Topological approaches for 3D object processing and applications

The great challenge in 3D object processing is to devise computationally efficient algorithms for recovering 3D models contaminated by noise and preserving their geometrical structure. The first problem addressed in this thesis is object denoising formulated in the discrete variational framework. We introduce a 3D mesh denoising method based on kernel density estimation. The proposed approach is able to reduce the over-smoothing effect and effectively remove undesirable noise while preserving prominent geometric features of a 3D mesh such as sharp features and fine details. The feasibility of the approach is demonstrated through extensive experiments. The rest of the thesis is devoted to a joint exploitation of geometry and topology of 3D objects for as parsimonious as possible representation of models and its subsequent application in object modeling, compression, and hashing problems. We introduce a 3D mesh compression technique using the centroidal mesh neighborhood information. The key idea is to apply eigen-decomposition to the mesh umbrella matrix, and then discard the smallest eigenvalues/eigenvectors in order to reduce the dimensionality of the new spectral basis so that most of the energy is concentrated in the low frequency coefficients. We also present a hashing technique for 3D models using spectral graph theory and entropic spanning trees by partitioning a 3D triangle mesh into an ensemble of submeshes, and then applying eigen-decomposition to the Laplace-Beltrami matrix of each sub-mesh, followed by computing the hash value of each sub-mesh. Moreover, we introduce several statistical distributions to analyze the topological properties of 3D objects. These probabilistic distributions provide useful information about the way 3D mesh models are connected. Illustrating experiments with synthetic and real data are provided to demonstrate the feasibility and the much improved performance of the proposed approaches in 3D object compression, hashing, and modeling

Surface Denoising based on Normal Filtering in a Robust Statistics Framework

During a surface acquisition process using 3D scanners, noise is inevitable and an important step in geometry processing is to remove these noise components from these surfaces (given as points-set or triangulated mesh). The noise-removal process (denoising) can be performed by filtering the surface normals first and by adjusting the vertex positions according to filtered normals afterwards. Therefore, in many available denoising algorithms, the computation of noise-free normals is a key factor. A variety of filters have been introduced for noise-removal from normals, with different focus points like robustness against outliers or large amplitude of noise. Although these filters are performing well in different aspects, a unified framework is missing to establish the relation between them and to provide a theoretical analysis beyond the performance of each method. In this paper, we introduce such a framework to establish relations between a number of widely-used nonlinear filters for face normals in mesh denoising and vertex normals in point set denoising. We cover robust statistical estimation with M-smoothers and their application to linear and non-linear normal filtering. Although these methods originate in different mathematical theories - which include diffusion-, bilateral-, and directional curvature-based algorithms - we demonstrate that all of them can be cast into a unified framework of robust statistics using robust error norms and their corresponding influence functions. This unification contributes to a better understanding of the individual methods and their relations with each other. Furthermore, the presented framework provides a platform for new techniques to combine the advantages of known filters and to compare them with available methods

Perturbation of the Eigenvectors of the Graph Laplacian: Application to Image Denoising

The original contributions of this paper are twofold: a new understanding of the influence of noise on the eigenvectors of the graph Laplacian of a set of image patches, and an algorithm to estimate a denoised set of patches from a noisy image. The algorithm relies on the following two observations: (1) the low-index eigenvectors of the diffusion, or graph Laplacian, operators are very robust to random perturbations of the weights and random changes in the connections of the patch-graph; and (2) patches extracted from smooth regions of the image are organized along smooth low-dimensional structures in the patch-set, and therefore can be reconstructed with few eigenvectors. Experiments demonstrate that our denoising algorithm outperforms the denoising gold-standards

Progressive Shape-Distribution-Encoder for 3D Shape Retrieval

Since there are complex geometric variations with 3D shapes, extracting efficient 3D shape features is one of the most challenging tasks in shape matching and retrieval. In this paper, we propose a deep shape descriptor by learning shape distributions at different diffusion time via a progressive shape-distribution-encoder (PSDE). First, we develop a shape distribution representation with the kernel density estimator to characterize the intrinsic geometry structures of 3D shapes. Then, we propose to learn a deep shape feature through an unsupervised PSDE. Specially, the unsupervised PSDE aims at modeling the complex non-linear transform of the estimated shape distributions between consecutive diffusion time. In order to characterize the intrinsic structures of 3D shapes more efficiently, we stack multiple PSDEs to form a network structure. Finally, we concatenate all neurons in the middle hidden layers of the unsupervised PSDE network to form an unsupervised shape descriptor for retrieval. Furthermore, by imposing an additional constraint on the outputs of all hidden layers, we propose a supervised PSDE to form a supervised shape descriptor, where for each hidden layer the similarity between a pair of outputs from the same class is as small as possible and the similarity between a pair of outputs from different classes is as large as possible. The proposed method is evaluated on three benchmark 3D shape datasets with large geometric variations, i.e., McGill, SHREC’10 ShapeGoogle and SHREC’14 Human datasets, and the experimental results demonstrate the superiority of the proposed method to the existing approaches

Geometric Approaches for 3D Shape Denoising and Retrieval

A key issue in developing an accurate 3D shape recognition system is to design an efficient shape descriptor for which an index can be built, and similarity queries can be answered efficiently. While the overwhelming majority of prior work on 3D shape analysis has concentrated primarily on rigid shape retrieval, many real objects such as articulated motions of humans are nonrigid and hence can exhibit a variety of poses and deformations. Motivated by the recent surge of interest in content-based analysis of 3D objects in computeraided design and multimedia computing, we develop in this thesis a unified theoretical and computational framework for 3D shape denoising and retrieval by incorporating insights gained from algebraic graph theory and spectral geometry. We first present a regularized kernel diffusion for 3D shape denoising by solving partial differential equations in the weighted graph-theoretic framework. Then, we introduce a computationally fast approach for surface denoising using the vertexcentered finite volume method coupled with the mesh covariance fractional anisotropy. Additionally, we propose a spectral-geometric shape skeleton for 3D object recognition based on the second eigenfunction of the Laplace-Beltrami operator in a bid to capture the global and local geometry of 3D shapes. To further enhance the 3D shape retrieval accuracy, we introduce a graph matching approach by assigning geometric features to each endpoint of the shape skeleton. Extensive experiments are carried out on two 3D shape benchmarks to assess the performance of the proposed shape retrieval framework in comparison with state-of-the-art methods. The experimental results show that the proposed shape descriptor delivers best-in-class shape retrieval performance

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference