We introduce an algorithm for unsupervised co-segmentation of a set of shapes so as to reveal the semantic shape parts and establish their correspondence across the set. The input set may exhibit significant shape variability where the shapes do not admit proper spatial alignment and the corresponding parts in any pair of shapes may be geometrically dissimilar. Our algorithm can handle such challenging input sets since, first, we perform co-analysis in a descriptor space, where a combination of shape descriptors relates the parts independently of their pose, location, and cardinality. Secondly, we exploit a key enabling feature of the input set, namely, dissimilar parts may be “linked ” through third-parties present in the set. The links are derived from the pairwise similarities between the parts ’ descriptors. To reveal such linkages, which may manifest themselves as anisotropic and non-linear structures in the descriptor space, we perform spectral clustering with the aid of diffusion maps. We show that with our approach, we are able to co-segment sets of shapes that possess significant variability, achieving results that are close to those of a supervised approach. Keywords: Co-segmentation, shape correspondence, spectral clustering, diffusion maps. Links: DL PDF 1

Figure 1: Horse algebra: the functional representation and map inference algorithm allow us to go beyond point-to-point maps. The source shape (top left corner) was mapped to the target shape (left) by posing descriptor-based functional constraints which do not disambiguate symmetries (i.e. without landmark constraints). By further adding correspondence constraints, we obtain a near isometric map which reverses orientation, mapping left to right (center). The representation allows for algebraic operations on shape maps, so we can subtract this map from the ambivalent map, to retrieve the orientation preserving near-isometry (right). Each column shows the first 20x20 block of the functional map representation (bottom), and the action of the map by transferring colors from the source shape to the target shape (top). We present a novel representation of maps between pairs of shapes that allows for efficient inference and manipulation. Key to our approach is a generalization of the notion of map that puts in correspondence real-valued functions rather than points on the shapes. By choosing a multi-scale basis for the function space on each shape, such as the eigenfunctions of its Laplace-Beltrami operator, we obtain a representation of a map that is very compact, yet fully suitable for global inference. Perhaps more remarkably, most

Large collections of 3D models from the same object class (e.g., chairs, cars, animals) are now commonly available via many public repositories, but exploring the range of shape variations across such collections remains a challenging task. In this work, we present a new exploration interface that allows users to browse collections based on similarities and differences between shapes in users-pecified regions of interest (ROIs). To support this interactive system, we introduce a novel analysis method for computing similarity relationships between points on 3D shapes across a collection. We encode the inherent ambiguity in these relationships using fuzzy point correspondences and propose a robust and efficient computational framework that estimates fuzzy correspondences using only a sparse set of pairwise model alignments. We evaluate our analysis method on a range of correspondence benchmarks and report substantial improvements in both speed and accuracy over existing alternatives. In addition, we demonstrate how fuzzy correspondences enable key features in our exploration tool, such as automated view alignment, ROI-based similarity search, and faceted browsing.

Figure 1: Overview of our active co-analysis: (a) We start with an initial unsupervised co-segmentation of the input set. (b) During active learning, the system automatically suggests constraints which would refine results and the user interactively adds constraints as appropriate. In this example, the user adds a cannot-link constraint (in red) and a must-link constraint (in blue) between segments. (c) The constraints are propagated to the set and the co-segmentation is refined. The process from (b) to (c) is repeated until the desired result is obtained. 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 usergiven 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.

As large repositories of 3D shape collections continue to grow, understanding the data, especially encoding the inter-model similarity and their variations, is of central importance. For example, many data-driven approaches now rely on access to semantic segmentation information, accurate inter-model point-to-point correspondence, and deformation models that characterize the model collections. Existing approaches, however, are either supervised requiring manual labeling; or employ super-linear matching algorithms and thus are unsuited for analyzing large collections spanning many thousands of models. We propose an automatic algorithm that starts with an initial template model and then jointly optimizes for part segmentation, point-to-point surface correspondence, and a compact deformation model to best explain the input model collection. As output, the algorithm produces a set of probabilistic part-based templates that groups the original models into clusters of models capturing their styles and variations. We evaluate our algorithm on several standard datasets and demonstrate its scalability by analyzing much larger collections of up to thousands of shapes.

Shape structure is about the arrangement and relations between shape parts. Structure-aware shape processing goes beyond local geometry and low level processing, and analyzes and processes shapes at a high level. It focuses more on the global inter and intra semantic relations among the parts of shape rather than on their local geometry. With recent developments in easy shape acquisition, access to vast repositories of 3D models, and simple-to-use desktop fabrication possibilities, the study of structure in shapes has become a central research topic in shape analysis, editing, and modeling. A whole new line of structure-aware shape processing algorithms has emerged that base their operation on an attempt to understand such structure in shapes. The algorithms broadly consist of two key phases: an analysis phase, which extracts structural information from input data; and a (smart) processing phase, which utilizes the extracted information for exploration, editing, and synthesis of novel shapes. In this survey paper, we organize, summarize, and present the key concepts and methodological approaches towards efficient structure-aware shape processing. We discuss common models of structure, their implementation in terms of mathematical formalism and algorithms, and explain the key principles in the context of a number of state-of-the-art approaches. Further, we attempt to list the key open problems and challenges, both at the technical and at the conceptual level, to make it easier for new researchers to better explore and contribute to this topic. Our goal is to both give the practitioner an overview of available structure-aware shape processing techniques, as well as identify future research questions in this important, emerging, and fascinating research area.

Figure 1: Without any user intervention, our framework automatically turns a freehand sketch drawing depicting multiple scene objects (left) to semantically valid, well arranged scenes of 3D models (right). (The ground and walls were manually added.) This work presents Sketch2Scene, a framework that automatically turns a freehand sketch drawing inferring multiple scene objects to semantically valid, well arranged scenes of 3D models. Unlike the existing works on sketch-based search and composition of 3D models, which typically process individual sketched objects one by one, our technique performs co-retrieval and co-placement of 3D relevant models by jointly processing the sketched objects. This is enabled by summarizing functional and spatial relationships among models in a large collection of 3D scenes as structural groups. Our technique greatly reduces the amount of user intervention needed for sketch-based modeling of 3D scenes and fits well into the traditional production pipeline involving concept design followed by 3D modeling. A pilot study indicates that the 3D scenes automatically synthesized by our technique in seconds are comparable to those manually created by an artist in hours in terms of visual aesthetics. Links: DL PDF 1

We present a novel sparse modeling approach to non-rigid shape match-ing using only the ability to detect repeatable regions. As the input to our algorithm, we are given only two sets of regions in two shapes; no descriptors are provided so the correspondence between the regions is not know, nor we know how many regions correspond in the two shapes. We show that even with such scarce information, it is possible to estab-lish very accurate correspondence between the shapes by using methods from the field of sparse modeling, being this, the first non-trivial use of sparse models in shape correspondence. We formulate the problem of per-muted sparse coding, in which we solve simultaneously for an unknown permutation ordering the regions on two shapes and for an unknown cor-respondence in functional representation. We also propose a robust vari-ant capable of handling incomplete matches. Numerically, the problem is solved efficiently by alternating the solution of a linear assignment and a sparse coding problem. The proposed methods are evaluated qualitatively and quantitatively on standard benchmarks containing both synthetic and scanned objects. 1

As 3D acquisition devices and modeling tools become widely avail-able there is a growing need for automatic algorithms that analyze the semantics and functionality of digitized shapes. Most recent research has focused on analyzing geometric structures of shapes. Our work is motivated by the observation that a majority of man-made shapes are designed to be used by people. Thus, in order to fully understand their semantics, one needs to answer a fundamen-tal question: “how do people interact with these objects? ” As an initial step towards this goal, we offer a novel algorithm for auto-matically predicting a static pose that a person would need to adopt in order to use an object. Specifically, given an input 3D shape, the goal of our analysis is to predict a corresponding human pose, in-cluding contact points and kinematic parameters. This is especially challenging for man-made objects that commonly exhibit a lot of variance in their geometric structure. We address this challenge by observing that contact points usually share consistent local geomet-ric features related to the anthropometric properties of correspond-ing parts and that human body is subject to kinematic constraints and priors. Accordingly, our method effectively combines local re-gion classification and global kinematically-constrained search to successfully predict poses for various objects. We also evaluate our algorithm on six diverse collections of 3D polygonal models (chairs, gym equipment, cockpits, carts, bicycles, and bipedal de-vices) containing a total of 147 models. Finally, we demonstrate that the poses predicted by our algorithm can be used in several shape analysis problems, such as establishing correspondences be-tween objects, detecting salient regions, finding informative view-points, and retrieving functionally-similar shapes.

Figure 1: A set of example poses are decomposed into rigid bone transformations B and a sparse, convex bone-vertex weight map W (left hand side) by our block coordinate descent algorithm (right hand side). During the process, the example poses (indicated as blue dots) can be reconstructed more accurately by alternatively updating W and B while the other is kept fixed. This paper introduces the Smooth Skinning Decomposition with Rigid Bones (SSDR), an automated algorithm to extract the linear blend skinning (LBS) from a set of example poses. The SSDR model can effectively approximate the skin deformation of nearly articulated models as well as highly deformable models by a low number of rigid bones and a sparse, convex bone-vertex weight map. Formulated as a constrained optimization problem where the least squared error of the reconstructed vertices by LBS is minimized, the SSDR model can be solved by a block coordinate descent-based algorithm to iteratively update the weight map and the bone transformations. By employing the sparseness and convex constraints on the weight map, the SSDR model can be used for traditional skinning decomposition tasks such as animation compression and hardware-accelerated rendering. Moreover, by imposing the orthogonal constraints on the bone rotation matrices (rigid bones), the SSDR model can also be applied in motion editing, skeleton extraction, and collision detection tasks. Through qualitative and quantitative evaluations, we show the SSDR model can measurably outperform the state-of-the-art skinning decomposition schemes in terms of accuracy and applicability.