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Register a new userFrame Field Operators
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Added by
David Palmer
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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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Field-guided parametrization methods have proven effective for quad meshing of surfaces; these methods compute smooth cross fields to guide the meshing process and then integrate the fields to construct a discrete mesh. A key challenge in extending these methods to three dimensions, however, is representation of field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog---an octahedral frame field---takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically-sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These new odeco frames, so-called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program--based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness.
We introduce a novel approach to measure the behavior of a geometric operator before and after coarsening. By comparing eigenvectors of the input operator and its coarsened counterpart, we can quantitatively and visually analyze how well the spectral properties of the operator are maintained. Using this measure, we show that standard mesh simplification and algebraic coarsening techniques fail to maintain spectral properties. In response, we introduce a novel approach for spectral coarsening. We show that it is possible to significantly reduce the sampling density of an operator derived from a 3D shape without affecting the low-frequency eigenvectors. By marrying techniques developed within the algebraic multigrid and the functional maps literatures, we successfully coarsen a variety of isotropic and anisotropic operators while maintaining sparsity and positive semi-definiteness. We demonstrate the utility of this approach for applications including operator-sensitive sampling, shape matching, and graph pooling for convolutional neural networks.
We have recently developed an algorithm for vector field visualization with oriented streamlines, able to depict the flow directions everywhere in a dense vector field and the sense of the local orientations. The algorithm has useful applications in the visualization of the director field in nematic liquid crystals. Here we propose an improvement of the algorithm able to enhance the visualization of the local magnitude of the field. This new approach of the algorithm is compared with the same procedure applied to the Line Integral Convolution (LIC) visualization.
In this note we investigate the operators associated with frame sequences in a Hilbert space $H$, i.e., the synthesis operator $T:ell ^{2}(mathbb{N}) to H$, the analysis operator $T^{ast}:Hto $ $% ell ^{2}(mathbb{N}) $ and the associated frame operator $S=TT^{ast}$ as operators defined on (or to) the whole space rather than on subspaces. Furthermore, the projection $P$ onto the range of $T$, the projection $Q$ onto the range of $T^{ast}$ and the Gram matrix $G=T^{ast}T$ are investigated. For all these operators, we investigate their pseudoinverses, how they interact with each other, as well as possible classification of frame sequences with them. For a tight frame sequence, we show that some of these operators are connected in a simple way.
This paper introduces progressive algorithms for the topological analysis of scalar data. Our approach is based on a hierarchical representation of the input data and the fast identification of topologically invariant vertices, which are vertices that have no impact on the topological description of the data and for which we show that no computation is required as they are introduced in the hierarchy. This enables the definition of efficient coarse-to-fine topological algorithms, which leverage fast update mechanisms for ordinary vertices and avoid computation for the topologically invariant ones. We demonstrate our approach with two examples of topological algorithms (critical point extraction and persistence diagram computation), which generate interpretable outputs upon interruption requests and which progressively refine them otherwise. Experiments on real-life datasets illustrate that our progressive strategy, in addition to the continuous visual feedback it provides, even improves run time performance with regard to non-progressive algorithms and we describe further accelerations with shared-memory parallelism. We illustrate the utility of our approach in batch-mode and interactive setups, where it respectively enables the control of the execution time of complete topological pipelines as well as previews of the topological features found in a dataset, with progressive updates delivered within interactive times.
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