpretable manner, from articulations of a single instance to changes in shape across categories. In this work, we improve on a prior generative model of geometric disentanglement for 3D shapes, wherein the space of object geometry is factorized into rigid orientation, non-rigid pose, and intrinsic shape. The resulting model can be trained from raw 3D shapes, without correspondences, labels, or even rigid alignment, using a combination of classical spectral geometry and probabilistic disentanglement of a structured latent representation space. Our improvements include more sophisticated handling of rotational invariance and the use of a diffeomorphic flow network to bridge latent and spectral space. The geometric structuring of the latent space imparts an interpretable characterization of the deformation space of an object. Furthermore, it enables tasks like pose transfer and pose-aware retrieval without requiring supervision. We evaluate our model on its generative modelling, representation learning, and disentanglement performance, showing improved rotation invariance and intrinsic-extrinsic factorization quality over the prior model.
... Also, the signature based approaches including HKS (Sun et al., 2009) and WKS (Aubry et al., 2011) are also not very robust to rotation because they do not explicitly consider rotational robustness in their formulations. The HKS (Heat Kernel Signiture) (Sun et al., 2009;Gebal et al., 2009) ...
Existing state-of-the-art 3D point clouds understanding methods merely perform well in a fully supervised manner. To the best of our knowledge, there exists no unified framework which simultaneously solves the downstream high-level understanding tasks including both segmentation and detection, especially when labels are extremely limited. This work presents a general and simple framework to tackle point clouds understanding when labels are limited. The first contribution is that we have done extensive methodology comparisons of traditional and learnt 3D descriptors for the task of weakly supervised 3D scene understanding, and validated that our adapted traditional PFH-based 3D descriptors show excellent generalization ability across different domains. The second contribution is that we proposed a learning-based region merging strategy based on the affinity provided by both the traditional/learnt 3D descriptors and learnt semantics. The merging process takes both low-level geometric and high-level semantic feature correlations into consideration. Experimental results demonstrate that our framework has the best performance among the three most important weakly supervised point clouds understanding tasks including semantic segmentation, instance segmentation, and object detection even when very limited number of points are labeled. Our method, termed Region Merging 3D (RM3D), has superior performance on ScanNet data-efficient learning online benchmarks and other four large-scale 3D understanding benchmarks under various experimental settings, outperforming current arts by a margin for various 3D understanding tasks without complicated learning strategies such as active learning.
... These properties make them a powerful tool for non-rigid shape analysis. For example, they are used to efficiently compute shape descriptors, such as the the Diffusion Distance [Nadler et al. 2005], the Shape-DNA [Reuter et al. 2005[Reuter et al. , 2006, the Global Point Signature [Rustamov 2007], the Heat Kernel Signature [Sun et al. 2009], the Auto Diffusion Function [Gebal et al. 2009] and the Wave Kernel Signature [Aubry et al. 2011]. Moreover the eigenfunctions are the basis for Functional Maps [Kovnatsky et al. 2013;Litany et al. 2017;Ovsjanikov et al. 2012Ovsjanikov et al. , 2016Rustamov et al. 2013], isospectralization [Cosmo et al. 2019;] and spectral methods in Geometric Deep Learning [Boscaini et al. 2015;Bruna et al. 2014;Sharp et al. 2020]. ...
Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the Laplace--Beltrami operator, for example, are fundamental for methods in shape analysis and mesh processing. The Subspace Iteration Method is a robust solver for these problems. In practice, however, Lanczos schemes are often faster. In this paper, we introduce the Hierarchical Subspace Iteration Method (HSIM), a novel solver for sparse eigenproblems that operates on a hierarchy of nested vector spaces. The hierarchy is constructed such that on the coarsest space all eigenpairs can be computed with a dense eigensolver. HSIM uses these eigenpairs as initialization and iterates from coarse to fine over the hierarchy. On each level, subspace iterations, initialized with the solution from the previous level, are used to approximate the eigenpairs. This approach substantially reduces the number of iterations needed on the finest grid compared to the non-hierarchical Subspace Iteration Method. Our experiments show that HSIM can solve Laplace--Beltrami eigenproblems on meshes faster than state-of-the-art methods based on Lanczos iterations, preconditioned conjugate gradients and subspace iterations.
... The HKS (heat kernel signature) is a descriptor of the local shape which has a range of desirable characteristics, including strength, resistance, and small disturbances to isometric transformations. Gebal et al. [79,80] suggested the concept of HKS for 3D shape and segmentation under the name of the Auto diffusion method independently [81][82][83][84]. ...
The combined influence of new computational tools and techniques with an increase of massive data sets transforms many research fields and can lead to technological breakthroughs that billions of people can make use of it. The past few years have seen remarkable developments in machine learning and especially in deep learning (DL). Techniques developed within those two fields (DL and biology) can now analyze and learn in different formats from a large number of real-world examples. Even though there are a large number of deep learning algorithms, also implemented extensively and are increasing through frameworks and libraries. A large number of open-source applications from academia, business, start-ups, or wider open-source communities speeds up applications development in this area (DL and Biology). This chapter covers a summary of the new concepts and comparisons, as well as developments in deep learning and the use of the biological dataset. It also describes drug-treated and diseased cells capable of effectively scaling computations and efficiently in the era of cell biology. In this chapter, the author introduces deep learning and emerging biological developments, discussion of technology for specifically attraction of deep learning in the biology field. The chapter concludes considering deep learning and current attraction in biology, cell, images, and bioinformatics data set.
... Intrinsic spectral geometry processing has thus yielded numerous useful techniques for vision and graphics, often due to its isometry invariance. This includes semilocalized, articulation invariant feature extraction, such as the heat (Sun et al., 2009;Gebal et al., 2009) and wave (Aubry et al., 2011) kernel signatures, later extended to learned generalizations (Boscaini et al., 2015b). Similar techniques can be applied to a variety of downstream tasks for 3D shapes as well, including correspondence (Rodolà et al., 2017;Ovsjanikov et al., 2012), retrieval (Bronstein et al., 2011), segmentation (Reuter, 2010), analogies (Boscaini et al., 2015a), classification (Masoumi andHamza, 2017), and manipulation (Vallet and Lévy, 2008). ...
A complete representation of 3D objects requires characterizing the space of deformations in an interpretable manner, from articulations of a single instance to changes in shape across categories. In this work, we improve on a prior generative model of geometric disentanglement for 3D shapes, wherein the space of object geometry is factorized into rigid orientation, non-rigid pose, and intrinsic shape. The resulting model can be trained from raw 3D shapes, without correspondences, labels, or even rigid alignment, using a combination of classical spectral geometry and probabilistic disentanglement of a structured latent representation space. Our improvements include more sophisticated handling of rotational invariance and the use of a diffeomorphic flow network to bridge latent and spectral space. The geometric structuring of the latent space imparts an interpretable characterization of the deformation space of an object. Furthermore, it enables tasks like pose transfer and pose-aware retrieval without requiring supervision. We evaluate our model on its generative modelling, representation learning, and disentanglement performance, showing improved rotation invariance and intrinsic-extrinsic factorization quality over the prior model.
... Based on the seminal work of Coifman and Lafon [CL06], such approaches leverage the relation between the geometry of the underlying space and the diffusion process defined on it, as encoded especially by the spectrum of the Laplace-Beltrami operator (LBO, for short). This general strategy has been successfully exploited for the construction of point signatures [SOG09,GBAL09] and shape matching [OBCS ⇤ 12] among other tasks. More recently, progress in this field has shifted towards a more "local" notion of shape analysis [OLCO13,MRCB18], where descriptors are computed only on small and properly selected neighborhoods (Sect. ...
In this paper, we propose a new construction for the Mexican hat wavelets on shapes with applications to partial shape matching. Our approach takes its main inspiration from the well‐established methodology of diffusion wavelets. This novel construction allows us to rapidly compute a multi‐scale family of Mexican hat wavelet functions, by approximating the derivative of the heat kernel. We demonstrate that this leads to a family of functions that inherit many attractive properties of the heat kernel (e.g. local support, ability to recover isometries from a single point, efficient computation). Due to its natural ability to encode high‐frequency details on a shape, the proposed method reconstructs and transfers δ‐functions more accurately than the Laplace‐Beltrami eigenfunction basis and other related bases. Finally, we apply our method to the challenging problems of partial and large‐scale shape matching. An extensive comparison to the state‐of‐the‐art shows that it is comparable in performance, while both simpler and much faster than competing approaches. image
... Based on the seminal work of Coifman and Lafon [CL06], such approaches leverage the relation between the geometry of the underlying space and the diffusion process defined on it, as encoded especially by the spectrum of the Laplace-Beltrami operator (LBO, for short). This general strategy has been successfully exploited for the construction of point signatures [SOG09,GBAL09] and shape matching [OBCS ⇤ 12] among other tasks. More recently, progress in this field has shifted towards a more "local" notion of shape analysis [OLCO13,MRCB18], where descriptors are computed only on small and properly selected neighborhoods (Sect. ...
In this paper, we propose a new construction for Mexican hat wavelets on shapes with applications to partial shape matching. Our approach takes its main inspiration in the well-established methodology of diffusion wavelets. Our novel construction enables us to rapidly compute a multiscale family of Mexican hat wavelet functions, by approximating the derivative of the heat kernel. We demonstrate that this approach allows our functions to inherit many attractive properties of the heat kernel. Due to its natural ability to encode high-frequency details on a shape, our method allows us to reconstruct and transfer $\delta$-functions more accurately than the Laplace-Beltrami eigenfunction basis and other related bases. Finally, we apply our approach to the challenging problems of partial and large-scale shape matching. An extensive comparison to state-of-the-art approaches shows that our method, while comparable in performance, is both simpler and much faster than competing approaches.
... In our approach, we propose here a 3D salient detector based on the Auto Diffusion Function (ADF), introduced by Gbal et al. [6]. This function depends on a time variable in order to control the different levels of details. ...
In this paper, we propose a robust blind watermarking method for 3D object based on Non Negative Matrix Factorization (NMF). Our main idea is to use the NMF basis matrix guide the watermark embedding. We first segment the model into partitions based on its salient points. Then, we apply NMF algorithm to the vertex norms. The distribution of vertex is grouped into bins according to the basis matrix X. Finally, we insert watermark bits in the vertex norm distribution. Experimental results demonstrate that the proposed method achieved good visual quality and also robustness against various attacks.
... The Laplace-Beltrami operator of a 3D model is intrinsically determined by the model itself, which is an isometric invariant being independent to the model's representation. The corresponding eigenfunctions of this operator are also isometric invariants, and these eigenfunctions align well with the protrusions and features of the model [8]. Inspired by these properties of the Laplace-Beltrami eigenfunctions, we present a method for detecting the salient points of a 3D model based on the model's Laplace-Beltrami eigenfunctions. ...
Three-dimensional (3D) salient point detection is a fundamental problem in computer graphics and computer vision. We propose a new method using the Laplace–Beltrami eigenfunctions for detecting the salient points of 3D models. We compute the extrema of the low-frequency Laplace–Beltrami eigenfunctions and remove the redundancy of the extrema. Merging the extrema of the eigenfunctions, we keep the extrema appearing simultaneously on no less than a certain number of eigenfunctions as the candidate salient points. Clustering these extrema, we consider the representatives of the clusters as the final salient points. Our experimental results demonstrate that the proposed method is effective for 3D salient point detection. Besides that, some experiments are also conducted to verify that the proposed method is insensitive to small boundary noise.
... Spectral shape analysis The isometry invariance and the underlying continuous formulation make the Laplace-Beltrami spectrum and eigenfunctions well-suited as a basis for mesh-invariant and pose-invariant shape descriptors and signatures. Examples of such descriptors are the Shape-DNA [RWP05,RWP06], the diffusion distance [NLCK05], the global point signature [Rus07], the heat kernel signature [SOG09], the Auto Diffusion Function [GBAL09] and the wave kernel signature [ASC11]. These shape descriptors can be combined to form bags of features that can be used to design algorithms for pose and mesh invariant shape search and retrieval [BBGO11]. ...
The spectrum and eigenfunctions of the Laplace‐Beltrami operator are at the heart of effective schemes for a variety of problems in geometry processing. A burden attached to these spectral methods is that they need to numerically solve a large‐scale eigenvalue problem, which results in costly precomputation. In this paper, we address this problem by proposing a fast approximation algorithm for the lowest part of the spectrum of the Laplace‐Beltrami operator. Our experiments indicate that the resulting spectra well‐approximate reference spectra, which are computed with state‐of‐the‐art eigensolvers. Moreover, we demonstrate that for different applications that comparable results are produced with the approximate and the reference spectra and eigenfunctions. The benefits of the proposed algorithm are that the cost for computing the approximate spectra is just a fraction of the cost required for numerically solving the eigenvalue problems, the storage requirements are reduced and evaluation times are lower. Our approach can help to substantially reduce the computational burden attached to spectral methods for geometry processing.
... Arguably, a better option is to build deformation-invariant descriptors. As such, DaLI 15 (deformation and light invariant) [Simo-Serra et al., 2015a] uses methods from diffusion geometry [Gebal et al., 2009, Sun et al., 2009] to build a descriptor for 2D image patches which is invariant to nonrigid deformations and photometric changes. A patch is described in terms of the heat it dissipates onto its neighbourhood over time. ...
This paper presents an overview of the evolution of local features from handcrafted to deeplearning-
based methods, followed by a discussion of several benchmarks and papers evaluating such local
features. Our investigations are motivated by 3D reconstruction problems, where the precise location of the
features is important. As we describe these methods, we highlight and explain the challenges of feature extraction
and potential ways to overcome them. We first present handcrafted methods, followed by methods
based on classical machine learning and finally we discuss methods based on deep-learning. This largely
chronologically-ordered presentation will help the reader to fully understand the topic of image and region
description in order to make best use of it in modern computer vision applications. In particular, understanding
handcrafted methods and their motivation can help to understand modern approaches and how machine
learning is used to improve the results.We also provide references to most of the relevant literature and code.
... The analysis of reconstructed 3D bodies includes the surface skeleton extraction, the location of feature points, the analysis of local properties (such as curvature), the measurement of lengths, areas and volumes of specific curves or regions. The registration and alignment of 3D reconstructions with reference models is either based on the location of very few feature points characterized by local shape descriptors (e.g., curvature maps [72], auto diffusion function [73], integral invariants [74], salient ge- ometric features [75]), or on dense correspondence (e.g., multi dimensional scaling methods based on geodesic distances [76]). The correlation between 3D shape measures on the human body and health issues is studied in [77,78,79,80,81], in comparison with classical anthropometric measures and indices. ...
... Heat kernels are specific solutions to the heat transfer problems with unique point sources. These heat kernels can be computed by means of the eigenvectors of the Laplace-Beltrami operator [26,27] . Refs. [28,29] compute the heat potential (tendency to attract heat) of each mesh point in order to identify crucial heat sources which are then used to compute the heat kernels and the underlying segmentation. ...
Mesh segmentation and parameterization are crucial for Reverse Engineering (RE). Bijective parameterizations of the sub-meshes are a sine-qua-non test for segmentation. Current segmentation methods use either (1) topologic or (2) geometric criteria to partition the mesh. Reported topology-based segmentations produce large sub-meshes which reject parameterizations. Geometry-based segmentations are very sensitive to local variations in dihedral angle or curvatures, thus producing an exaggerated large number of small sub-meshes. Although small sub-meshes accept nearly isometric parameterizations, this significant granulation defeats the intent of synthesizing a usable Boundary Representation (compulsory for RE). In response to these limitations, this article presents an implementation of a hybrid geometry / topology segmentation algorithm for mechanical workpieces. This method locates heat transfer constraints (topological criterion) in low frequency neighborhoods of the mesh (geometric criterion) and solves for the resulting temperature distribution on the mesh. The mesh partition dictated by the temperature scalar map results in large, albeit parameterizable, sub-meshes. Our algorithm is tested with both benchmark repository and physical piece scans data. The experiments are successful, except for the well - known cases of topological cylinders, which require a user - introduced boundary along the cylinder generatrices.
... We start by detecting the limbs of a given model using an automatic and unsupervised 3D salient point detector. We use the local maximum of the Auto Diffusion Function (ADF) [24] defined on the mesh surface as: ...
In this paper, we propose a 3D non-rigid shape retrieval method based on canonical shape analysis. Our main idea is to transform the problem of non-rigid shape retrieval into a rigid shape retrieval problem via the well-known multidimensional scaling (MDS) approach and random walk on graphs. We first segment the non-rigid shape into local partitions based on its salient features. Then, we calculate a local MDS problem for each partition, where the local commute time distance is used as weighting function in order to preserve local shape details. Finally, we aggregate the set of local MDS problems as a global constrained problem. The constraint is formulated using the biharmonic function between local salient features. In contrast to MDS method, the proposed local MDS is computationally efficient, parameters free and gives isometry-invariant forms with minimum features distortion. Due to these advantageous properties, the proposed method achieved good retrieval accuracy on non-rigid shape benchmark datasets.
... Oltre alle geodetiche, sono possibili scelte più sofisticate, come le diffusion e commute-time distanze [32]. Sulla base del fatto che queste distanze sono ben approssimate dall'operatore di Laplace-Beltrami, sono state proposte diverse descrizioni spettrali per caratterizzare le feature geometriche di forme tridimensionali non rigide [33], come la ShapeDNA [34], la heat kernel signature [35,36], la wave kernel signature [37], la global point signature [38] e la specral graph wavelet signature [39]. ...
... For a long time, researchers tried to formalize good 3D shape signatures. Sun et al. [27] and Gebal et al. [11] introduced heat kernel signature. Aubry et al. [1] introduced wave kernel signature. ...
3D volumetric object generation/prediction from single 2D image is a quite challenging but meaningful task in 3D visual computing. In this paper, we propose a novel neural network architecture, named "3DensiNet", which uses density heat-map as an intermediate supervision tool for 2D-to-3D transformation. Specifically, we firstly present a 2D density heat-map to 3D volumetric object encoding-decoding network, which outperforms classical 3D autoencoder. Then we show that using 2D image to predict its density heat-map via a 2D to 2D encoding-decoding network is feasible. In addition, we leverage adversarial loss to fine tune our network, which improves the generated/predicted 3D voxel objects to be more similar to the ground truth voxel object. Experimental results on 3D volumetric prediction from 2D images demonstrates superior performance of 3DensiNet over other state-of-the-art techniques in handling 3D volumetric object generation/prediction from single 2D image.
... Many popular shape descriptors such as heat- [SOG09,GBAL09] and wave- [ASC11] kernel signatures, global point signatures [Rus07], and shape DNAs [RWP06] were constructed in the spectral domain. Coifman et al. introduced the notion of diffusion distances [CL06] on non-Euclidean domains, also constructed considering the spectral decomposition of heat operators. ...
The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback of such bases is their inherently global nature, as the Laplacian eigenfunctions carry geometric and topological structure of the entire manifold. In this paper, we introduce a new framework for local spectral shape analysis. We show how to efficiently construct localized orthogonal bases by solving an optimization problem that in turn can be posed as the eigendecomposition of a new operator obtained by a modification of the standard Laplacian. We study the theoretical and computational aspects of the proposed framework and showcase our new construction on the classical problems of shape approximation and correspondence. We obtain significant improvement compared to classical Laplacian eigenbases as well as other alternatives for constructing localized bases.
... In addition, those points are not altered by scaling, rotation, additive noise, articulation, and deformation. In this context, Haj-Mohamed and Belaid[12]presented an unsupervised and automatic 3D salient point detector founded on Auto Diffusion Function ADF introduced by Gbal et al.[13]. This scalar function is defined as a linear sequence of the Laplace-Beltrami Operator eigenfunctions. ...
In this paper, we present a novel robust blind 3D mesh watermarking approach. We embed signature bits into the vertex norms distribution. At first, the robust source locations are extracted by using a salient point detector, based on the Auto Diffusion Function (ADF). Afterwards, the mesh is segmented into different regions according to the detected salient points. Then, the same watermark bits are embedded statistically into each region. The experimental results show the robustness of our method against cropping and other common attacks. Due to the stability of salient points, we can retrieve the watermarked region and extract the watermark. In addition, the performance of our method is also demonstrated on the minimal surface distortion in the embedding process.
... Gȩbal et al. [12] for 3D shape skeletonization and segmentation under the name of auto diffusion function. To give rise to substantially more accurate matching than HKS, the wave kernel signature (WKS) [13] was proposed as an alternative in an effort to allow access to high-frequency information. ...
Spectral shape descriptors have been used extensively in a broad spectrum of geometry processing applications ranging from shape retrieval and segmentation to classification. In this pa- per, we propose a spectral graph wavelet approach for 3D shape classification using the bag-of-features paradigm. In an effort to capture both the local and global geometry of a 3D shape, we present a three-step feature description framework. First, local descriptors are extracted via the spectral graph wavelet transform having the Mexican hat wavelet as a generating ker- nel. Second, mid-level features are obtained by embedding lo- cal descriptors into the visual vocabulary space using the soft- assignment coding step of the bag-of-features model. Third, a global descriptor is constructed by aggregating mid-level fea- tures weighted by a geodesic exponential kernel, resulting in a matrix representation that describes the frequency of appearance of nearby codewords in the vocabulary. Experimental results on two standard 3D shape benchmarks demonstrate the effective- ness of the proposed classification approach in comparison with state-of-the-art methods.
... Our method borrows the non-local processing idea from this thread of work. Several other descriptors gather different kind of informations and are particularly efficient for taking into account the shape at different scales [GBAL09]. However, a complete survey of such descriptors is beyond the scope of this paper since they are not well suited for resampling locally the surface from them. ...
3D scanners provide a virtual representation of object surfaces at some given precision that depends on many factors such as the object material, the quality of the laser-ray or the resolution of the camera. This precision may even vary over the surface, depending for example on the distance to the scanner which results in uneven and unstructured point sets, with an uncertainty on the coordinates. To enhance the quality of the scanner output, one usually resorts to local surface interpolation between measured points. However, object surfaces often exhibit interesting statistical features such as repetitive geometric textures. Building on this property, we propose a new approach for surface super-resolution that detects repetitive patterns or self-similarities and exploits them to improve the scan resolution by aggregating scattered measures. In contrast with other surface super-resolution methods, our algorithm has two important advantages. First, when handling multiple scans, it does not rely on surface registration. Second, it is able to produce super-resolution from even a single scan. These features are made possible by a new local shape description able to capture differential properties of order above 2. By comparing those descriptors, similarities are detected and used to generate a high-resolution surface. Our results show a clear resolution gain over state-of-the-art interpolation methods.
... Finally, the heat kernel is stable under small perturbations of the underlying manifold. All these properties make the heat kernel a good candidate for the definition of informative functions and distances to be used for shape description, such as the heat kernel signature (HKS) (Sun et al. 2009;Gebal et al. 2009) and the diffusion function. The HKS at a time , denoted by , is defined as: ...
In GRAVITATE, two disparate specialities will come together in one working platform for the archaeologist: the fields of shape analysis, and of metadata search. These fields are relatively disjoint at the moment, and the research and development challenge of GRAVITATE is precisely to merge them for our chosen tasks. As shown in chapter 7 the small amount of literature that already attempts join 3D geometry and semantics is not related to the cultural heritage domain. Therefore, after the project is done, there should be a clear ‘before-GRAVITATE’ and ‘after-GRAVITATE’ split in how these two aspects of a cultural heritage artefact are treated. This state of the art report (SOTA) is ‘before-GRAVITATE’. Shape analysis and metadata description are described separately, as currently in the literature and we end the report with common recommendations in chapter 8 on possible or plausible cross-connections that suggest themselves. These considerations will be refined for the Roadmap for Research deliverable. Within the project, a jargon is developing in which ‘geometry’ stands for the physical properties of an artefact (not only its shape, but also its colour and material) and ‘metadata’ is used as a general shorthand for the semantic description of the provenance, location, ownership, classification, use etc. of the artefact. As we proceed in the project, we will find a need to refine those broad divisions, and find intermediate classes (such as a semantic description of certain colour patterns), but for now the terminology is convenient – not least because it highlights the interesting area where both aspects meet. On the ‘geometry’ side, the GRAVITATE partners are UVA, Technion, CNR/IMATI; on the metadata side, IT Innovation, British Museum and Cyprus Institute; the latter two of course also playing the role of internal users, and representatives of the Cultural Heritage (CH) data and target user’s group. CNR/IMATI’s experience in shape analysis and similarity will be an important bridge between the two worlds for geometry and metadata. The authorship and styles of this SOTA reflect these specialisms: the first part (chapters 3 and 4) purely by the geometry partners (mostly IMATI and UVA), the second part (chapters 5 and 6) by the metadata partners, especially IT Innovation while the joint overview on 3D geometry and semantics is mainly by IT Innovation and IMATI. The common section on Perspectives was written with the contribution of all.
This work investigates efficient score-based black-box adversarial attacks with a high Attack Success Rate (ASR) and good generalizability. We design a novel attack method based on a \textit{Hierarchical} \textbf{Di}sentangled \textbf{F}eature space and \textit{cross domain}, called \textbf{DifAttack++}, which differs significantly from the existing ones operating over the entire feature space. Specifically, DifAttack++ firstly disentangles an image's latent feature into an \textit{adversarial feature} (AF) and a \textit{visual feature} (VF) via an autoencoder equipped with our specially designed \textbf{H}ierarchical \textbf{D}ecouple-\textbf{F}usion (HDF) module, where the AF dominates the adversarial capability of an image, while the VF largely determines its visual appearance. We train such autoencoders for the clean and adversarial image domains respectively, meanwhile realizing feature disentanglement, by using pairs of clean images and their Adversarial Examples (AEs) generated from available surrogate models via white-box attack methods. Eventually, in the black-box attack stage, DifAttack++ iteratively optimizes the AF according to the query feedback from the victim model until a successful AE is generated, while keeping the VF unaltered. Extensive experimental results demonstrate that our method achieves superior ASR and query efficiency than SOTA methods, meanwhile exhibiting much better visual quality of AEs. The code is available at https://github.com/csjunjun/DifAttack.git.
Background
In recent geometric design, many effective toolkits for geometric modeling and optimization have been proposed and applied in practical cases, while effective and efficient designing of shapes that have desirable topological properties remains to be a challenge. The development of computational topology, especially persistent homology, permits convenient usage of topological invariants in shape analysis, geometric modeling, and shape optimization. Persistence diagram, the useful topological summary of persistent homology, provides a stable representation of multiscale homology invariants in the presence of noise in original data. Recent works show the wide use of persistent homology tools in geometric design.
Objective
In this paper, we review the geometric design based on computational topological tools in three aspects: the extraction of topological features and representations, topology-aware shape modeling, and topology-based shape optimization.
Methods
By tracking the development of each aspect and comparing the methods using classical topological invariants, motivations, and key approaches of important related works based on persistent homology are clarified.
Results
We review geometric design through topological extraction, topological design, and shape optimization based on topological preservation. Related works show the successful applications of computational topology tools of geometric design.
Conclusion
Solutions for the proposed core problems will affect the geometric design and its applications. In the future, the development of computational topology may boost computer-aided topological design.
Basis functions provide a simple and effective way to interpolate signals on a surface with an arbitrary dimension, topology, and discretisation. However, the definition of local and shape-aware basis functions is still an open problem, which has been addressed through optimisation methods that are time-consuming, over-constrained, or admit more than one solution. In this context, we review the definition, properties, computation, and applications of the class of Laplacian spectral basis functions, which are defined as solutions to the harmonic equation, the diffusion equation, and PDEs involving the Laplace-Beltrami operator. The resulting functions are efficiently computed through the solution of sparse linear systems and satisfy different properties, such as smoothness, locality, and multi-scale shape encoding. Finally, the discussion of the properties of the Laplacian spectral basis functions is integrated with an analysis of their behaviour with respect to different measures (i.e., conformal and area-preserving metrics, transformation distance) and of their applications to spectral geometry processing and shape analysis.
The research on 3D scene viewpoints has been a frontier problem in computer graphics and virtual reality technology. In a pioneering study, it had been extensively used in virtual scene understanding, image-based modeling, and visualization computing. With the development of computer graphics and the human-computer interaction, the viewpoint evaluation becomes more significant for the comprehensive understanding of complex scenes. The high-quality viewpoints could navigate observers to the region of interest, help subjects to seek the hidden relations of hierarchical structure, and improve the efficiency of virtual exploration. These studies later contributed to research such as robot vision, dynamic scene planning, virtual driving and artificial intelligence navigation.The introduction of visual perception had The introduction of visual perception had contributed to the inspiration of viewpoints research, and the combination with machine learning made significant progress in the viewpoints selection. The viewpoints research also has been significant in the optimization of global lighting, visualization calculation, 3D supervising rendering, and reconstruction of a virtual scene. Additionally, it has a huge potential in novel fields such as 3D model retrieval, virtual tactile analysis, human visual perception research, salient point calculation, ray tracing optimization, molecular visualization, and intelligent scene computing. Keywords: View point, Three-dimensional scene, Visual perception, Mesh saliency, Curvature
Step functions are non-smooth and piecewise constant functions with a finite number of pieces. Each of these pieces indicates a local region contained in the entire domain. Several geometry processing applications involve step functions defined on non-Euclidean domains, such as shape segmentation, partial matching and self-similarity detection. Standard signal processing cannot handle this class of functions. The classical Fourier series, for instance, does not give a good representation of these non-smooth functions. In this paper, we define a new frame for the sparse approximation and transfer of the step functions defined on manifolds. The definition of our frame is completely spectral and provides a concise representation through an efficient computation. We exploit the sparse representation that takes full advantage of the proposed frame. Furthermore, our frame is built specifically to enhance its use in combination with the functional maps, a powerful tool for transferring signals between manifolds. This functional approach makes the proposed framework stable to isometric and non-isometric deformations. A large set of experiments confirms that the proposed frame improves the sparse approximation and transfer of step functions.
This paper presents an optimized wave kernel signature (OWKS) using a modified particle swarm optimization (MPSO) algorithm. The variance parameter and its setting mode play a central role in this kernel. In order to circumvent a purely arbitrary choice of the internal parameters of the WKS algorithm, we present a four-step feature descriptor framework in an effort to further improve the classical wave kernel signature (WKS) by acting on its variance parameter. The advantage of the enhanced method comes from the tuning of the variance parameter using MPSO and the selection of the first vector from the constructed OWKS at its first energy scale, thus giving rise to substantially better matching and retrieval accuracy for deformable 3D shape. The special choice of this vector is to extremely reinforce the stability for efficient salient features extraction method from the 3D meshes. Experimental results demonstrate the effectiveness of our proposed shape classification and retrieval approach in comparison with state-of-the-art methods. For instance, in terms of the nearest neighbor (NN) metric, the OWKS achieves a 96.9% score, with performance improvements of 83.5 and 90.4% over the baseline methods WKS and heat kernel signature, respectively.
This paper presents an optimized wave kernel signature (OWKS) using a modified particle swarm optimization (MPSO) algorithm. The variance parameter and its setting mode play a central role in this kernel. In order to circumvent a purely arbitrary choice of the internal parameters of the WKS algorithm, we present a four-step feature descriptor framework in an effort to further improve the classical wave kernel signature (WKS) by acting on its variance parameter. The advantage of the enhanced method comes from the tuning of the variance parameter using MPSO and the selection of the first vector from the constructed OWKS at its first energy scale, thus giving rise to substantially better matching and retrieval accuracy for deformable 3D shape. The special choice of this vector is to extremely reinforce the stability for efficient salient features extraction method from the 3D meshes. Experimental results demonstrate the effectiveness of our proposed shape classification and retrieval approach in comparison with state-of-the-art methods. For instance, in terms of the nearest neighbor (NN) metric, the OWKS achieves a 96.9% score, with performance improvements of 83.5 and 90.4% over the baseline methods WKS and heat kernel signature, respectively.
Computing solutions to linear systems is a fundamental building block of many geometry processing algorithms. In many cases the Cholesky factorization of the system matrix is computed to subsequently solve the system, possibly for many right-hand sides, using forward and back substitution. We demonstrate how to exploit sparsity in both the right-hand side and the set of desired solution values to obtain significant speedups. The method is easy to implement and potentially useful in any scenarios where linear problems have to be solved locally. We show that this technique is useful for geometry processing operations, in particular we consider the solution of diffusion problems. All problems profit significantly from sparse computations in terms of runtime, which we demonstrate by providing timings for a set of numerical experiments.
Quantitative shape comparison is a fundamental problem in computer vision, geometry processing
and medical imaging. In this paper, we present a spectral graph wavelet approach for shape analysis of
carpal bones of the human wrist. We employ spectral graph wavelets to represent the cortical surface
of a carpal bone via the spectral geometric analysis of the Laplace-Beltrami operator in the discrete domain.
We propose global spectral graph wavelet (GSGW) descriptor that is isometric invariant, efficient
to compute, and combines the advantages of both low-pass and band-pass filters. We perform experiments
on shapes of the carpal bones of ten women and ten men from a publicly-available database of
wrist bones. Using one-way multivariate analysis of variance (MANOVA) and permutation testing, we
show through extensive experiments that the proposed GSGW framework gives a much better performance
compared to the global point signature (GPS) embedding approach for comparing shapes of the
carpal bones across populations.
The heat kernel is a fundamental geometric object associated to every Riemannian manifold, used across applications in computer vision, graphics, and machine learning. In this article, we propose a novel computational approach to estimating the heat kernel of a statistically sampled manifold (e.g. meshes or point clouds), using its representation as the transition density function of Brownian motion on the manifold. Our approach first constructs a set of local approximations to the manifold via moving least squares. We then simulate Brownian motion on the manifold by stochastic numerical integration of the associated Ito diffusion system. By accumulating a number of these trajectories, a kernel density estimation method can then be used to approximate the transition density function of the diffusion process, which is equivalent to the heat kernel. We analyse our algorithm on the 2-sphere, as well as on shapes in 3D. Our approach is readily parallelizable and can handle manifold samples of large size as well as surfaces of high co-dimension, since all the computations are local. We relate our method to the standard approaches in diffusion geometry and discuss directions for future work.
In this paper, we present a novel approach to compute 3D canonical forms which is useful for non-rigid 3D shape retrieval. We resort to using the feature space to get a compact representation of points in a small-dimensional Euclidean space. Our aim is to improve the classical Multi-Dimensional Scaling MDS algorithm to avoid the super-quadratic computational complexity. To this end, we compute the canonical form of the local geodesic distance matrix between pairs of a small subset of vertices in local feature patches. To preserve local shape details, we drive the mesh deformation by the local weighted commute time. When used as a spatial relationship between local features, the invariant properties of the Biharmonic distance improve the final results.
Salient region detection without prior knowledge is a challenging task, especially for 3D deformable shapes. This paper presents a novel framework that relies on clustering of a data set derived from the scale space of the auto diffusion function. It consists of three major techniques: scalar field construction, shape segmentation initialization and salient region detection. We define the scalar field using the auto diffusion function at consecutive time scales to reveal shape features. Initial segmentation of a shape is obtained using persistence-based clustering, which is performed on the scalar field at a large time scale to capture the global shape structure. We propose two measures to assess the clustering both on a global and local level using persistent homology. From these measures, salient regions are detected during the evolution of the scalar field. Experimental results on three popular datasets demonstrate the superior performance of the proposed framework in region detection.
Given a triangle mesh representing a closed manifold surface of arbitrary genus, a method is proposed to automatically extract the Reeb graph of the manifold with respect to the height function. The method is based on a slicing strategy that traces contours while inserting them directly in the mesh as constraints. Critical areas, which identify isolated and non-isolated critical points of the surface, are recognized and coded in the extended Reeb graph (ERG). The remeshing strategy guarantees that topological features are correctly maintained in the graph, and the tiling of ERG nodes reproduces the original shape at a minimal, but topologically correct,
geometric level.
Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projec-tions derived from appropriately defined mesh operators to carry out desired tasks. Early works in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis of mesh geometry based on a combinatorial Laplacian aids our understanding of the low-pass filtering approach to mesh smoothing. Over the past ten years or so, the list of applications in the area of geometry processing which utilize the eigenstructures of a variety of mesh operators in different manners have been growing steadily. Many works presented so far draw parallels from developments in fields such as graph theory, computer vision, machine learning, graph drawing, numerical linear algebra, and high-performance computing. This state-of-the-art report aims to provide a com-prehensive survey on the spectral approach, focusing on its power and versatility in solving geometry processing problems and attempting to bridge the gap between relevant research in computer graphics and other fields. Nec-essary theoretical background will be provided and existing works will be classified according to different criteria — the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embed-dings used — and described in adequate length. Finally, despite much empirical success, there still remain many open questions pertaining to the spectral approach, which we will discuss in the report as well.
Given atriangle mesh representing aclosed manifold surface of arbitrary genus, amethod is proposed to automatically extract the Reeb graph of the manifold with respect to the height function. The method is based on aslicing strategy that traces contours while inserting them directly in the mesh as constraints. Critical areas, which identify isolated and non-isolated critical points of the surface, are recognized and coded in the extended Reeb graph (ERG). The remeshing strategy guarantees that topological features are correctly maintained in the graph, and the tiling of ERG nodes reproduces the original shape at aminimal, but topologically correct, geometric level.
Shape analysis plays a pivotal role in a large number of applications, ranging from traditional geometry processing to more recent 3D content management. In this scenario, spectral methods are extremely promising as they provide a natural library of tools for shape analysis, intrinsically defined by the shape itself. In particular, the eigenfunctions of the Laplace–Beltrami operator yield a set of real-valued functions that provide interesting insights in the structure and morphology of the shape. In this paper, we first analyze different discretizations of the Laplace–Beltrami operator (geometric Laplacians, linear and cubic FEM operators) in terms of the correctness of their eigenfunctions with respect to the continuous case. We then present the family of segmentations induced by the nodal sets of the eigenfunctions, discussing its meaningfulness for shape understanding.
Matching articulated shapes represented by voxel-sets reduces to maximal sub-graph isomorphism when each set is described by a weighted graph. Spectral graph the- ory can be used to map these graphs onto lower dimen- sional spaces and match shapes by aligning their embed- dings in virtue of their invariance to change of pose. Clas- sical graph isomorphism schemes relying on the ordering of the eigenvalues to align the eigenspaces fail when handling large data-sets or noisy data. We derive a new formula- tion that finds the best alignment between two congruent K-dimensional sets of points by selecting the best subset of eigenfunctions of the Laplacian matrix. The selection is done by matching eigenfunction signatures built with his- tograms, and the retained set provides a smart initializa- tion for the alignment problem with a considerable impact on the overall performance. Dense shape matching casted into graph matching reduces then, to point registration of embeddings under orthogonal transformations; the regis- tration is solved using the framework of unsupervised clus- tering and the EM algorithm. Maximal subset matching of non identical shapes is handled by defining an appropriate outlier class. Experimental results on challenging examples show how the algorithm naturally treats changes of topol- ogy, shape variations and different sampling densities.
The aim of this paper is to describe a conceptual model for surface representation based on topological coding, which de.nes
a sketch of a surface usable for classi.cation or compression purposes. Theoretical approaches based on di.erential topology
and geometry have been used for surface coding, for example Morse theory and Reeb graphs. To use these approaches in discrete
geometry, it is necessary to adapt concepts developed for smooth manifolds to discrete surface models, as for example piecewise
linear approximations. A typical problem is represented by degenerate critical points, that is non-isolated critical points
such as plateaux and .at areas of the surface. Methods proposed in literature either do not consider the problem or propose
local adjustments of the surface, which solve the theoretical problem but may lead to a wrong interpretation of the shape
by introducing artefacts, which do not correspond to any shape feature. In this paper, an Extended Reeb Graph representation
(ERG) is proposed, which can handle degenerate critical points, and an algorithm is presented for its construction.
This paper introduces a method to extract fingerprints of any surface or solid object by taking the eigenvalues of its respective Laplace-Beltrami operator. Using an object's spectrum (i.e. the family of its eigenvalues) as a fingerprint for its shape is motivated by the fact that the related eigenvalues are isometry invariants of the object. Employing the Laplace-Beltrami spectra (not the spectra of the mesh Laplacian) as fingerprints of surfaces and solids is a novel approach in the field of geometric modeling and computer graphics. Those spectra can be calculated for any representation of the geometric object (e.g. NURBS or any parametrized or implicitly represented surface or even for polyhedra). Since the spectrum is an isometry invariant of the respective object this fingerprint is also independent of the spatial position. Additionally the eigenvalues can be normalized so that scaling factors for the geometric object can be obtained easily. Therefore checking if two objects are isometric needs no prior alignment (registration / localization) of the objects, but only a comparison of their spectra. With the help of such fingerprints it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids.
Curve-skeleton,is a very useful 1D structure to abstract the geometry,and topology,of a 3D object. Extraction of curve-skeletons is a fundamental,problem in computer graphics, visualization, image processing and computer vision. There many useful applications including virtual colonoscopies, collision detection, computer animation, surface reconstruction and shape matching etc. In the literature [1][2], most previous methods require a volumetric discrete representation of the input model. However, transforming them into volumetric,representations,may,raise discretization,error in both geometry,and connectivity. In this work [3], we propose a novel technique to extract skeletons directly from the mesh domain
Resampling raw surface meshes is one of the most fundamental operations used by nearly all digital geometry processing systems. The vast majority of this work has focused on triangular remeshing, yet quadrilateral meshes are preferred for many surface PDE prob- lems, especially fluid dynamics, and are best suited for defining Catmull-Clark subdivision surfaces. We describe a fundamentally new approach to the quadrangulation of manifold polygon meshes using Laplacian eigenfunctions, the natural harmonics of the sur- face. These surface functions distribute their extrema evenly across a mesh, which connect via gradient flow into a quadrangular base mesh. An iterative relaxation algorithm simultaneously refines this initial complex to produce a globally smooth parameterization of the surface. From this, we can construct a well-shaped quadrilat- eral mesh with very few extraordinary vertices. The quality of this mesh relies on the initial choice of eigenfunction, for which we de- scribe algorithms and hueristics to efficiently and effectively select the harmonic most appropriate for the intended application.
Modern techniques for interrogating the topology (e.g. genus) of a single meshed surface find handles via a breadth-first search for cycles that do not separate the mesh into two disjoint components. Morse theory can find handles with a simple search for the minima, maxima and saddles of a real function evaluated at the vertices. Bad choices for this function can generate numerous critical vertices that complicate the search for non-separating cycles. We show that the limit of the iteration of a constrained Laplacian on a mesh yields a "fair" Morse function with a single minumum and maximum, and two critical points for every handle. We then describe an algorithm that uses this function and its critical points to find a bouquet of non-separating cycles that cut the mesh into a flattenable domain homeomorphic to a disk. These cycles can be further optimized for applications in texturing, parameterization, geometry images and filtering topological noise.
Mesh segmentation has become a necessary ingredient in many applications in computer graphics. This paper proposes a novel hierarchical mesh segmentation algorithm, which is based on new methods for prominent feature point and core extraction. The algorithm has several benefits. First, it is invariant both to the pose of the model and to different proportions between the model’s components. Second, it produces correct hierarchical segmentations of meshes, both in the coarse levels of the hierarchy and in the fine levels, where tiny segments are extracted. Finally, the boundaries between the segments go along the natural seams of the models.
In this paper we propose a novel approach of computing skeletons of robust topology for simply connected surfaces with boundary by constructing Reeb graphs from the eigenfunctions of an anisotropic Laplace-Beltrami operator. Our work brings together the idea of Reeb graphs and skeletons by incorporating a flux-based weight function into the Laplace-Beltrami operator. Based on the intrinsic geometry of the surface, the resulting Reeb graph is pose independent and captures the global profile of surface geometry. Our algorithm is very efficient and it only takes several seconds to compute on neuroanatomical structures such as the cingulate gyrus and corpus callosum. In our experiments, we show that the Reeb graphs serve well as an approximate skeleton with consistent topology while following the main body of conventional skeletons quite accurately.
Given a manifold surface M and a continuous function f : M rarr R, the Reeb graph of (M, f) is a widely-used high-level descriptor of M and its usefulness has been demonstrated for a variety of applications, which range from shape parameterization and abstraction to deformation and comparison. In this context, we propose a novel computation of the Reeb graph that is based on the analysis of the iso-contours solely at saddle points and does not require sampling or sweeping the image of f. Furthermore, the proposed approach does not use global sorting steps of the function values and exploits only a local information on f, without handling it as a whole. By combining the minimal number of nodes in the Reeb graph with the use of a small amount of memory footprint and temporary data structures, the overall computation takes O(sn)-time, where n is the number of vertices of the triangulation of M and s is the number of saddles of f. Finally, the technique can be easily extended to compute the Reeb graphs of time-varying functions.
In this paper, we propose to address the semantic- oriented 3D mesh hierarchical segmentation problem, using enhanced topological skeletons. This high level information drives both the feature boundary computation as well as the feature hierarchy definition. Proposed hierarchical scheme is based on the key idea that the topology of a feature is a more important decomposition criterion than its geometry. First, the enhanced topological skeleton of the input triangulated surface is constructed. Then it is used to delimit the core of the object and to identify junction areas. This second step results in a fine segmentation of the object. Finally, a fine to coarse strategy enables a semantic- oriented hierarchical composition of features, subdividing human limbs into arms and hands for example. Method performance is evaluated according to seven criteria enumerated in latest segmentation surveys [3]. Thanks to the high level description it uses as an input, presented approach results, with low computation times, in robust and meaningful compatible hierarchical decompositions.
One of the challenges in geometry processing is to automatically reconstruct a higher-level representation from raw geometric data. For instance, computing a parameterization of an object helps attaching information to it and converting between various representations. More generally, this family of problems may be thought of in terms of constructing structured function bases attached to surfaces. In this paper, we study a specific type of hierarchical function bases, defined by the eigenfunctions of the Laplace-Beltrami operator. When applied to a sphere, this function basis corresponds to the classical spherical harmonics. On more general objects, this defines a function basis well adapted to the geometry and the topology of the object. Based on physical analogies (vibration modes), we first give an intuitive view before explaining the underlying theory. We then explain in practice how to compute an approximation of the eigenfunctions of a differential operator, and show possible applications in geometry processing
All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion and electro-magnetic fields on the surface. A good parameterization is crucial for simulation and visualization. This paper provides an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algorithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implementation is based on a finite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserves angular structure, and can play an important role in various applications including texture mapping, remeshing, morphing and simulation. The method is demonstrated by visualizing the Riemann surface structure of real surfaces represented as triangle meshes.
We present a new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R 3, S 3 and H 3. The algorithm makes no restr iction on the genus and can handl e singular triangulations.Additionally, we present an algorithm that, starting from a discrete harmonic map, gives a conjugate harmonic map. This can be applied to the identity map on a minimal surface to produce its conjugate minimal surface, a procedure that often yields unstable solutions to a free boundary value problem for minimal surfaces. Symmetry properties of boundary curves are respected during conjugation.
Given a Morse function f over a 2-manifold with or without boundary,
the Reeb graph is obtained by contracting the connected components
of the level sets to points.
We prove tight upper and lower bounds on the
number of loops in the Reeb graph
that depend on the genus, the number of boundary components,
and whether or not the 2-manifold is orientable.
We also give an algorithm that constructs the Reeb graph in time
O(n log n), where n is the number of edges in the
triangulation used to represent the 2-manifold and the Morse function.
The proceedings of the 2001 Neural Information Processing Systems (NIPS) Conference.
The annual conference on Neural Information Processing Systems (NIPS) is the flagship conference on neural computation. The conference is interdisciplinary, with contributions in algorithms, learning theory, cognitive science, neuroscience, vision, speech and signal processing, reinforcement learning and control, implementations, and diverse applications. Only about 30 percent of the papers submitted are accepted for presentation at NIPS, so the quality is exceptionally high. These proceedings contain all of the papers that were presented at the 2001 conference.
Bradford Books imprint
With the ongoing development of 3D technologies, 3D shapes are becoming an interactive media of major importance. Their commonest representation, the surface mesh, suffers however from high variability towards standard shape-preserving surface transformations.It is necessary thus to design intrinsic shape modeling techniques. In this thesis, we explore topological modeling by studying Reeb graph based structures. In particular, we introduce a novel shape abstraction, called the enhanced topological skeleton, which enables not only the study of the topological evolution of Morse functions' level sets but also that of their geometrical evolution. We show the utility of this intrinsic shape representation in three research problems related to Computer Graphics and Computer Vision. First, we introduce the notion of geometrical calculus on Reeb graphs for the stable and automatic computation of control skeletons for interactive shape handling. Then, by introducing the notions of Reeb chart and Reeb pattern, we propose a new method for partial 3D shape similarity estimation. We show this approach outperforms the competing methods of the international SHape REtrieval Contest 2007 by a gain of 14%. Finally, we present two techniques for the functional decomposition computation of a 3D shape, both from human perception based heuristics and from the analysis of time-varying 3D data. For each of these research problems, concrete applicative examples are presented to assess the utility of our approach.
Following the transition presented in chapter 8, it is now quite natural to study the geodesic behaviour of a Riemannian manifold. For local metric geometry it is natural because geodesics are locally the shortest paths. This point of view was treated in §6.5. But geodesics of any length are of interest for the geometer. Another strong motivation for the study of geodesic dynamics comes from mechanics. Since Riemannian manifolds provide a very general setting for Hamiltonian mechanics, with their geodesics being the desired Hamiltonian trajectories, we are of course interested in their behaviour for any interval of time (any length). This perspective mixes dynamics and geometry and is extremely popular today. People always want to predict the future, more or less exactly. We will comment more on this below. Dynamics plays an ever larger role in geometry, even in very simple contexts such as the study of pentagons and Pappus theorems (see Schwartz 1993,1998 [1114, 1115] and the important work d’Ambra & Gromov 1991 [426]. Note also that Gromov 1987 [622] introduced dynamical systems into the study of discrete groups.
Modern techniques for interrogating the topology (e.g. genus) of a single meshed surface find handles via a breadth-first search for cycles that do not separate the mesh into two disjoint components. Morse theory can find handles with a simple search for the minima, maxima and saddles of a real function evaluated at the vertices. Bad choices for this function can generate numerous critical vertices that complicate the search for non-separating cycles. We show that the limit of the iteration of a constrained Laplacian on a mesh yields a "fair" Morse function with a single minumum and maximum, and two critical points for every handle. We then describe an algorithm that uses this function and its critical points to find a bouquet of non-separating cycles that cut the mesh into a flattenable domain homeomorphic to a disk. These cycles can be further optimized for applications in texturing, parameterization, geometry images and filtering topological noise.
This text on analysis of Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The Atiyah-Singer index theorem and its applications are developed (without complete proofs) via the heat equation method. Zeta functions for Laplacians and analytic torsion are also treated, and the recently uncovered relation between index theory and analytic torsion is laid out. The text is aimed at students who have had a first course in differentiable manifolds, and the Riemannian geometry used is developed from the beginning. There are over 100 exercises with hints.
Harmonic fields have been shown to provide effective guidance for a number of geometry processing problems. In this paper, we propose a method for fast updating of harmonic fields defined on polygonal meshes, enabling real-time insertion and deletion of constraints. Our approach utilizes the penalty method to enforce constraints in harmonic field computation. It maintains the symmetry of the Laplacian system and takes advantage of fast multi-rank updating and downdating of Cholesky factorization, achieving both speed and numerical stability. We demonstrate how the interactivity induced by fast harmonic field update can be utilized in several applications, including harmonic-guided quadrilateral remeshing, vector field design, interactive geometric detail modeling, and handle-driven shape editing and animation transfer with a dynamic handle set.
In this paper, we propose a new quadrilateral remeshing method for manifolds of arbitrary genus that is at once general, flexible, and efficient. Our technique is based on the use of smooth harmonic scalar fields defined over the mesh. Given such a field, we compute its gradient field and a second vector field that is everywhere orthogonal to the gradient. We then trace integral lines through these vector fields to sample the mesh. The two nets of integral lines together are used to form the polygons of the output mesh. Curvature-sensitive spacing of the lines provides for anisotropic meshes that adapt to the local shape. Our scalar field construction allows users to exercise extensive control over the structure of the final mesh. The entire process is performed without computing an explicit parameterization of the surface, and is thus applicable to manifolds of any genus without the need for cutting the surface into patches.
Surface editing operations commonly require geometric details of the surface to be preserved as much as possible. We argue that geometric detail is an intrinsic property of a surface and that, consequently, surface editing is best performed by operating over an intrinsic surface representation. We provide such a representation of a surface, based on the Laplacian of the mesh, by encoding each vertex relative to its neighborhood. The Laplacian of the mesh is enhanced to be invariant to locally linearized rigid transformations and scaling. Based on this Laplacian representation, we develop useful editing operations: interactive free-form deformation in a region of interest based on the transformation of a handle, transfer and mixing of geometric details between two surfaces, and transplanting of a partial surface mesh onto another surface. The main computation involved in all operations is the solution of a sparse linear system, which can be done at interactive rates. We demonstrate the effectiveness of our approach in several examples, showing that the editing operations change the shape while respecting the structural geometric detail.
A deformation invariant representation of surfaces, the GPS embedding, is introduced using the eigenvalues and eigenfunctions of the Laplace-Beltrami differential operator. Notably, since the definition of the GPS embedding completely avoids the use of geodesic distances, and is based on objects of global character, the obtained repre- sentation is robust to local topology changes. The GPS embedding captures enough information to handle various shape processing tasks as shape classification, segmentation, and correspondence. To demonstrate the practical relevance of the GPS embedding, we introduce a deformation invariant shape descriptor called G2-distributions, and demonstrate their discriminative power, invariance under natural deformations, and robustness.
We introduce a framework for triangle shape optimization and feature preserving smoothing of triangular meshes that is guided by the vertex Laplacians, specifically, the uniformly weighted Laplacian and the discrete mean curvature normal. Vertices are relocated so that they approximate prescribed Laplacians and positions in a weighted least-squares sense; the resulting linear system leads to an efficient, non-iterative solution. We provide different weighting schemes and demonstrate the effectiveness of the framework on a number of detailed and highly irregular meshes; our technique successfully improves the quality of the triangulation while remaining faithful to the original surface geometry, and it is also capable of smoothing the surface while preserving geometric features.
Although considerable attention in recent years has been given to the problem of symmetry detection in general shapes, few methods have been developed that aim to detect and quantify the intrinsic symmetry of a shape rather than its extrinsic, or pose-dependent symmetry. In this paper, we present a novel approach for efficiently computing symmetries of a shape which are invariant up to isometry preserving transformations. We show that the intrinsic symmetries of a shape are transformed into the Euclidean symmetries in the signature space defined by the eigenfunctions of the Laplace-Beltrami operator. Based on this observation, we devise an algorithm which detects and computes the isometric mappings from the shape onto itself. We show that our approach is both computationally efficient and robust with respect to small non-isometric deformations, even if they include topological changes.
This paper presents a novel segmentation method to assist the rigging of articulated bodies. The method computes a coarse-to-fine hierarchy of segments ordered by the level of detail. The results are invariant to deformations, and numerically robust to noise, irregular tessellations, and topological short-circuits. The segmentation is based on two key ideas. First, it exploits the multiscale properties of the diffusion distance on surfaces, and then it introduces a new definition of medial structures, composing a bijection between medial structures and segments. Our method computes this bijection through a simple and fast iterative approach, and applies it to triangulated meshes.
Abstract We propose a mesh segmentation algorithm via recursive bisection where at each step, a sub-mesh embedded in 3D is first spectrally projected into the plane and then a contour is extracted from the planar embedding. We rely on two operators to compute the projection: the well-known graph Laplacian and a geometric operator designed to emphasize concavity. The two embeddings reveal distinctive shape semantics of the 3D model and complement each other in capturing the structural or geometrical aspect of a segmentation. Transforming the shape analysis problem to the 2D domain also facilitates our segmentability analysis and sampling tasks. We propose a novel measure of the segmentability of a shape, which is used as the stopping criterionfor our segmentation. The measure is derived from simple area- and perimeter-based convexity measures. We achieve invariance to shape bending through multi-dimensional scaling (MDS) based on the notion of inner distance. We also utilize inner distances to develop a novel sampling scheme to extract two samples along a contour which correspond to two vertices residing on different parts of the sub-mesh. The two samples are used to derive a spectral linear ordering of the mesh faces. We obtain a final cut via a linear search over the face sequence based on part salience, where a choice of weights for different factors of part salience is guided by the result from segmentability analysis.
We introduce a novel algorithm that decomposes a deformable shape into meaningful parts requiring only a single input pose. Using modal analysis, we are able to identify parts of the shape that tend to move rigidly. We define a deformation energy on the shape, enabling modal analysis to find the typical deformations of the shape. We then find a decomposition of the shape such that the typical deformations can be well approximated with deformation fields that are rigid in each part of the decomposition. We optimize for the best decomposition, which captures how the shape deforms. A hierarchical refinement scheme makes it possible to compute more detailed decompositions for some parts of the shape.
Although our algorithm does not require user intervention, it is possible to control the process by directly changing the deformation energy, or interactively refining the decomposition as necessary. Due to the construction of the energy function and the properties of modal analysis, the computed decompositions are robust to changes in pose as well as meshing, noise, and even imperfections such as small holes in the surface.
Mesh vertices are usually represented with absolute coordinates. In some applications, this leads to problems for local operations because of global misalignment. We investigate the idea of describing mesh geometry in a differential way. These differential coordinates describe local properties of the geometry rather than the absolute position in space. The main application discussed here is the insertion of shape features from one mesh into another, given the meshes have the same connectivity. We regard this as local control over mesh morphing. Differential coordinates also prove useful for free-form modeling of meshes.
Surface editing operations commonly require geometric details of the surface to be preserved as much as possible. We argue that geometric detail is an intrinsic property of a surface and that, consequently, surface editing is best performed by operating over an intrinsic surface representation. We provide such a representation of a surface, based on the Laplacian of the mesh, by encoding each vertex relative to its neighborhood. The Laplacian of the mesh is enhanced to be invariant to locally linearized rigid transformations and scaling. Based on this Laplacian representation, we develop useful editing operations: interactive free-form deformation in a region of interest based on the transformation of a handle, transfer and mixing of geometric details between two surfaces, and transplanting of a partial surface mesh onto another surface. The main computation involved in all operations is the solution of a sparse linear system, which can be done at interactive rates. We demonstrate the effectiveness of our approach in several examples, showing that the editing operations change the shape while respecting the structural geometric detail.
This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). Our approach is to develop DEC using only discrete combinatorial and geometric operations on a simplicial complex and its geometric dual. The derivation of these may require that the objects on the discrete mesh, but not the mesh itself, are interpolated. Our theory includes not only discrete equivalents of differential forms, but also discrete vector fields and the operators acting on these objects. Definitions are given for discrete versions of all the usual operators of exterior calculus. The presence of forms and vector fields allows us to address their various interactions, which are important in applications. In many examples we find that the formulas derived from DEC are identitical to the existing formulas in the literature. We also show that the circumcentric dual of a simplicial complex plays a useful role in the metric dependent part of this theory. The appearance of dual complexes leads to a proliferation of the operators in the discrete theory. One potential application of DEC is to variational problems which come equipped with a rich exterior calculus structure. On the discrete level, such structures will be enhanced by the availability of DEC. One of the objectives of this thesis is to fill this gap. There are many constraints in numerical algorithms that naturally involve differential forms. Preserving such features directly on the discrete level is another goal, overlapping with our goals for variational problems. In this thesis we have tried to push a purely discrete point of view as far as possible. We argue that this can only be pushed so far, and that interpolation is a useful device. For example, we found that interpolation of functions and vector fields is a very convenient. In future work we intend to continue this interpolation point of view, extending it to higher degree forms, especially in the context of the sharp, Lie derivative and interior product operators. Some preliminary ideas on this point of view are presented in the thesis. We also present some preliminary calculations of formulas on regular nonsimplicial complexes
Current approaches to skeleton generation are based on topological and geometrical information only; this can be insufficient for realistic character animation, since the location of the joints does not usually match the real bone structure of the model. This paper proposes the use of anatomical information to enhance the skeleton. Using a harmonic function, this information can be recovered from the skeleton itself, which is guaranteed not to have undesired endpoints. The skeleton is computed as a Reeb graph of such a function over the surface of the model. Starting from one point selected on the head of the character, the entire process is fast, automatic and robust; it generates skeletons whose joints can be associated with the character's anatomy. Results are provided, including a quantitative validation of the generated skeletons.
We present a spectral approach to automatically and efficiently obtain discrete free-boundary conformal parameterizations of triangle mesh patches, without the common artifacts due to positional constraints on vertices and without undue bias introduced by sampling irregularity. High-quality parameterizations are computed through a constrained minimization of a discrete weighted conformal energy by finding the largest eigenvalue/eigenvector of a generalized eigenvalue problem involving sparse, symmetric matrices. We demonstrate that this novel and robust approach improves on previous linear techniques both quantitatively and qualitatively.
Shiga toxin-producing Escherichia coli are emerging as a significant source of food-borne infectious disease all over the world. Illness caused by Shiga toxin-producing E. coli can range from self limited, watery diarrhea to life-threatening manifestations such as hemorrhagic colitis, hemolytic uremic syndrome or thrombotic thrombocytopenic purpura and death. Shiga toxin-producing E. coli can potentially enter the human food chain from a number of animal sources, most commonly by contamination of meat with feces or intestinal contents after slaughter or cross-contamination of unpasteurized milk products. Because of the low infectious dose of the O157:H7 Shiga toxin-producing E. coli strain, laboratory diagnosis of Shiga toxin-producing E. coli in food samples has developed a great importance. This review will focus on the microorganism, giving priority to illness prevention and Shiga toxin-producing E. coli detection in food.
Reverse engineering (RE) deals with an enormous number of
irregular and scattered digitized points that require intensive
processing in order to reconstruct the surfaces of an object. Surface
reconstruction of freeform objects is based on geometrical and
topological criteria. Current fitting methods reconstruct an object
using a bottom-up approach, from points to a dense mesh, and finally
into smoothed connected freeform sub-surfaces. This type of
reconstruction, however, can cause topological problems that lead to
undesired surface fitting results. Such problems are particularly common
with concave shapes. To avoid problems of this type, the paper proposes
a new method that automatically detects the topological structure of an
object as a base for surface fitting. The topological reconstruction
method described in the paper is based on two stages: (1) creating 3D
non-self-intersecting iso-curves from a 3D triangular mesh and (2)
extracting a topological graph. The feasibility of the proposed
topological reconstruction method is demonstrated on several examples
using freeform objects with complex topologies
Drawing on the correspondence between the graph Laplacian, the Laplace-Beltrami operator on a manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded in a higher dimensional space. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Several applications are considered.
Cutting up a complex object into simpler sub-objects is a fundamental problem in various disciplines. In image processing, images are segmented while in computational geometry, solid polyhedra are decomposed. In recent years, in computer graphics, polygonal meshes are decomposed into sub-meshes. In this paper we propose a novel hierarchical mesh decomposition algorithm. Our algorithm computes a decomposition into the meaningful components of a given mesh, which generally refers to segmentation at regions of deep concavities. The algorithm also avoids over-segmentation and jaggy boundaries between the components. Finally, we demonstrate the utility of the algorithm in control-skeleton extraction.
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which are important in applications. Previous attempts at discrete exterior calculus have addressed only differential forms. We also introduce the notion of a circumcentric dual of a simplicial complex. The importance of dual complexes in this field has been well understood, but previous researchers have used barycentric subdivision or barycentric duals. We show that the use of circumcentric duals is crucial in arriving at a theory of discrete exterior calculus that admits both vector fields and forms.
Mesh segmentation via spectral em-bedding and contour analysis Computer Graphics Forum (Spe-cial Issue of Eurographics
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] LIU R., ZHANG H.: Mesh segmentation via spectral em-bedding and contour analysis. Computer Graphics Forum (Spe-cial Issue of Eurographics 2007) 26, 3 (2007), 385–394.
Discrete exterior calculus. http://arxiv.org/abs/ math?papernum=0508341 Harmonic functions for quadrilateral remeshing of arbitrary manifolds
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- S Kircher
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[DHLM05] DESBRUN M., HIRANI A. N., LEOK M., MARSDEN J. E.: Discrete exterior calculus. http://arxiv.org/abs/ math?papernum=0508341, Aug 2005. [DKG05] DONG S., KIRCHER S., GARLAND M.: Harmonic functions for quadrilateral remeshing of arbitrary manifolds. Comput. Aided Geom. Des. 22, 5 (2005), 392–423.
Reeb graph based 3D shape modeling and applications Topol-ogy driven 3D mesh hierarchical segmentation
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TIERNY J.: Reeb graph based 3D shape modeling and applications. PhD thesis, TELECOM Lille 1, Universite des Sci-ences et Technologies de Lille, 2008. [TVD07] TIERNY J., VANDEBORRE J.-P., DAOUDI M.: Topol-ogy driven 3D mesh hierarchical segmentation. In IEEE Inter-national Conference on Shape Modeling and Applications (SMI 2007) (Lyon, France, 2007), pp. 215–220.
Topology recognition of 3d closed freeform objects based on topological graphs Anisotropic laplace-beltrami eigenmaps: Bridging reeb graphs and skeletons
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- D Steiner
- Fischer A
- Y Shi
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- Krishna S
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- I Dinov
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[SF01] STEINER D., FISCHER A.: Topology recognition of 3d closed freeform objects based on topological graphs. In PG '01: Proceedings of the 9th Pacific Conference on Computer Graph-ics and Applications (Washington, DC, USA, 2001), IEEE Com-puter Society, p. 82. [SLK * 08] SHI Y., LAI R., KRISHNA S., SICOTTE N., DINOV I., TOGA A.: Anisotropic laplace-beltrami eigenmaps: Bridging reeb graphs and skeletons. pp. 1–7.