Florian Bernard

Paul Roetzer, Paul Swoboda, Daniel Cremers et al.

We present a scalable combinatorial algorithm for globally optimizing over the space of geometrically consistent mappings between 3D shapes. We use the mathematically elegant formalism proposed by Windheuser et al. (ICCV 2011) where 3D shape matching was formulated as an integer linear program over the space of orientation-preserving diffeomorphisms. Until now, the resulting formulation had limited practical applicability due to its complicated constraint structure and its large size. We propose a novel primal heuristic coupled with a Lagrange dual problem that is several orders of magnitudes faster compared to previous solvers. This allows us to handle shapes with substantially more triangles than previously solvable. We demonstrate compelling results on diverse datasets, and, even showcase that we can address the challenging setting of matching two partial shapes without availability of complete shapes. Our code is publicly available at http://github.com/paul0noah/sm-comb .

Paul Roetzer, Zorah Lähner, Florian Bernard

We consider the problem of finding a continuous and non-rigid matching between a 2D contour and a 3D mesh. While such problems can be solved to global optimality by finding a shortest path in the product graph between both shapes, existing solutions heavily rely on unrealistic prior assumptions to avoid degenerate solutions (e.g. knowledge to which region of the 3D shape each point of the 2D contour is matched). To address this, we propose a novel 2D-3D shape matching formalism based on the conjugate product graph between the 2D contour and the 3D shape. Doing so allows us for the first time to consider higher-order costs, i.e. defined for edge chains, as opposed to costs defined for single edges. This offers substantially more flexibility, which we utilise to incorporate a local rigidity prior. By doing so, we effectively circumvent degenerate solutions and thereby obtain smoother and more realistic matchings, even when using only a one-dimensional feature descriptor. Overall, our method finds globally optimal and continuous 2D-3D matchings, has the same asymptotic complexity as previous solutions, produces state-of-the-art results for shape matching and is even capable of matching partial shapes. Our code is publicly available (https://github.com/paul0noah/sm-2D3D).

Maolin Gao, Paul Roetzer, Marvin Eisenberger et al.

We propose a novel mixed-integer programming (MIP) formulation for generating precise sparse correspondences for highly non-rigid shapes. To this end, we introduce a projected Laplace-Beltrami operator (PLBO) which combines intrinsic and extrinsic geometric information to measure the deformation quality induced by predicted correspondences. We integrate the PLBO, together with an orientation-aware regulariser, into a novel MIP formulation that can be solved to global optimality for many practical problems. In contrast to previous methods, our approach is provably invariant to rigid transformations and global scaling, initialisation-free, has optimality guarantees, and scales to high resolution meshes with (empirically observed) linear time. We show state-of-the-art results for sparse non-rigid matching on several challenging 3D datasets, including data with inconsistent meshing, as well as applications in mesh-to-point-cloud matching.

Dongliang Cao, Florian Bernard

3D shape matching is a long-standing problem in computer vision and computer graphics. While deep neural networks were shown to lead to state-of-the-art results in shape matching, existing learning-based approaches are limited in the context of multi-shape matching: (i) either they focus on matching pairs of shapes only and thus suffer from cycle-inconsistent multi-matchings, or (ii) they require an explicit template shape to address the matching of a collection of shapes. In this paper, we present a novel approach for deep multi-shape matching that ensures cycle-consistent multi-matchings while not depending on an explicit template shape. To this end, we utilise a shape-to-universe multi-matching representation that we combine with powerful functional map regularisation, so that our multi-shape matching neural network can be trained in a fully unsupervised manner. While the functional map regularisation is only considered during training time, functional maps are not computed for predicting correspondences, thereby allowing for fast inference. We demonstrate that our method achieves state-of-the-art results on several challenging benchmark datasets, and, most remarkably, that our unsupervised method even outperforms recent supervised methods.

Dongliang Cao, Florian Bernard

The matching of 3D shapes has been extensively studied for shapes represented as surface meshes, as well as for shapes represented as point clouds. While point clouds are a common representation of raw real-world 3D data (e.g. from laser scanners), meshes encode rich and expressive topological information, but their creation typically requires some form of (often manual) curation. In turn, methods that purely rely on point clouds are unable to meet the matching quality of mesh-based methods that utilise the additional topological structure. In this work we close this gap by introducing a self-supervised multimodal learning strategy that combines mesh-based functional map regularisation with a contrastive loss that couples mesh and point cloud data. Our shape matching approach allows to obtain intramodal correspondences for triangle meshes, complete point clouds, and partially observed point clouds, as well as correspondences across these data modalities. We demonstrate that our method achieves state-of-the-art results on several challenging benchmark datasets even in comparison to recent supervised methods, and that our method reaches previously unseen cross-dataset generalisation ability.

Zhakshylyk Nurlanov, Frank R. Schmidt, Florian Bernard

Many challenges from natural world can be formulated as a graph matching problem. Previous deep learning-based methods mainly consider a full two-graph matching setting. In this work, we study the more general partial matching problem with multi-graph cycle consistency guarantees. Building on a recent progress in deep learning on graphs, we propose a novel data-driven method (URL) for partial multi-graph matching, which uses an object-to-universe formulation and learns latent representations of abstract universe points. The proposed approach advances the state of the art in semantic keypoint matching problem, evaluated on Pascal VOC, CUB, and Willow datasets. Moreover, the set of controlled experiments on a synthetic graph matching dataset demonstrates the scalability of our method to graphs with large number of nodes and its robustness to high partiality.

Paul Roetzer, Ahmed Abbas, Dongliang Cao et al.

In this work we propose to combine the advantages of learningbased and combinatorial formalisms for 3D shape matching. While learningbased methods lead to state-of-the-art matching performance, they do not ensure geometric consistency, so that obtained matchings are locally non-smooth. On the contrary, axiomatic, optimisation-based methods allow to take geometric consistency into account by explicitly constraining the space of valid matchings. However, existing axiomatic formalisms do not scale to practically relevant problem sizes, and require user input for the initialisation of non-convex optimisation problems. We work towards closing this gap by proposing a novel combinatorial solver that combines a unique set of favourable properties: our approach (i) is initialisation free, (ii) is massively parallelisable and powered by a quasi-Newton method, (iii) provides optimality gaps, and (iv) delivers improved matching quality with decreased runtime and globally optimal results for many instances.

Viktoria Ehm, Paul Roetzer, Marvin Eisenberger et al.

Finding correspondences between 3D shapes is a crucial problem in computer vision and graphics, which is for example relevant for tasks like shape interpolation, pose transfer, or texture transfer. An often neglected but essential property of matchings is geometric consistency, which means that neighboring triangles in one shape are consistently matched to neighboring triangles in the other shape. Moreover, while in practice one often has only access to partial observations of a 3D shape (e.g. due to occlusion, or scanning artifacts), there do not exist any methods that directly address geometrically consistent partial shape matching. In this work we fill this gap by proposing to integrate state-of-the-art deep shape features into a novel integer linear programming partial shape matching formulation. Our optimization yields a globally optimal solution on low resolution shapes, which we then refine using a coarse-to-fine scheme. We show that our method can find more reliable results on partial shapes in comparison to existing geometrically consistent algorithms (for which one first has to fill missing parts with a dummy geometry). Moreover, our matchings are substantially smoother than learning-based state-of-the-art shape matching methods.

Johan Thunberg, Florian Bernard

We propose a novel non-negative spherical relaxation for optimization problems over binary matrices with injectivity constraints, which in particular has applications in multi-matching and clustering. We relax respective binary matrix constraints to the (high-dimensional) non-negative sphere. To optimize our relaxed problem, we use a conditional power iteration method to iteratively improve the objective function, while at same time sweeping over a continuous scalar parameter that is (indirectly) related to the universe size (or number of clusters). Opposed to existing procedures that require to fix the integer universe size before optimization, our method automatically adjusts the analogous continuous parameter. Furthermore, while our approach shares similarities with spectral multi-matching and spectral clustering, our formulation has the strong advantage that we do not rely on additional post-processing procedures to obtain binary results. Our method shows compelling results in various multi-matching and clustering settings, even when compared to methods that use the ground truth universe size (or number of clusters).

Viktoria Ehm, Maolin Gao, Paul Roetzer et al.

Finding correspondences between 3D shapes is an important and long-standing problem in computer vision, graphics and beyond. A prominent challenge are partial-to-partial shape matching settings, which occur when the shapes to match are only observed incompletely (e.g. from 3D scanning). Although partial-to-partial matching is a highly relevant setting in practice, it is rarely explored. Our work bridges the gap between existing (rather artificial) 3D full shape matching and partial-to-partial real-world settings by exploiting geometric consistency as a strong constraint. We demonstrate that it is indeed possible to solve this challenging problem in a variety of settings. For the first time, we achieve geometric consistency for partial-to-partial matching, which is realized by a novel integer non-linear program formalism building on triangle product spaces, along with a new pruning algorithm based on linear integer programming. Further, we generate a new inter-class dataset for partial-to-partial shape-matching. We show that our method outperforms current SOTA methods on both an established intra-class dataset and our novel inter-class dataset.

Maolin Gao, Zorah Lähner, Johan Thunberg et al.

Finding correspondences between shapes is a fundamental problem in computer vision and graphics, which is relevant for many applications, including 3D reconstruction, object tracking, and style transfer. The vast majority of correspondence methods aim to find a solution between pairs of shapes, even if multiple instances of the same class are available. While isometries are often studied in shape correspondence problems, they have not been considered explicitly in the multi-matching setting. This paper closes this gap by proposing a novel optimisation formulation for isometric multi-shape matching. We present a suitable optimisation algorithm for solving our formulation and provide a convergence and complexity analysis. Our algorithm obtains multi-matchings that are by construction provably cycle-consistent. We demonstrate the superior performance of our method on various datasets and set the new state-of-the-art in isometric multi-shape matching.

Florian Bernard, Zeeshan Khan Suri, Christian Theobalt

We present a convex mixed-integer programming formulation for non-rigid shape matching. To this end, we propose a novel shape deformation model based on an efficient low-dimensional discrete model, so that finding a globally optimal solution is tractable in (most) practical cases. Our approach combines several favourable properties: it is independent of the initialisation, it is much more efficient to solve to global optimality compared to analogous quadratic assignment problem formulations, and it is highly flexible in terms of the variants of matching problems it can handle. Experimentally we demonstrate that our approach outperforms existing methods for sparse shape matching, that it can be used for initialising dense shape matching methods, and we showcase its flexibility on several examples.

Florian Bernard, Johan Thunberg, Paul Swoboda et al.

The matching of multiple objects (e.g. shapes or images) is a fundamental problem in vision and graphics. In order to robustly handle ambiguities, noise and repetitive patterns in challenging real-world settings, it is essential to take geometric consistency between points into account. Computationally, the multi-matching problem is difficult. It can be phrased as simultaneously solving multiple (NP-hard) quadratic assignment problems (QAPs) that are coupled via cycle-consistency constraints. The main limitations of existing multi-matching methods are that they either ignore geometric consistency and thus have limited robustness, or they are restricted to small-scale problems due to their (relatively) high computational cost. We address these shortcomings by introducing a Higher-order Projected Power Iteration method, which is (i) efficient and scales to tens of thousands of points, (ii) straightforward to implement, (iii) able to incorporate geometric consistency, (iv) guarantees cycle-consistent multi-matchings, and (iv) comes with theoretical convergence guarantees. Experimentally we show that our approach is superior to existing methods.

Florian Bernard, Johan Thunberg, Jorge Goncalves et al.

In this work we study permutation synchronisation for the challenging case of partial permutations, which plays an important role for the problem of matching multiple objects (e.g. images or shapes). The term synchronisation refers to the property that the set of pairwise matchings is cycle-consistent, i.e. in the full matching case all compositions of pairwise matchings over cycles must be equal to the identity. Motivated by clustering and matrix factorisation perspectives of cycle-consistency, we derive an algorithm to tackle the permutation synchronisation problem based on non-negative factorisations. In order to deal with the inherent non-convexity of the permutation synchronisation problem, we use an initialisation procedure based on a novel rotation scheme applied to the solution of the spectral relaxation. Moreover, this rotation scheme facilitates a convenient Euclidean projection to obtain a binary solution after solving our relaxed problem. In contrast to state-of-the-art methods, our approach is guaranteed to produce cycle-consistent results. We experimentally demonstrate the efficacy of our method and show that it achieves better results compared to existing methods.

Florian Bernard, Christian Theobalt, Michael Moeller

In this work we study convex relaxations of quadratic optimisation problems over permutation matrices. While existing semidefinite programming approaches can achieve remarkably tight relaxations, they have the strong disadvantage that they lift the original $n {\times} n$-dimensional variable to an $n^2 {\times} n^2$-dimensional variable, which limits their practical applicability. In contrast, here we present a lifting-free convex relaxation that is provably at least as tight as existing (lifting-free) convex relaxations. We demonstrate experimentally that our approach is superior to existing convex and non-convex methods for various problems, including image arrangement and multi-graph matching.

Johan Thunberg, Florian Bernard, Jorge Goncalves

Transitive consistency is an intrinsic property for collections of linear invertible transformations between Euclidean coordinate frames. In practice, when the transformations are estimated from data, this property is lacking. This work addresses the problem of synchronizing transformations that are not transitively consistent. Once the transformations have been synchronized, they satisfy the transitive consistency condition - a transformation from frame $A$ to frame $C$ is equal to the composite transformation of first transforming A to B and then transforming B to C. The coordinate frames correspond to nodes in a graph and the transformations correspond to edges in the same graph. Two direct or centralized synchronization methods are presented for different graph topologies; the first one for quasi-strongly connected graphs, and the second one for connected graphs. As an extension of the second method, an iterative Gauss-Newton method is presented, which is later adapted to the case of affine and Euclidean transformations. Two distributed synchronization methods are also presented for orthogonal matrices, which can be seen as distributed versions of the two direct or centralized methods; they are similar in nature to standard consensus protocols used for distributed averaging. When the transformations are orthogonal matrices, a bound on the optimality gap can be computed. Simulations show that the gap is almost right, even for noise large in magnitude. This work also contributes on a theoretical level by providing linear algebraic relationships for transitively consistent transformations. One of the benefits of the proposed methods is their simplicity - basic linear algebraic methods are used, e.g., the Singular Value Decomposition (SVD). For a wide range of parameter settings, the methods are numerically validated.