Coarse‐to‐Fine Isometric Shape Correspondence by Tracking Symmetric Flips

We address the symmetric flip problem that is inherent to multi‐resolution isometric shape matching algorithms. To this effect, we extend our previous work which handles the dense isometric correspondence problem in the original 3D Euclidean space via coarse‐to‐fine combinatorial matching. The key idea is based on keeping track of all optimal solutions, which may be more than one due to symmetry especially at coarse levels, throughout denser levels of the shape matching process. We compare the resulting dense correspondence algorithm with state‐of‐the‐art techniques over several 3D shape benchmark datasets. The experiments show that our method, which is fast and scalable, is performance‐wise better than or on a par with the best performant algorithms existing in the literature for isometric (or nearly isometric) shape correspondence. Our key idea of tracking symmetric flips can be considered as a meta‐approach that can be applied to other multi‐resolution shape matching algorithms, as we also demonstrate by experiments.     

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.     

Scale Normalization for Isometric Shape Matching

We address the scale problem inherent to isometric shape correspondence in a combinatorial matching framework. We consider a particular setting of the general correspondence problem where one of the two shapes to be matched is an isometric (or nearly isometric) part of the other up to an arbitrary scale. We resolve the scale ambiguity by finding a coarse matching between shape extremities based on a novel scale‐invariant isometric distortion measure. The proposed algorithm also supports (partial) dense matching, that alleviates the symmetric flip problem due to initial coarse sampling. We test the performance of our matching algorithm on several shape datasets in comparison to state of the art. Our method proves useful, not only for partial matching, but also for complete matching of semantically similar hybrid shape pairs whose maximum geodesic distances may not be compatible, a case that would fail most of the conventional isometric shape matchers.     

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.     

Structured Regularization of Functional Map Computations

We consider the problem of non‐rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace‐Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel, theoretically well‐justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one‐parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly‐used refinement techniques.     

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.     

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.     

Stable Region Correspondences Between Non‐Isometric Shapes

We consider the problem of finding meaningful correspondences between 3D models that are related but not necessarily very similar. When the shapes are quite different, a point‐to‐point map is not always appropriate, so our focus in this paper is a method to build a set of correspondences between shape regions or parts. The proposed approach exploits a variety of feature functions on the shapes and makes use of the key observation that points in matching parts have similar ranks in the sorting of the corresponding feature values. Our algorithm proceeds in two steps. We first build an affinity matrix between points on the two shapes, based on feature rank similarity over many feature functions. We then define a notion of stability of a pair of regions, with respect to this affinity matrix, obtained as a fixed point of a nonlinear operator. Our method yields a family of corresponding maximally stable regions between the two shapes that can be used to define shape parts. We observe that this is an instance of the biclustering problem and that it is related to solving a constrained maximal eigenvalue problem. We provide an algorithm to solve this problem that mimics the power method. We show the robustness of its output to noisy input features as well its convergence properties. The obtained part correspondences are shown to be almost perfect matches in the isometric case, and also semantically appropriate even in non‐isometric cases. We provide numerous examples and applications of this technique, for example to sharpening correspondences in traditional shape matching algorithms.     

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.     

A low-dimensional representation for robust partial isometric correspondences computation

Intrinsic shape matching has become the standard approach for pose invariant correspondence estimation among deformable shapes. Most existing approaches assume global consistency. While global isometric matching is well understood, only a few heuristic solutions are known for partial matching. Partial matching is particularly important for robustness to topological noise, which is a common problem in real-world scanner data. We introduce a new approach to partial isometric matching based on the observation that isometries are fully determined by local information: a map of a single point and its tangent space fixes an isometry. We develop a new representation for partial isometric maps based on equivalence classes of correspondences between pairs of points and their tangent-spaces. We apply our approach to register partial point clouds and compare it to the state-of-the-art methods, where we obtain significant improvements over global methods for real-world data and stronger guarantees than previous partial matching algorithms.     

Analysis and Visualization of Maps Between Shapes

In this paper we propose a method for analysing and visualizing individual maps between shapes, or collections of such maps. Our method is based on isolating and highlighting areas where the maps induce significant distortion of a given measure in a multi‐scale way. Unlike the majority of prior work, which focuses on discovering maps in the context of shape matching, our main focus is on evaluating, analysing and visualizing a given map, and the distortion(s) it introduces, in an efficient and intuitive way. We are motivated primarily by the fact that most existing metrics for map evaluation are quadratic and expensive to compute in practice, and that current map visualization techniques are suitable primarily for global map understanding, and typically do not highlight areas where the map fails to meet certain quality criteria in a multi‐scale way. We propose to address these challenges in a unified way by considering the functional representation of a map, and performing spectral analysis on this representation. In particular, we propose a simple multi‐scale method for map evaluation and visualization, which provides detailed multi‐scale information about the distortion induced by a map, which can be used alongside existing global visualization techniques.     

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。     

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.     

A new graph matching method for point-set correspondence using the EM algorithm and Softassign

Finding correspondences between two point-sets is a common step in many vision applications (e.g., image matching or shape retrieval). We present a graph matching method to solve the point-set correspondence problem, which is posed as one of mixture modelling. Our mixture model encompasses a model of structural coherence and a model of affine-invariant geometrical errors. Instead of absolute positions, the geometrical positions are represented as relative positions of the points with respect to each other. We derive the Expectation–Maximization algorithm for our mixture model. In this way, the graph matching problem is approximated, in a principled way, as a succession of assignment problems which are solved using Softassign. Unlike other approaches, we use a true continuous underlying correspondence variable. We develop effective mechanisms to detect outliers. This is a useful technique for improving results in the presence of clutter. We evaluate the ability of our method to locate proper matches as well as to recognize object categories in a series of registration and recognition experiments. Our method compares favourably to other graph matching methods as well as to point-set registration methods and outlier rejectors.     

Polar NURBS Surface with Curvature Continuity

Polar NURBS surface is a kind of periodic NURBS surface, one boundary of which shrinks to a degenerate polar point. The specific topology of its control‐point mesh offers the ability to represent a cap‐like surface, which is common in geometric modeling. However, there is a critical and challenging problem that hinders its application: curvature continuity at the extraordinary singular pole. We first propose a sufficient and necessary condition of curvature continuity at the pole. Then, we present constructive methods for the two key problems respectively: how to construct a polar NURBS surface with curvature continuity and how to reform an ordinary polar NURBS surface to curvature continuous. The algorithms only depend on the symbolic representation and operations of NURBS, and they introduce no restrictions on the degree or the knot vectors. Examples and comparisons demonstrate the applications of the curvature‐continuous polar NURBS surface in hole‐filling and free‐shape modeling.     

A fast modal space transform for robust nonrigid shape retrieval

Nonrigid or deformable 3D objects are common in many application domains. Retrieval of such objects in large databases based on shape similarity is still a challenging problem. In this paper, we take advantages of functional operators as characterizations of shape deformation, and further propose a framework to design novel shape signatures for encoding nonrigid geometries. Our approach constructs a context-aware integral kernel operator on a manifold, then applies modal analysis to map this operator into a low-frequency functional representation, called fast functional transform, and finally computes its spectrum as the shape signature. In a nutshell, our method is fast, isometry-invariant, discriminative, smooth and numerically stable with respect to multiple types of perturbations. Experimental results demonstrate that our new shape signature for nonrigid objects can outperform all methods participating in the nonrigid track of the SHREC’11 contest. It is also the second best performing method in the real human model track of SHREC’14.     

基于Voronoi地图表示方法的同步定位与地图创建

针对基于混合米制地图机器人同步定位与地图创建 (Simultaneous localization and mapping, SLAM)中地图划分方法不完善的问题, 提出了基于Voronoi地图表示方法的同步定位与地图创建算法VorSLAM. 该算法在全局坐标系下创建特征地图, 并根据此特征地图使用Voronoi图唯一地划分地图空间, 在每一个划分内部创建一个相对于特征的局部稠密地图. 特征地图与各个局部地图最终一起连续稠密地描述了环境. Voronoi地图表示方法解决了地图划分的唯一性问题, 理论证明局部地图可以完整描述该划分所对应的环境轮廓. 该地图表示方法一个基本特点是特征与局部地图一一对应, 每个特征都关联一个定义在该特征上的局部地图. 基于该特点, 提出了一个基于形状匹配的数据关联算法, 用以解决传统数据关联算法出现的多重关联问题. 一个公寓弧形走廊的实验验证了VorSLAM算法和基于形状匹配的数据关联方法的有效性.