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  1. Abstract

    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.

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  2. Abstract

    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 asymmetric 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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  4. Abstract

    Recently proposed as a stable means of evaluating geometric compactness, theisoperimetric profileof 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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  5. Abstract

    We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.

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  6. Abstract

    The California Current System is a productive eastern boundary region off the coasts of Washington, Oregon, and California. There is strong seasonality to the region, with high levels of rainfall and river input to the coastal ocean during the winter season, and coastal and Ekman upwelling during the spring and summer. Iron (Fe) input to the coastal ocean during the winter months can be stored in the continental shelf mud belts and then be delivered to the surface ocean by upwelling in the spring and summer. There have been a number of studies providing strong evidence of Fe‐limitation of diatom growth occurring in regions of the California Current System off of California, and the occurrence of Fe‐limitation has been linked with narrow continental shelf mud belt width and low river input. We provide evidence for potential Fe‐limitation of diatoms off the southern coast of Oregon in July 2014, just off the shelf break near Cape Blanco in a region with moderate shelf width and river input. Since eastern boundary regions account for a disproportionally large amount of global primary production, this observation of potential Fe‐limitation in an unexpected near‐shore region of the California Current System has implications for global models of primary productivity. In order to re‐evaluate the factors impacting Fe availability, we utilize satellite imagery to compare with historical datasets, and show that unexpected levels of Fe can often be explained by eddies, plumes of upwelled water moving offshore, or lack of recent upwelling.

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