Browsing by Author "Azencot, Omri"

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    Consistent Shape Matching via Coupled Optimization
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Azencot, Omri; Dubrovina, Anastasia; Guibas, Leonidas; Bommes, David and Huang, Hui
    We propose a new method for computing accurate point-to-point mappings between a pair of triangle meshes given imperfect initial correspondences. Unlike the majority of existing techniques, we optimize for a map while leveraging information from the inverse map, yielding results which are highly consistent with respect to composition of mappings. Remarkably, our method considers only a linear number of candidate points on the target shape, allowing us to work directly with high resolution meshes, and to avoid a delicate and possibly error-prone up-sampling procedure. Key to this dimensionality reduction is a novel candidate selection process, where the mapped points drift over the target shape, finalizing their location based on intrinsic distortion measures. Overall, we arrive at an iterative scheme where at each step we optimize for the map and its inverse by solving two relaxed Quadratic Assignment Problems using off-the-shelf optimization tools. We provide quantitative and qualitative comparison of our method with several existing techniques, and show that it provides a powerful matching tool when accurate and consistent correspondences are required.
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    A Data-Driven Approach to Functional Map Construction and Bases Pursuit
    (The Eurographics Association and John Wiley & Sons Ltd., 2021) Azencot, Omri; Lai, Rongjie; Digne, Julie and Crane, Keenan
    We propose a method to simultaneously compute scalar basis functions with an associated functional map for a given pair of triangle meshes. Unlike previous techniques that put emphasis on smoothness with respect to the Laplace-Beltrami operator and thus favor low-frequency eigenfunctions, we aim for a basis that allows for better feature matching. This change of perspective introduces many degrees of freedom into the problem allowing to better exploit non-smooth descriptors. To effectively search in this high-dimensional space of solutions, we incorporate into our minimization state-of-the-art regularizers. We solve the resulting highly non-linear and non-convex problem using an iterative scheme via the Alternating Direction Method of Multipliers. At each step, our optimization involves simple to solve linear or Sylvester-type equations. In practice, our method performs well in terms of convergence, and we additionally show that it is similar to a provably convergent problem. We show the advantages of our approach by extensively testing it on multiple datasets in a few applications including shape matching, consistent quadrangulation and scalar function transfer.
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    Elastic Correspondence between Triangle Meshes
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Ezuz, Danielle; Heeren, Behrend; Azencot, Omri; Rumpf, Martin; Ben-Chen, Mirela; Alliez, Pierre and Pellacini, Fabio
    We propose a novel approach for shape matching between triangular meshes that, in contrast to existing methods, can match crease features. Our approach is based on a hybrid optimization scheme, that solves simultaneously for an elastic deformation of the source and its projection on the target. The elastic energy we minimize is invariant to rigid body motions, and its non-linear membrane energy component favors locally injective maps. Symmetrizing this model enables feature aligned correspondences even for non-isometric meshes. We demonstrate the advantage of our approach over state of the art methods on isometric and non-isometric datasets, where we improve the geodesic distance from the ground truth, the conformal and area distortions, and the mismatch of the mean curvature functions. Finally, we show that our computed maps are applicable for surface interpolation, consistent cross-field computation, and consistent quadrangular remeshing of a set of shapes.
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    Operator Representations in Geometry Processing
    (2017) Azencot, Omri
    This thesis introduces fundamental equations as well as discrete tools and numerical methods for carrying out various geometrical tasks on three-dimensional surfaces via operators. An example for an operator is the Laplacian which maps real-valued functions to their sum of second derivatives. More generally, many mathematical objects feature an operator interpretation, and in this work, we consider a few of them in the context of geometry processing and numerical simulation problems. The operator point of view is useful in applications since high-level algorithms can be devised for the problems at hand with operators serving as the main building blocks. While this approach has received some attention in the past, it has not reached its full potential, as the following thesis tries to hint. The contribution of this document is twofold. First, it describes the analysis and discretization of derivations and related operators such as covariant derivative, Lie bracket, pushforward and flow on triangulated surfaces. These operators play a fundamental role in numerous computational science and engineering problems, and thus enriching the readily available differential tools with these novel components offers multiple new avenues to explore. Second, these objects are then used to solve certain differential equations on curved domains such as the advection equation, the Navier– Stokes equations and the thin films equations. Unlike previous work, our numerical methods are intrinsic to the surface—that is, independent of a particular geometry flattening. In addition, the suggested machinery preserves structure—namely, a central quantity to the problem, as the total mass, is exactly preserved. These two properties typically provide a good balance between computation times and quality of results. From a broader standpoint, recent years have brought an expected increase in computation power along with extraordinary advances in the theory and methodology of geometry acquisition and processing. Consequently, many approaches which were infeasible before, became viable nowadays. In this view, the operator perspective and its application to differential equations, as depicted in this work, provides an interesting alternative, among the other approaches, for working with complex problems on non-flat geometries. In the following chapters, we study in which cases operators are applicable, while providing a fair comparison to state-of-the-art methods.