Partial Functional Correspondence

In this paper, we propose a method for computing partial functional correspondence between non‐rigid shapes. We use perturbation analysis to show how removal of shape parts changes the Laplace–Beltrami eigenfunctions, and exploit it as a prior on the spectral representation of the correspondence. Corresponding parts are optimization variables in our problem and are used to weight the functional correspondence; we are looking for the largest and most regular (in the Mumford–Shah sense) parts that minimize correspondence distortion. We show that our approach can cope with very challenging correspondence settings.     

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.     

Fully Spectral Partial Shape Matching

We propose an efficient procedure for calculating partial dense intrinsic correspondence between deformable shapes performed entirely in the spectral domain. Our technique relies on the recently introduced partial functional maps formalism and on the joint approximate diagonalization (JAD) of the Laplace‐Beltrami operators previously introduced for matching non‐isometric shapes. We show that a variant of the JAD problem with an appropriately modified coupling term (surprisingly) allows to construct quasi‐harmonic bases localized on the latent corresponding parts. This circumvents the need to explicitly compute the unknown parts by means of the cumbersome alternating minimization used in the previous approaches, and allows performing all the calculations in the spectral domain with constant complexity independent of the number of shape vertices. We provide an extensive evaluation of the proposed technique on standard non‐rigid correspondence benchmarks and show state‐of‐the‐art performance in various settings, including partiality and the presence of topological noise.     

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.     

Deformation‐Driven Shape Correspondence

Non‐rigid 3D shape correspondence is a fundamental and difficult problem. Most applications which require a correspondence rely on manually selected markers. Without user assistance, the performances of existing automatic correspondence methods depend strongly on a good initial shape alignment or shape prior, and they generally do not tolerate large shape variations. We present an automatic feature correspondence algorithm capable of handling large, non‐rigid shape variations, as well as partial matching. This is made possible by leveraging the power of state‐of‐the‐art mesh deformation techniques and relying on a combinatorial tree traversal for correspondence search. The search is deformation‐driven, prioritized by a self‐distortion energy measured on meshes deformed according to a given correspondence. We demonstrate the ability of our approach to naturally match shapes which differ in pose, local scale, part decomposition, and geometric detail through numerous examples.     

Partial Shape Matching Using Transformation Parameter Similarity

In this paper, we present a method for non‐rigid, partial shape matching in vector graphics. Given a user‐specified query region in a 2D shape, similar regions are found, even if they are non‐linearly distorted. Furthermore, a non‐linear mapping is established between the query regions and these matches, which allows the automatic transfer of editing operations such as texturing. This is achieved by a two‐step approach. First, pointwise correspondences between the query region and the whole shape are established. The transformation parameters of these correspondences are registered in an appropriate transformation space. For transformations between similar regions, these parameters form surfaces in transformation space, which are extracted in the second step of our method. The extracted regions may be related to the query region by a non‐rigid transform, enabling non‐rigid shape matching.     

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.     

Global Correspondence Optimization for Non‐Rigid Registration of Depth Scans

We present a registration algorithm for pairs of deforming and partial range scans that addresses the challenges of non‐rigid registration within a single non‐linear optimization. Our algorithm simultaneously solves for correspondences between points on source and target scans, confidence weights that measure the reliability of each correspondence and identify non‐overlapping areas, and a warping field that brings the source scan into alignment with the target geometry. The optimization maximizes the region of overlap and the spatial coherence of the deformation while minimizing registration error. All optimization parameters are chosen automatically; hand‐tuning is not necessary. Our method is not restricted to part‐in‐whole matching, but addresses the general problem of partial matching, and requires no explicit prior correspondences or feature points. We evaluate the performance and robustness of our method using scan data acquired by a structured light scanner and compare our method with existing non‐rigid registration algorithms.     

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.     

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.     

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.     

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.     

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.     

Non‐Rigid Registration Under Isometric Deformations

We present a robust and efficient algorithm for the pairwise non‐rigid registration of partially overlapping 3D surfaces. Our approach treats non‐rigid registration as an optimization problem and solves it by alternating between correspondence and deformation optimization. Assuming approximately isometric deformations, robust correspondences are generated using a pruning mechanism based on geodesic consistency. We iteratively learn an appropriate deformation discretization from the current set of correspondences and use it to update the correspondences in the next iteration. Our algorithm is able to register partially similar point clouds that undergo large deformations, in just a few seconds. We demonstrate the potential of our algorithm in various applications such as example based articulated segmentation, and shape interpolation.     

A Condition Number for Non‐Rigid Shape Matching

Despite the large amount of work devoted in recent years to the problem of non‐rigid shape matching, practical methods that can successfully be used for arbitrary pairs of shapes remain elusive. In this paper, we study the hardness of the problem of shape matching, and introduce the notion of the shape condition number, which captures the intuition that some shapes are inherently more difficult to match against than others. In particular, we make a connection between the symmetry of a given shape and the stability of any method used to match it while optimizing a given distortion measure. We analyze two commonly used classes of methods in deformable shape matching, and show that the stability of both types of techniques can be captured by the appropriate notion of a condition number. We also provide a practical way to estimate the shape condition number and show how it can be used to guide the selection of landmark correspondences between shapes. Thus we shed some light on the reasons why general shape matching remains difficult and provide a way to detect and mitigate such difficulties in practice.     

Full 3D Plant Reconstruction via Intrusive Acquisition

Digitally capturing vegetation using off‐the‐shelf scanners is a challenging problem. Plants typically exhibit large self‐occlusions and thin structures which cannot be properly scanned. Furthermore, plants are essentially dynamic, deforming over the time, which yield additional difficulties in the scanning process. In this paper, we present a novel technique for acquiring and modelling of plants and foliage. At the core of our method is an intrusive acquisition approach, which disassembles the plant into disjoint parts that can be accurately scanned and reconstructed offline. We use the reconstructed part meshes as 3D proxies for the reconstruction of the complete plant and devise a global‐to‐local non‐rigid registration technique that preserves specific plant characteristics. Our method is tested on plants of various styles, appearances and characteristics. Results show successful reconstructions with high accuracy with respect to the acquired data.     

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.     

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.