Oliver van Kaick

A Survey on Shape Correspondence

We review methods designed to compute correspondences between geometric shapes represented by triangle meshes, contours or point sets. This survey is motivated in part by recent developments in space–time registration, where one seeks a correspondence between non‐rigid and time‐varying surfaces, and semantic shape analysis, which underlines a recent trend to incorporate shape understanding into the analysis pipeline. Establishing a meaningful correspondence between shapes is often difficult because it generally requires an understanding of the structure of the shapes at both the local and global levels, and sometimes the functionality of the shape parts as well. Despite its inherent complexity, shape correspondence is a recurrent problem and an essential component of numerous geometry processing applications. In this survey, we discuss the different forms of the correspondence problem and review the main solution methods, aided by several classification criteria arising from the problem definition. The main categories of classification are defined in terms of the input and output representation, objective function and solution approach. We conclude the survey by discussing open problems and future perspectives.

Active co-analysis of a set of shapes

Unsupervised co-analysis of a set of shapes is a difficult problem since the geometry of the shapes alone cannot always fully describe the semantics of the shape parts. In this paper, we propose a semi-supervised learning method where the user actively assists in the co-analysis by iteratively providing inputs that progressively constrain the system. We introduce a novel constrained clustering method based on a spring system which embeds elements to better respect their inter-distances in feature space together with the user-given set of constraints. We also present an active learning method that suggests to the user where his input is likely to be the most effective in refining the results. We show that each single pair of constraints affects many relations across the set. Thus, the method requires only a sparse set of constraints to quickly converge toward a consistent and error-free semantic labeling of the set.

NON-RIGID SPECTRAL CORRESPONDENCE OF TRIANGLE MESHES

We present an algorithm for finding a meaningful vertex-to-vertex correspondence between two triangle meshes, which is designed to handle general non-rigid transformations. Our algorithm operates on embeddings of the two shapes in the spectral domain so as to normalize them with respect to uniform scaling and rigid-body transformation. Invari-ance to shape bending is achieved by relying on approximate geodesic point proximities on a mesh to capture its shape. To deal with moderate stretching, we first raise the issue of “eigenmode switching” and discuss heuristics to bring the eigenmodes to alignment. For additional non-rigid discrepancies in the spectral embeddings, we propose to use non-rigid alignment via thin-plate splines. This is combined with a refinement step based on geodesic proximities to improve dense correspondence. We show empirically that our algorithm outperforms previous spectral methods, as well as schemes that compute correspondence in the spatial domain via non-rigid iterative closest points or the use of local shape descriptors, e.g., 3D shape context. Finally, to speed up our algorithm, we examine the effect of using subsampling and Nystrom method.

Spectral Methods for Mesh Processing and Analysis

Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections 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 comprehensive 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. Necessary 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 embeddings 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.

Prior Knowledge for Part Correspondence

Classical approaches to shape correspondence base their computation purely on the properties, in particular geometric similarity, of the shapes in question. Their performance still falls far short of that of humans in challenging cases where corresponding shape parts may differ significantly in geometry or even topology. We stipulate that in these cases, shape correspondence by humans involves recognition of the shape parts where prior knowledge on the parts would play a more dominant role than geometric similarity. We introduce an approach to part correspondence which incorporates prior knowledge imparted by a training set of pre‐segmented, labeled models and combines the knowledge with content‐driven analysis based on geometric similarity between the matched shapes. First, the prior knowledge is learned from the training set in the form of per‐label classifiers. Next, given two query shapes to be matched, we apply the classifiers to assign a probabilistic label to each shape face. Finally, by means of a joint labeling scheme, the probabilistic labels are used synergistically with pairwise assignments derived from geometric similarity to provide the resulting part correspondence. We show that the incorporation of knowledge is especially effective in dealing with shapes exhibiting large intra‐class variations. We also show that combining knowledge and content analyses outperforms approaches guided by either attribute alone.

Co-hierarchical analysis of shape structures

We introduce an unsupervised co-hierarchical analysis of a set of shapes, aimed at discovering their hierarchical part structures and revealing relations between geometrically dissimilar yet functionally equivalent shape parts across the set. The core problem is that of representative co-selection. For each shape in the set, one representative hierarchy (tree) is selected from among many possible interpretations of the hierarchical structure of the shape. Collectively, the selected tree representatives maximize the within-cluster structural similarity among them. We develop an iterative algorithm for representative co-selection. At each step, a novel cluster-and-select scheme is applied to a set of candidate trees for all the shapes. The tree-to-tree distance for clustering caters to structural shape analysis by focusing on spatial arrangement of shape parts, rather than their geometric details. The final set of representative trees are unified to form a structural co-hierarchy. We demonstrate co-hierarchical analysis on families of man-made shapes exhibiting high degrees of geometric and finer-scale structural variabilities.

Shape Segmentation by Approximate Convexity Analysis

We present a shape segmentation method for complete and incomplete shapes. The key idea is to directly optimize the decomposition based on a characterization of the expected geometry of a part in a shape. Rather than setting the number of parts in advance, we search for the smallest number of parts that admit the geometric characterization of the parts. The segmentation is based on an intermediate-level analysis, where first the shape is decomposed into approximate convex components, which are then merged into consistent parts based on a nonlocal geometric signature. Our method is designed to handle incomplete shapes, represented by point clouds. We show segmentation results on shapes acquired by a range scanner, and an analysis of the robustness of our method to missing regions. Moreover, our method yields results that are comparable to state-of-the-art techniques evaluated on complete shapes.

Meta-representation of shape families

We introduce a meta-representation that represents the essence of a family of shapes. The meta-representation learns the configurations of shape parts that are common across the family, and encapsulates this knowledge with a system of geometric distributions that encode relative arrangements of parts. Thus, instead of predefined priors, what characterizes a shape family is directly learned from the set of input shapes. The meta-representation is constructed from a set of co-segmented shapes with known correspondence. It can then be used in several applications where we seek to preserve the identity of the shapes as members of the family. We demonstrate applications of the meta-representation in exploration of shape repositories, where interesting shape configurations can be examined in the set; guided editing, where models can be edited while maintaining their familial traits; and coupled editing, where several shapes can be collectively deformed by directly manipulating the distributions in the meta-representation. We evaluate the efficacy of the proposed representation on a variety of shape collections.

A Comparative Evaluation of Metrics for Fast Mesh Simplification

Triangle mesh simplification is of great interest in a variety of knowledge domains, since it allows manipulation and visualization of large models, and it is the starting point for the design of many multiresolution representations. A crucial point in the structure of a simplification method is the definition of an appropriate metric for guiding the decimation process, with the purpose of generating low error approximations at different levels of resolution. This paper proposes two new alternative metrics for mesh simplification, with the aim of producing high‐quality results with reduced execution time and memory usage, and being simple to implement. A set of different established metrics is also described and a comparative evaluation of these metrics against the two new metrics is performed. A single implementation is used in the experiments, in order to enable the evaluation of these metrics independently from other simplification aspects. Results obtained from the simplification of a number of models, using the different metrics, are compared.

Contour Correspondence via Ant Colony Optimization

We formulate contour correspondence as a Quadratic Assignment Problem (QAP), incorporating proximity information. By maintaining the neighborhood relation between points this way, we show that better matching results are obtained in practice. We propose the first Ant Colony Optimization (ACO) algorithm specifically aimed at solving the QAP-based shape correspondence problem. Our ACO framework is flexible in the sense that it can handle general point correspondence, but also allows extensions, such as order preservation, for the more specialized contour matching problem. Various experiments are presented which demonstrate that this approach yields high-quality correspondence results and is computationally efficient when compared to other methods.