SGP16: Eurographics Symposium on Geometry Processing (CGF 35-5)

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https://diglib.eg.org/handle/10.2312/2631427

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Advection-Based Function Matching on Surfaces
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Azencot, Omri; Vantzos, Orestis; Ben-Chen, Mirela; Maks Ovsjanikov and Daniele Panozzo
A tangent vector field on a surface is the generator of a smooth family of maps from the surface to itself, known as the flow. Given a scalar function on the surface, it can be transported, or advected, by composing it with a vector field's flow. Such transport is exhibited by many physical phenomena, e.g., in fluid dynamics. In this paper, we are interested in the inverse problem: given source and target functions, compute a vector field whose flow advects the source to the target. We propose a method for addressing this problem, by minimizing an energy given by the advection constraint together with a regularizing term for the vector field. Our approach is inspired by a similar method in computational anatomy, known as LDDMM, yet leverages the recent framework of functional vector fields for discretizing the advection and the flow as operators on scalar functions. The latter allows us to efficiently generalize LDDMM to curved surfaces, without explicitly computing the flow lines of the vector field we are optimizing for. We show two approaches for the solution: using linear advection with multiple vector fields, and using non-linear advection with a single vector field. We additionally derive an approximated gradient of the corresponding energy, which is based on a novel vector field transport operator. Finally, we demonstrate applications of our machinery to intrinsic symmetry analysis, function interpolation and map improvement.
Complex Transfinite Barycentric Mappings with Similarity Kernels
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Chen, Renjie; Gotsman, Craig; Maks Ovsjanikov and Daniele Panozzo
Transfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define inter-polants of values given on a smooth planar contour. When the data is two-dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some of the basic complex-valued algebra needed to describe these planar maps, and use it to define similarity kernels, a natural alternative to the usual barycentric kernels. We develop the theory behind similarity kernels, explore their properties, and show that the transfinite versions of the popular three-point barycentric coordinates (Laplace, mean value and Wachspress) have surprisingly simple similarity kernels. We furthermore show how similarity kernels may be used to invert injective transfinite barycentric mappings using an iterative algorithm which converges quite rapidly. This is useful for rendering images deformed by planar barycentric mappings.
Construction of Topologically Correct and Manifold Isosurfaces
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Grosso, Roberto; Maks Ovsjanikov and Daniele Panozzo
We present a simple method to describe the geometry and topologically classify the intersection of level sets of trilinear interpolants with a reference unit cell. The solutions of three quadratic equations are used to correctly triangulate the level set within the cell satisfying the conditions imposed by the asymptotic decider. This way the triangulation of isosurfaces across cells borders is manifold and topologically correct. The algorithm presented is intuitive and easy to implement.
Data-Driven Bending Elasticity Design by Shell Thickness
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Zhang, Xiaoting; Le, Xinyi; Wu, Zihao; Whiting, Emily; Wang, Charlie C. L.; Maks Ovsjanikov and Daniele Panozzo
We present a method to design the deformation behavior of 3D printed models by an interactive tool, where the variation of bending elasticity at different regions of a model is realized by a change in shell thickness. Given a soft material to be used in 3D printing, we propose an experimental setup to acquire the bending behavior of this material on tubes with different diameters and thicknesses. The relationship between shell thickness and bending elasticity is stored in an echo state network using the acquired dataset. With the help of the network, an interactive design tool is developed to generate non-uniformly hollowed models to achieve desired bending behaviors. The effectiveness of this method is verified on models fabricated by different 3D printers by studying whether their physical deformation can match the designed target shape.
Exploration of Empty Space among Spherical Obstacles via Additively Weighted Voronoi Diagram
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Manak, Martin; Maks Ovsjanikov and Daniele Panozzo
Properties of granular materials or molecular structures are often studied on a simple geometric model - a set of 3D balls. If the balls simultaneously change in size by a constant speed, topological properties of the empty space outside all these balls may also change. Capturing the changes and their subsequent classification may reveal useful information about the model. This has already been solved for balls of the same size, but only an approximate solution has been reported for balls of different sizes. These solutions work on simplicial complexes derived from the dual structure of the ordinary Voronoi diagram of ball centers and use the mathematical concept of simplicial homology groups. If the balls have different radii, it is more appropriate to use the additively weighted Voronoi diagram (also known as the Apollonius diagram) instead of the ordinary diagram, but the dual structure is no longer a simplicial complex, so the previous approaches cannot be used directly. In this paper, a method is proposed to overcome this problem. The method works with Voronoi edges and vertices instead of the dual structure. Additional bridge edges are introduced to overcome disconnected cases. The output is a tree graph of events where cavities are created or merged during a simulated shrinking of the balls. This graph is then reorganized and filtered according to some criteria to get a more concise information about the development of the empty space in the model.
Identifying Style of 3D Shapes using Deep Metric Learning
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Lim, Isaak; Gehre, Anne; Kobbelt, Leif; Maks Ovsjanikov and Daniele Panozzo
We present a method that expands on previous work in learning human perceived style similarity across objects with different structures and functionalities. Unlike previous approaches that tackle this problem with the help of hand-crafted geometric descriptors, we make use of recent advances in metric learning with neural networks (deep metric learning). This allows us to train the similarity metric on a shape collection directly, since any low- or high-level features needed to discriminate between different styles are identified by the neural network automatically. Furthermore, we avoid the issue of finding and comparing sub-elements of the shapes. We represent the shapes as rendered images and show how image tuples can be selected, generated and used efficiently for deep metric learning. We also tackle the problem of training our neural networks on relatively small datasets and show that we achieve style classification accuracy competitive with the state of the art. Finally, to reduce annotation effort we propose a method to incorporate heterogeneous data sources by adding annotated photos found online in order to expand or supplant parts of our training data.
Incorporating Sharp Features in the General Solid Sweep Framework
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Adsul, Bharat; Machchhar, Jinesh; Sohoni, Milind; Maks Ovsjanikov and Daniele Panozzo
This paper extends a recently proposed robust computational framework for constructing the boundary representation (brep) of the volume swept by a given smooth solid moving along a one parameter family h of rigid motions. Our extension allows the input solid to have sharp features, and thus it is a significant and useful generalization of that work. This naturally requires a precise description of the geometry of the surface generated by the sweep of a sharp edge supported by two intersecting smooth faces. We uncover the geometry along with the related issues like parametrization and singularities via a novel mathematical analysis. Correct trimming of such a surface is achieved by an analysis of the interplay between the cone of normals at a sharp point and its trajectory under h. The overall topology is explained by a key lifting theorem which allows us to compute the adjacency relations amongst entities in the swept volume by relating them to corresponding adjacencies in the input solid. Moreover, global issues related to body-check such as orientation, singularities and self-intersections are efficiently resolved. Examples from a pilot implementation illustrate the efficiency and effectiveness of our framework.
Iterative Closest Conformal Maps between Planar Domains
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Segall, Aviv; Ben-Chen, Mirela; Maks Ovsjanikov and Daniele Panozzo
Conformal maps between planar domains are an important tool in geometry processing, used for shape deformation and image warping. The Riemann mapping theorem guarantees that there exists a conformal map between any two simply connected planar domains, yet computing this map efficiently remains challenging. In practice, one of the main algorithmic questions is the correspondence between the boundaries of the domains. On the one hand, there exist a number of conformal maps between any two domains, thus many potential boundary correspondences, yet on the other, given full boundary prescription a conformal map might not exist. Furthermore, an approximate boundary fitting can be enough for many applications. We therefore propose an alternating minimization algorithm for finding a boundary-approximating conformal map given only an initial global alignment of the two input domains.We utilize the Cauchy-Green complex barycentric coordinates to parameterize the space of conformal maps from the source domain, and thus compute a continuous map without requiring the discretization of the domain, and without mapping to intermediate domains. This yields a very efficient method which allows to interactively modify additional user-provided constraints, such as point-to-point and stroke-to-stroke correspondences. Furthermore, we show how to easily generalize this setup to quasi-conformal maps, thus enriching the space of mappings and reducing the area distortion. We compare our algorithm to state-of-the-art methods for mapping between planar domains, and demonstrate that we achieve less distorted maps on the same inputs. Finally, we show applications of our approach to stroke based deformation and constrained texture mapping.
Learning 3D Scene Synthesis from Annotated RGB-D Images
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Kermani, Zeinab Sadeghipour; Liao, Zicheng; Tan, Ping; Zhang, Hao (Richard); Maks Ovsjanikov and Daniele Panozzo
We present a data-driven method for synthesizing 3D indoor scenes by inserting objects progressively into an initial, possibly, empty scene. Instead of relying on few hundreds of hand-crafted 3D scenes, we take advantage of existing large-scale annotated RGB-D datasets, in particular, the SUN RGB-D database consisting of 10,000+ depth images of real scenes, to form the prior knowledge for our synthesis task. Our object insertion scheme follows a co-occurrence model and an arrangement model, both learned from the SUN dataset. The former elects a highly probable combination of object categories along with the number of instances per category while a plausible placement is defined by the latter model. Compared to previous works on probabilistic learning for object placement, we make two contributions. First, we learn various classes of higher-order objectobject relations including symmetry, distinct orientation, and proximity from the database. These relations effectively enable considering objects in semantically formed groups rather than by individuals. Second, while our algorithm inserts objects one at a time, it attains holistic plausibility of the whole current scene while offering controllability through progressive synthesis. We conducted several user studies to compare our scene synthesis performance to results obtained by manual synthesis, stateof- the-art object placement schemes, and variations of parameter settings for the arrangement model.
Mesh Statistics for Robust Curvature Estimation
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Váša, Libor; Vaněček, Petr; Prantl, Martin; Skorkovská, Věra; Martínek, Petr; Kolingerová, Ivana; Maks Ovsjanikov and Daniele Panozzo
While it is usually not difficult to compute principal curvatures of a smooth surface of sufficient differentiability, it is a rather difficult task when only a polygonal approximation of the surface is available, because of the inherent ambiguity of such representation. A number of different approaches has been proposed in the past that tackle this problem using various techniques. Most papers tend to focus on a particular method, while an comprehensive comparison of the different approaches is usually missing. We present results of a large experiment, involving both common and recently proposed curvature estimation techniques, applied to triangle meshes of varying properties. It turns out that none of the approaches provides reliable results under all circumstances. Motivated by this observation, we investigate mesh statistics, which can be computed from vertex positions and mesh connectivity information only, and which can help in deciding which estimator will work best for a particular case. Finally, we propose a meta-estimator, which makes a choice between existing algorithms based on the value of the mesh statistics, and we demonstrate that such meta-estimator, despite its simplicity, provides considerably more robust results than any existing approach.
Near-Isometric Level Set Tracking
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Tao, Michael; Solomon, Justin; Butscher, Adrian; Maks Ovsjanikov and Daniele Panozzo
Implicit representations of geometry have found applications in shape modeling, simulation, and other graphics pipelines. These representations, however, do not provide information about the paths of individual points as shapes move and undergo deformation. For this reason, we reconsider the problem of tracking points on level set surfaces, with the goal of designing an algorithm that - unlike previous work - can recover rotational motion and nearly isometric deformation. We track points on level sets of a time-varying function using approximate Killing vector fields (AKVFs), the velocity fields of near-isometric motions. To this end, we provide suitable theoretical and discrete constructions for computing AKVFs in a narrow band surrounding an animated level set surface. Furthermore, we propose time integrators well-suited to integrating AKVFs in time to track points. We demonstrate the theoretical and practical advantages of our proposed algorithms on synthetic and practical tasks.
Non-Rigid Puzzles
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Litany, Or; Rodolà, Emanuele; Bronstein, Alex M.; Bronstein, Michael M.; Cremers, Daniel; Maks Ovsjanikov and Daniele Panozzo
Shape correspondence is a fundamental problem in computer graphics and vision, with applications in various problems including animation, texture mapping, robotic vision, medical imaging, archaeology and many more. In settings where the shapes are allowed to undergo non-rigid deformations and only partial views are available, the problem becomes very challenging. To this end, we present a non-rigid multi-part shape matching algorithm. We assume to be given a reference shape and its multiple parts undergoing a non-rigid deformation. Each of these query parts can be additionally contaminated by clutter, may overlap with other parts, and there might be missing parts or redundant ones. Our method simultaneously solves for the segmentation of the reference model, and for a dense correspondence to (subsets of) the parts. Experimental results on synthetic as well as real scans demonstrate the effectiveness of our method in dealing with this challenging matching scenario.
Polycube Simplification for Coarse Layouts of Surfaces and Volumes
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Cherchi, Gianmarco; Livesu, Marco; Scateni, Riccardo; Maks Ovsjanikov and Daniele Panozzo
Representing digital objects with structured meshes that embed a coarse block decomposition is a relevant problem in applications like computer animation, physically-based simulation and Computer Aided Design (CAD). One of the key ingredients to produce coarse block structures is to achieve a good alignment between the mesh singularities (i.e., the corners of each block). In this paper we improve on the polycube-based meshing pipeline to produce both surface and volumetric coarse block-structured meshes of general shapes. To this aim we add a new step in the pipeline. Our goal is to optimize the positions of the polycube corners to produce as coarse as possible base complexes. We rely on re-mapping the positions of the corners on an integer grid and then using integer numerical programming to reach the optimal. To the best of our knowledge this is the first attempt to solve the singularity misalignment problem directly in polycube space. Previous methods for polycube generation did not specifically address this issue. Our corner optimization strategy is efficient and requires a negligible extra running time for the meshing pipeline. In the paper we show that our optimized polycubes produce coarser block structured surface and volumetric meshes if compared with previous approaches. They also induce higher quality hexahedral meshes and are better suited for spline fitting because they reduce the number of splines necessary to cover the domain, thus improving both the efficiency and the overall level of smoothness throughout the volume.
Scale-Invariant Directional Alignment of Surface Parametrizations
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Campen, Marcel; Ibing, Moritz; Ebke, Hans-Christian; Zorin, Denis; Kobbelt, Leif; Maks Ovsjanikov and Daniele Panozzo
Various applications of global surface parametrization benefit from the alignment of parametrization isolines with principal curvature directions. This is particularly true for recent parametrization-based meshing approaches, where this directly translates into a shape-aware edge flow, better approximation quality, and reduced meshing artifacts. Existing methods to influence a parametrization based on principal curvature directions suffer from scale-dependence, which implies the necessity of parameter variation, or try to capture complex directional shape features using simple 1D curves. Especially for non-sharp features, such as chamfers, fillets, blends, and even more for organic variants thereof, these abstractions can be unfit. We present a novel approach which respects and exploits the 2D nature of such directional feature regions, detects them based on coherence and homogeneity properties, and controls the parametrization process accordingly. This approach enables us to provide an intuitive, scale-invariant control parameter to the user. It also allows us to consider non-local aspects like the topology of a feature, enabling further improvements. We demonstrate that, compared to previous approaches, global parametrizations of higher quality can be generated without user intervention.
Stable Region Correspondences Between Non-Isometric Shapes
(The Eurographics Association and John Wiley & Sons Ltd., 2016) Ganapathi-Subramanian, Vignesh; Thibert, Boris; Ovsjanikov, Maks; Guibas, Leonidas; Maks Ovsjanikov and Daniele Panozzo
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.

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Browsing SGP16: Eurographics Symposium on Geometry Processing (CGF 35-5) by Subject "I.3.5 [Computer Graphics]"

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