Abstract We present a framework to integrate secondary motion into the existing animation pipelines. Skinning provides fast computation for real‐time animation and intuitive control over the deformation. Despite the benefits, traditional skinning methods lack secondary dynamics such as the jiggling of fat tissues. We address the rigidity of skinning methods by physically simulating the deformation handles with spring forces. Most studies introduce secondary motion into skinning by employing FEM simulation on volumetric mesh vertices, coupling their computational complexity with mesh resolution. Unlike these approaches, we do not require any volumetric mesh input. Our method scales to higher mesh resolutions by directly simulating deformation handles. The simulated handles, namely the spring bones, enrich rigid skinning deformation with a diverse range of secondary animation for subjects including rigid bodies, elastic bodies, soft tissues, and cloth simulation. In essence, we leverage the benefits of physical simulations in the scope of deformation handles to achieve controllable real‐time dynamics on a wide range of subjects while remaining compatible with existing skinning pipelines. Our method avoids tetrahedral remeshing and it is significantly faster compared to FEM‐based volumetric mesh simulations.

Abstract We compute the kernel of a shape embedded in 3D as a polygon mesh, which is defined as the set of all points that have a clear line of sight to every point of the mesh. The KerGen algorithm, short for Kernel Generation, employs efficient plane‐plane and line‐plane intersections, alongside point classifications based on their positions relative to planes. This approach allows for the incremental addition of kernel vertices and edges to the resulting set in a simple and systematic way. The output is a polygon mesh that represents the surface of the kernel. Extensive comparisons with the existing methods, CGAL and Polyhedron Kernel, demonstrate the remarkable timing performance of our novel additive kernel computation method. Yet another advantage of our additive process is the availability of the partial kernel at any stage, making it useful for specific geometry processing applications such as star decomposition and castable shape reconstruction.

The majority of the simplification methods focus on preserving the appearance of the mesh, ignoring the spectral properties of the differential operators derived from the mesh. The spectrum of the Laplace-Beltrami operator is essential for a large subset of applications in geometry processing. Coarsening a mesh without considering its spectral properties might result in incorrect calculations on the simplified mesh. Given a 3D triangular mesh, this article aims to simplify the mesh using edge collapses, while focusing on preserving the spectral properties of the associated cotangent Laplace-Beltrami operator. Unlike the existing spectrum-preserving coarsening methods, we consider solely the eigenvalues of the operator in order to preserve the spectrum. The presented method is partition based, that is the input mesh is divided into smaller patches which are simplified individually. We evaluate our method on a variety of meshes, by using functional maps and quantitative norms, to measure how well the eigenvalues and eigenvectors of the Laplace-Beltrami operator computed on the input mesh are maintained by the output mesh. We demonstrate that the achieved spectrum preservation is at least as effective as the existing spectral coarsening methods.

We propose a fully-automatic method that computes from scratch point-to-point dense correspondences between isometric shapes under topological noise. While relying on pairwise distance preservation constraints is common and generally sufficient to handle isometric deformations, presence of topological noise needs further actions that we present as our main contributions. First, instead of comparing distances over two paths on two input surfaces, we cast fuzzy votes at the path endpoints based on topologically-robust heat diffusion from path vertices. Second, we make the matching even more stable to topological noise by introducing the so-called reodesics, which are locally shortest geodesics that go through robust matches. In addition to the five standard datasets for isometric shape correspondence with and without topological noise, we employ and release a sixth one geared specifically towards topological noise evaluation with ground-truth information. We demonstrate our qualitative and quantitative advantages over seven recent state-of-the-art methods on these six datasets.

Generating 3D models from 2D images or sketches is a widely studied important problem in computer graphics. We describe the first method to generate a 3D human model from a single sketched stick figure. In contrast to the existing human modeling techniques, our method does not require a statistical body shape model. We exploit Variational Autoencoders to develop a novel framework capable of transitioning from a simple 2D stick figure sketch, to a corresponding 3D human model. Our network learns the mapping between the input sketch and the output 3D model. Furthermore, our model learns the embedding space around these models. We demonstrate that our network can generate not only 3D models, but also 3D animations through interpolation and extrapolation in the learned embedding space. In addition to 3D human models, we produce 3D horse models in order to show the generalization ability of our framework. Extensive experiments show that our model learns to generate compatible 3D models and animations with 2D sketches.

Abstract We propose a novel approach for reconstructing plausible three‐dimensional (3D) human body models from small number of 3D points which represent body parts. We leverage a database of 3D models of humans varying from each other by physical attributes such as age, gender, weight, and height. First we divide the bodies in database into seven semantic regions. Then, for each input region consisting of maximum 40 points, we search the database for the best matching body part. For the matching criterion, we use the distance between novel point‐based features of input points and body parts in the database. We then combine the matched parts from different bodies into one body, with the help of Laplacian deformation, which results in a plausible human body. To evaluate our results objectively, we pick points from each part of the ground‐truth human body models, then reconstruct them using our method and compare the resulting bodies with the corresponding ground‐truths. Also, our results are compared with registration‐based results. In addition, we run our algorithm with noisy data to test the robustness of our method and run it with input points whose body parts are manually edited, which produces plausible human bodies that do not even exist in our database. Our experiments verify qualitatively and quantitatively that the proposed approach reconstructs human bodies with different physical attributes from a small number of points using a small database.

Generating 3D models from 2D images or sketches is a widely studied important problem in computer graphics. We describe the first method to generate a 3D human model from a single sketched stick figure. In contrast to the existing human modeling techniques, our method requires neither a statistical body shape model nor a rigged 3D character model. We exploit Variational Autoencoders to develop a novel framework capable of transitioning from a simple 2D stick figure sketch, to a corresponding 3D human model. Our network learns the mapping between the input sketch and the output 3D model. Furthermore, our model learns the embedding space around these models. We demonstrate that our network can generate not only 3D models, but also 3D animations through interpolation and extrapolation in the learned embedding space. Extensive experiments show that our model learns to generate reasonable 3D models and animations.

Canonical forms attempt to factor out a non-rigid shape’s pose, giving a pose-neutral shape. This opens up the possibility of using methods originally designed for rigid shape retrieval for the task of non-rigid shape retrieval. We extend our recent benchmark for testing canonical form algorithms. Our new benchmark is used to evaluate a greater number of state-of-the-art canonical forms, on five recent non-rigid retrieval datasets, within two different retrieval frameworks. A total of fifteen different canonical form methods are compared. We find that the difference in retrieval accuracy between different canonical form methods is small, but varies significantly across different datasets. We also find that efficiency is the main difference between the methods.

We exploit the permutation creation ability of genetic optimization to find the permutation of one point set that puts it into correspondence with another one. To this end, we provide a genetic algorithm for the 3D shape correspondence problem, which is the main contribution of this article. As another significant contribution, we present an adaptive sampling approach that relocates the matched points based on the currently available correspondence via an alternating optimization. The point sets to be matched are sampled from two isometric (or nearly isometric) shapes. The sparse one-to-one correspondence, i.e., bijection, that we produce is validated both in terms of running time and accuracy in a comprehensive test suite that includes four standard shape benchmarks and state-of-the-art techniques.

We present a shape deformation algorithm that unfolds any given 3D shape into a canonical pose that is invariant to nonrigid transformations. Unlike classical approaches, such as least-squares multidimensional scaling, we preserve the geometric details of the input shape in the resulting shape, which in turn leads to a content-based nonrigid shape retrieval application with higher accuracy. Our optimization framework, fed with a triangular or a tetrahedral mesh in 3D, tries to move each vertex as far away from each other as possible subject to finite element regularization constraints. Intuitively this effort minimizes the bending over the shape while preserving the details. Avoiding geodesic distances in our computation renders the method robust to topological noise. Compared to state-of-the-art approaches, our method is simpler to implement, faster, more accurate in shape retrieval, and less sensitive to topological errors.

Abstract 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.

Abstract 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.

Abstract 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.

Abstract 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.

We present a purely isometric method that establishes 3D correspondence between two (nearly) isometric shapes. Our method evenly samples high-curvature vertices from the given mesh representations, and then seeks an injective mapping from one vertex set to the other that minimizes the isometric distortion. We formulate the problem of shape correspondence as combinatorial optimization over the domain of all possible mappings, which then reduces in a probabilistic setting to a log-likelihood maximization problem that we solve via the Expectation-Maximization (EM) algorithm. The EM algorithm is initialized in the spectral domain by transforming the sampled vertices via classical Multidimensional Scaling (MDS). Minimization of the isometric distortion, and hence maximization of the log-likelihood function, is then achieved in the original 3D euclidean space, for each iteration of the EM algorithm, in two steps: by first using bipartite perfect matching, and then a greedy optimization algorithm. The optimal mapping obtained at convergence can be one-to-one or many-to-one upon choice. We demonstrate the performance of our method on various isometric (or nearly isometric) pairs of shapes for some of which the ground-truth correspondence is available.

Abstract We present a dense correspondence method for isometric shapes, which is accurate yet computationally efficient. We minimize the isometric distortion directly in the 3D Euclidean space, i.e., in the domain where isometry is originally defined, by using a coarse‐to‐fine sampling and combinatorial matching algorithm. Our method does not require any initialization and aims to find an accurate solution in the minimum‐distortion sense for perfectly isometric shapes. We demonstrate the performance of our method on various isometric (or nearly isometric) pairs of shapes.

We present an automatic method that establishes 3D correspondence between isometric shapes. Our goal is to find an optimal correspondence between two given (nearly) isometric shapes, that minimizes the amount of deviation from isometry. We cast the problem as a complete surface correspondence problem. Our method first divides the given shapes to be matched into surface patches of equal area and then seeks for a mapping between the patch centers which we refer to as base vertices. Hence the correspondence is established in a fast and robust manner at a relatively coarse level as imposed by the patch radius. We optimize the isometry cost in two steps. In the first step, the base vertices are transformed into spectral domain based on geodesic affinity, where the isometry errors are minimized in polynomial time by complete bipartite graph matching. The resulting correspondence serves as a good initialization for the second step of optimization in which we explicitly minimize the isometry cost via an iterative greedy algorithm in the original 3D Euclidean space. We demonstrate the performance of our method on various isometric (or nearly isometric) pairs of shapes for some of which the ground-truth correspondence is available.