Browsing by Author "Solomon, Justin"

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    Frame Field Operators
    (The Eurographics Association and John Wiley & Sons Ltd., 2021) Palmer, David; Stein, Oded; Solomon, Justin; Digne, Julie and Crane, Keenan
    Differential operators are widely used in geometry processing for problem domains like spectral shape analysis, data interpolation, parametrization and mapping, and meshing. In addition to the ubiquitous cotangent Laplacian, anisotropic second-order operators, as well as higher-order operators such as the Bilaplacian, have been discretized for specialized applications. In this paper, we study a class of operators that generalizes the fourth-order Bilaplacian to support anisotropic behavior. The anisotropy is parametrized by a symmetric frame field, first studied in connection with quadrilateral and hexahedral meshing, which allows for fine-grained control of local directions of variation. We discretize these operators using a mixed finite element scheme, verify convergence of the discretization, study the behavior of the operator under pullback, and present potential applications.
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    Hexahedral Mesh Repair via Sum-of-Squares Relaxation
    (The Eurographics Association and John Wiley & Sons Ltd., 2020) Marschner, Zoë; Palmer, David; Zhang, Paul; Solomon, Justin; Jacobson, Alec and Huang, Qixing
    The validity of trilinear hexahedral (hex) mesh elements is a prerequisite for many applications of hex meshes, such as finite element analysis. A commonly used check for hex mesh validity evaluates mesh quality on the corners of the parameter domain of each hex, an insufficient condition that neglects invalidity elsewhere in the element, but is straightforward to compute. Hex mesh quality optimizations using this validity criterion suffer by being unable to detect invalidities in a hex mesh reliably, let alone fix them. We rectify these challenges by leveraging sum-of-squares relaxations to pinpoint invalidities in a hex mesh efficiently and robustly. Furthermore, we design a hex mesh repair algorithm that can certify validity of the entire hex mesh. We demonstrate our hex mesh repair algorithm on a dataset of meshes that include hexes with both corner and face-interior invalidities and demonstrate that where naïve algorithms would fail to even detect invalidities, we are able to repair them. Our novel methodology also introduces the general machinery of sum-of-squares relaxation to geometry processing, where it has the potential to solve related problems.
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    Medial Axis Isoperimetric Profiles
    (The Eurographics Association and John Wiley & Sons Ltd., 2020) Zhang, Paul; DeFord, Daryl; Solomon, Justin; Jacobson, Alec and Huang, Qixing
    Recently proposed as a stable means of evaluating geometric compactness, the isoperimetric profile of a planar domain measures the minimum perimeter needed to inscribe a shape with prescribed area varying from 0 to the area of the domain. While this profile has proven valuable for evaluating properties of geographic partitions, existing algorithms for its computation rely on aggressive approximations and are still computationally expensive. In this paper, we propose a practical means of approximating the isoperimetric profile and show that for domains satisfying a ''thick neck'' condition, our approximation is exact. For more general domains, we show that our bound is still exact within a conservative regime and is otherwise an upper bound. Our method is based on a traversal of the medial axis which produces efficient and robust results. We compare our technique with the state-of-the-art approximation to the isoperimetric profile on a variety of domains and show significantly tighter bounds than were previously achievable.
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    SGP 2023 CGF 42-5: Frontmatter
    (The Eurographics Association and John Wiley & Sons Ltd., 2023) Memari, Pooran; Solomon, Justin; Memari, Pooran; Solomon, Justin
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    Wassersplines for Neural Vector Field-Controlled Animation
    (The Eurographics Association and John Wiley & Sons Ltd., 2022) Zhang, Paul; Smirnov, Dmitriy; Solomon, Justin; Dominik L. Michels; Soeren Pirk
    Much of computer-generated animation is created by manipulating meshes with rigs. While this approach works well for animating articulated objects like animals, it has limited flexibility for animating less structured free-form objects. We introduce Wassersplines, a novel trajectory inference method for animating unstructured densities based on recent advances in continuous normalizing flows and optimal transport. The key idea is to train a neurally-parameterized velocity field that represents the motion between keyframes. Trajectories are then computed by advecting keyframes through the velocity field. We solve an additional Wasserstein barycenter interpolation problem to guarantee strict adherence to keyframes. Our tool can stylize trajectories through a variety of PDE-based regularizers to create different visual effects. We demonstrate our tool on various keyframe interpolation problems to produce temporally-coherent animations without meshing or rigging.