Cross‐Collection Map Inference by Intrinsic Alignment of Shape Spaces

Inferring maps between shapes is a long standing problem in geometry processing. The less similar the shapes are, the harder it is to compute a map, or even define criteria to evaluate it. In many cases, shapes appear as part of a collection, e.g. an animation or a series of faces or poses of the same character, where the shapes are similar enough, such that maps within the collection are easy to obtain. Our main observation is that given two collections of shapes whose “shape space” structure is similar, it is possible to find a correspondence between the collections, and then compute a cross‐collection map. The cross‐map is given as a functional correspondence, and thus it is more appropriate in cases where a bijective point‐to‐point map is not well defined. Our core idea is to treat each collection as a point‐sampling from a low‐dimensional shape‐space manifold, and use dimensionality reduction techniques to find a low‐dimensional Euclidean embedding of this sampling. To measure distances on the shape‐space manifold, we use the recently introduced shape differences, which lead to a similar low‐dimensional structure of the shape spaces, even if the shapes themselves are quite different. This allows us to use standard affine registration for point‐clouds to align the shape‐spaces, and then find a functional cross‐map using a linear solve. We demonstrate the results of our algorithm on various shape collections and discuss its properties.     

Consistent Partial Matching of Shape Collections via Sparse Modeling

Recent efforts in the area of joint object matching approach the problem by taking as input a set of pairwise maps, which are then jointly optimized across the whole collection so that certain accuracy and consistency criteria are satisfied. One natural requirement is cycle‐consistency—namely the fact that map composition should give the same result regardless of the path taken in the shape collection. In this paper, we introduce a novel approach to obtain consistent matches without requiring initial pairwise solutions to be given as input. We do so by optimizing a joint measure of metric distortion directly over the space of cycle‐consistent maps; in order to allow for partially similar and extra‐class shapes, we formulate the problem as a series of quadratic programs with sparsity‐inducing constraints, making our technique a natural candidate for analysing collections with a large presence of outliers. The particular form of the problem allows us to leverage results and tools from the field of evolutionary game theory. This enables a highly efficient optimization procedure which assures accurate and provably consistent solutions in a matter of minutes in collections with hundreds of shapes.     

Consistent Shape Maps via Semidefinite Programming

Recent advances in shape matching have shown that jointly optimizing the maps among the shapes in a collection can lead to significant improvements when compared to estimating maps between pairs of shapes in isolation. These methods typically invoke a cycle‐consistency criterion — the fact that compositions of maps along a cycle of shapes should approximate the identity map. This condition regularizes the network and allows for the correction of errors and imperfections in individual maps. In particular, it encourages the estimation of maps between dissimilar shapes by compositions of maps along a path of more similar shapes. In this paper, we introduce a novel approach for obtaining consistent shape maps in a collection that formulates the cycle‐consistency constraint as the solution to a semidefinite program (SDP). The proposed approach is based on the observation that, if the ground truth maps between the shapes are cycle‐consistent, then the matrix that stores all pair‐wise maps in blocks is low‐rank and positive semidefinite. Motivated by recent advances in techniques for low‐rank matrix recovery via semidefinite programming, we formulate the problem of estimating cycle‐consistent maps as finding the closest positive semidefinite matrix to an input matrix that stores all the initial maps. By analyzing the Karush‐Kuhn‐Tucker (KKT) optimality condition of this program, we derive theoretical guarantees for the proposed algorithm, ensuring the correctness of the recovery when the errors in the inputs maps do not exceed certain thresholds. Besides this theoretical guarantee, experimental results on benchmark datasets show that the proposed approach outperforms state‐of‐the‐art multiple shape matching methods.     

Deblurring and Denoising of Maps between Shapes

Shape correspondence is an important and challenging problem in geometry processing. Generalized map representations, such as functional maps, have been recently suggested as an approach for handling difficult mapping problems, such as partial matching and matching shapes with high genus, within a generic framework. While this idea was shown to be useful in various scenarios, such maps only provide low frequency information on the correspondence. In many applications, such as texture transfer and shape interpolation, a high quality pointwise map that can transport high frequency data between the shapes is required. We name this problem map deblurring and propose a robust method, based on a smoothness assumption, for its solution. Our approach is suitable for non‐isometric shapes, is robust to mesh tessellation and accurately recovers vertex‐to‐point, or precise, maps. Using the same framework we can also handle map denoising, namely improvement of given pointwise maps from various sources. We demonstrate that our approach outperforms the state‐of‐the‐art for both deblurring and denoising of maps on benchmarks of non‐isometric shapes, and show an application to high quality intrinsic symmetry computation.     

Partial 3‐D Correspondence from Shape Extremities

We present a 3‐D correspondence method to match the geometric extremities of two shapes which are partially isometric. We consider the most general setting of the isometric partial shape correspondence problem, in which shapes to be matched may have multiple common parts at arbitrary scales as well as parts that are not similar. Our rank‐and‐vote‐and‐combine algorithm identifies and ranks potentially correct matches by exploring the space of all possible partial maps between coarsely sampled extremities. The qualified top‐ranked matchings are then subjected to a more detailed analysis at a denser resolution and assigned with confidence values that accumulate into a vote matrix. A minimum weight perfect matching algorithm is finally iterated to combine the accumulated votes into an optimal (partial) mapping between shape extremities, which can further be extended to a denser map. We test the performance of our method on several data sets and benchmarks in comparison with state of the art.     

Shape Matching via Quotient Spaces

We introduce a novel method for non‐rigid shape matching, designed to address the symmetric ambiguity problem present when matching shapes with intrinsic symmetries. Unlike the majority of existing methods which try to overcome this ambiguity by sampling a set of landmark correspondences, we address this problem directly by performing shape matching in an appropriate quotient space, where the symmetry has been identified and factored out. This allows us to both simplify the shape matching problem by matching between subspaces, and to return multiple solutions with equally good dense correspondences. Remarkably, both symmetry detection and shape matching are done without establishing any landmark correspondences between either points or parts of the shapes. This allows us to avoid an expensive combinatorial search present in most intrinsic symmetry detection and shape matching methods. We compare our technique with state‐of‐the‐art methods and show that superior performance can be achieved both when the symmetry on each shape is known and when it needs to be estimated.     

Regularized Pointwise Map Recovery from Functional Correspondence

The concept of using functional maps for representing dense correspondences between deformable shapes has proven to be extremely effective in many applications. However, despite the impact of this framework, the problem of recovering the point‐to‐point correspondence from a given functional map has received surprisingly little interest. In this paper, we analyse the aforementioned problem and propose a novel method for reconstructing pointwise correspondences from a given functional map. The proposed algorithm phrases the matching problem as a regularized alignment problem of the spectral embeddings of the two shapes. Opposed to established methods, our approach does not require the input shapes to be nearly‐isometric, and easily extends to recovering the point‐to‐point correspondence in part‐to‐whole shape matching problems. Our numerical experiments demonstrate that the proposed approach leads to a significant improvement in accuracy in several challenging cases.     

Multiple Shape Correspondence by Dynamic Programming

We present a multiple shape correspondence method based on dynamic programming, that computes consistent bijective maps between all shape pairs in a given collection of initially unmatched shapes. As a fundamental distinction from previous work, our method aims to explicitly minimize the overall distortion, i.e., the average isometric distortion of the resulting maps over all shape pairs. We cast the problem as optimal path finding on a graph structure where vertices are maps between shape extremities. We exploit as much context information as possible using a dynamic programming based algorithm to approximate the optimal solution. Our method generates coarse multiple correspondences between shape extremities, as well as denser correspondences as by‐product. We assess the performance on various mesh sequences of (nearly) isometric shapes. Our experiments show that, for isometric shape collections with non‐uniform triangulation and noise, our method can compute relatively dense correspondences reasonably fast and outperform state of the art in terms of accuracy.     

Continuous Matching via Vector Field Flow

We present a new method for non‐rigid shape matching designed to enforce continuity of the resulting correspondence. Our method is based on the recently proposed functional map representation, which allows efficient manipulation and inference but often fails to provide a continuous point‐to‐point mapping. We address this problem by exploiting the connection between the operator representation of mappings and flows of vector fields. In particular, starting from an arbitrary continuous map between two surfaces we find an optimal flow that makes the final correspondence operator as close as possible to the initial functional map. Our method also helps to address the symmetric ambiguity problem inherent in many intrinsic correspondence methods when matching symmetric shapes. We provide practical and theoretical results showing that our method can be used to obtain an orientation preserving or reversing map starting from a functional map that represents the mixture of the two. We also show how this method can be used to improve the quality of maps produced by existing shape matching methods, and compare the resulting map's continuity with results obtained by other operator‐based techniques.     

2D-Shape Analysis Using Conformal Mapping

The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmüller spaces. In this space every simple closed curve in the plane (a “shape”) is represented by a ‘fingerprint’ which is a diffeomorphism of the unit circle to itself (a differentiable and invertible, periodic function). More precisely, every shape defines to a unique equivalence class of such diffeomorphisms up to right multiplication by a Möbius map. The fingerprint does not change if the shape is varied by translations and scaling and any such equivalence class comes from some shape. This coset space, equipped with the infinitesimal Weil-Petersson (WP) Riemannian norm is a metric space. In this space, the shortest path between each two shapes is unique, and is given by a geodesic connecting them. Their distance from each other is given by integrating the WP-norm along that geodesic. In this paper we concentrate on solving the “welding” problem of “sewing” together conformally the interior and exterior of the unit circle, glued on the unit circle by a given diffeomorphism, to obtain the unique 2D shape associated with this diffeomorphism. This will allow us to go back and forth between 2D shapes and their representing diffeomorphisms in this “space of shapes”. We then present an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes. The group of diffeomorphisms of S¹ acts as a group of isometries on the space of shapes and we show how this can be used to define shape transformations, like for instance ‘adding a protruding limb’ to any shape.     

Data‐Driven Sparse Priors of 3D Shapes

We present a sparse optimization framework for extracting sparse shape priors from a collection of 3D models. Shape priors are defined as point‐set neighborhoods sampled from shape surfaces which convey important information encompassing normals and local shape characterization. A 3D shape model can be considered to be formed with a set of 3D local shape priors, while most of them are likely to have similar geometry. Our key observation is that the local priors extracted from a family of 3D shapes lie in a very low‐dimensional manifold. Consequently, a compact and informative subset of priors can be learned to efficiently encode all shapes of the same family. A comprehensive library of local shape priors is first built with the given collection of 3D models of the same family. We then formulate a global, sparse optimization problem which enforces selecting representative priors while minimizing the reconstruction error. To solve the optimization problem, we design an efficient solver based on the Augmented Lagrangian Multipliers method (ALM). Extensive experiments exhibit the power of our data‐driven sparse priors in elegantly solving several high‐level shape analysis applications and geometry processing tasks, such as shape retrieval, style analysis and symmetry detection.     

Use of an automatic content analysis tool: A technique for seeing both local and global scope

This paper examines what can be learned about bodies of literature using a concept mapping tool, Leximancer. Statistical content analysis and concept mapping were used to analyse bodies of literature from different domains in three case studies. In the first case study, concept maps were generated and analysed for two closely related document sets—a thesis on language games and the background literature for the thesis. The aim for the case study was to show how concept maps might be used to analyse related document collections for coverage. The two maps overlapped on the concept of “language”; however, there was a stronger focus in the thesis on “simulations” and “agents.” Other concepts were not as strong in the thesis map as expected. The study showed how concept maps can help to establish the coverage of the background literature in a thesis. In the second case study, three sets of documents from the domain of conceptual and spatial navigation were collected, each discussing a separate topic: navigational strategies, the brain's role in navigation, and concept mapping. The aim was to explore emergent patterns in a set of related concept maps that may not be apparent from reading the literature alone. Separate concept maps were generated for each topic and also for the combined set of literature. It was expected that each of the topics would be situated in different parts of the combined map, with the concept of “navigation” central to the map. Instead, the concept of “spatial” was centrally situated and the areas of the map for the brain and for navigational strategies overlaid the same region. The unexpected structure provided a new perspective on the coverage of the documents. In the third and final case study, a set of documents on sponges—a domain unfamiliar to the reader—was collected from the Internet and then analysed with a concept map. The aim of this case study was to present how a concept map could aid in quickly understanding a new, technically intensive domain. Using the concept map to identify significant concepts and the Internet to look for their definitions, a basic understanding of key terms in the domain was obtained relatively quickly. It was concluded that using concept maps is effective for identifying trends within documents and document collections, for performing differential analysis on documents, and as an aid for rapidly gaining an understanding in a new domain by exploring the local detail within the global scope of the textual corpus.     

自监督深度残差函数映射网络的3维模型对应关系计算

目的 针对传统非刚性3维模型的对应关系计算方法需要模型间真实对应关系监督的缺点,提出一种自监督深度残差函数映射网络（self-supervised deep residual functional maps network,SSDRFMN）。方法 首先将局部坐标系与直方图结合以计算3维模型的特征描述符,即方向直方图签名（signature of histograms of orientations,SHOT）描述符;其次将源模型与目标模型的SHOT描述符输入SSDRFMN,利用深度函数映射（deep functional maps,DFM）层计算两个模型间的函数映射矩阵,并通过模糊对应层将函数映射关系转换为点到点的对应关系;最后利用自监督损失函数计算模型间的测地距离误差,对计算出的对应关系进行评估。结果 实验结果表明,在MPI-FAUST数据集上,本文算法相比于有监督的深度函数映射（supervised deep functional maps,SDFM）算法,人体模型对应关系的测地误差减小了1.45;相比于频谱上采样（spectral upsampling,SU）算法减小了1.67。在TOSCA数据集上,本文算法相比于SDFM算法,狗、猫和狼等模型的对应关系的测地误差分别减小了3.13、0.98和1.89;相比于SU算法分别减小了2.81、2.22和1.11,并有效克服了已有深度函数映射方法需要模型间的真实对应关系来监督的缺点,使得该方法可以适用于不同的数据集,可扩展性大幅增强。结论 本文通过自监督深度残差函数映射网络训练模型的方向直方图签名描述符,提升了模型对应关系的准确率。本文方法可以适应于不同的数据集,相比传统方法,普适性较好。     

All‐Hex Mesh Generation via Volumetric PolyCube Deformation

While hexahedral mesh elements are preferred by a variety of simulation techniques, constructing quality all‐hex meshes of general shapes remains a challenge. An attractive hex‐meshing approach, often referred to as sub‐mapping, uses a low distortion mapping between the input model and a PolyCube (a solid formed from a union of cubes), to transfer a regular hex grid from the PolyCube to the input model. Unfortunately, the construction of suitable PolyCubes and corresponding volumetric maps for arbitrary shapes remains an open problem. Our work introduces a new method for computing low‐distortion volumetric PolyCube deformations of general shapes and for subsequent all‐hex remeshing. For a given input model, our method simultaneously generates an appropriate PolyCube structure and mapping between the input model and the PolyCube. From these we automatically generate good quality all‐hex meshes of complex natural and man‐made shapes.     

Adjoint Map Representation for Shape Analysis and Matching

In this paper, we propose to consider the adjoint operators of functional maps, and demonstrate their utility in several tasks in geometry processing. Unlike a functional map, which represents a correspondence simply using the pull‐back of function values, the adjoint operator reflects both the map and its distortion with respect to given inner products. We argue that this property of adjoint operators and especially their relation to the map inverse under the choice of different inner products, can be useful in applications including bi‐directional shape matching, shape exploration, and pointwise map recovery among others. In particular, in this paper, we show that the adjoint operators can be used within the cycle‐consistency framework to encode and reveal the presence or lack of consistency between distortions in a collection, in a way that is complementary to the previously used purely map‐based consistency measures. We also show how the adjoint can be used for matching pairs of shapes, by accounting for maps in both directions, can help in recovering point‐to‐point maps from their functional counterparts, and describe how it can shed light on the role of functional basis selection.     

Microtiles: Extracting Building Blocks from Correspondences

In this paper, we develop a theoretical framework for characterizing shapes by building blocks. We address two questions: First, how do shape correspondences induce building blocks? For this, we introduce a new representation for structuring partial symmetries (partial self‐correspondences), which we call “microtiles”. Starting from input correspondences that form point‐wise equivalence relations, microtiles are obtained by grouping connected components of points that share the same set of symmetry transformations. The decomposition is unique, requires no parameters beyond the input correspondences, and encodes the partial symmetries of all subsets of the input. The second question is: What is the class of shapes that can be assembled from these building blocks? Here, we specifically consider r‐similarity as correspondence model, i.e., matching of local r‐neighborhoods. Our main result is that the microtiles of the partial r‐symmetries of an object S can build all objects that are (r+ε)‐similar to S for any ε >0. Again, the construction is unique. Furthermore, we give necessary conditions for a set of assembly rules for the pairwise connection of tiles. We describe a practical algorithm for computing microtile decompositions under rigid motions, a corresponding prototype implementation, and conduct a number of experiments to visualize the structural properties in practice.     

As‐Killing‐As‐Possible Vector Fields for Planar Deformation

Cartoon animation, image warping, and several other tasks in two‐dimensional computer graphics reduce to the formulation of a reasonable model for planar deformation. A deformation is a map from a given shape to a new one, and its quality is determined by the type of distortion it introduces. In many applications, a desirable map is as isometric as possible. Finding such deformations, however, is a nonlinear problem, and most of the existing solutions approach it by minimizing a nonlinear energy. Such methods are not guaranteed to converge to a global optimum and often suffer from robustness issues. We propose a new approach based on approximate Killing vector fields (AKVFs), first introduced in shape processing. AKVFs generate near‐isometric deformations, which can be motivated as direction fields minimizing an “as‐rigid‐as‐possible” (ARAP) energy to first order. We first solve for an AKVF on the domain given user constraints via a linear optimization problem and then use this AKVF as the initial velocity field of the deformation. In this way, we transfer the inherent nonlinearity of the deformation problem to finding trajectories for each point of the domain having the given initial velocities. We show that a specific class of trajectories — the set of logarithmic spirals — is especially suited for this task both in practice and through its relationship to linear holomorphic vector fields. We demonstrate the effectiveness of our method for planar deformation by comparing it with existing state‐of‐the‐art deformation methods.