Browsing by Author "Born, Janis"

Now showing 1 - 3 of 3
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    Layout Embedding via Combinatorial Optimization
    (The Eurographics Association and John Wiley & Sons Ltd., 2021) Born, Janis; Schmidt, Patrick; Kobbelt, Leif; Mitra, Niloy and Viola, Ivan
    We consider the problem of injectively embedding a given graph connectivity (a layout) into a target surface. Starting from prescribed positions of layout vertices, the task is to embed all layout edges as intersection-free paths on the surface. Besides merely geometric choices (the shape of paths) this problem is especially challenging due to its topological degrees of freedom (how to route paths around layout vertices). The problem is typically addressed through a sequence of shortest path insertions, ordered by a greedy heuristic. Such insertion sequences are not guaranteed to be optimal: Early path insertions can potentially force later paths into unexpected homotopy classes. We show how common greedy methods can easily produce embeddings of dramatically bad quality, rendering such methods unsuitable for automatic processing pipelines. Instead, we strive to find the optimal order of insertions, i.e. the one that minimizes the total path length of the embedding. We demonstrate that, despite the vast combinatorial solution space, this problem can be effectively solved on simply-connected domains via a custom-tailored branch-and-bound strategy. This enables directly using the resulting embeddings in downstream applications which cannot recover from initializations in a wrong homotopy class. We demonstrate the robustness of our method on a shape dataset by embedding a common template layout per category, and show applications in quad meshing and inter-surface mapping.
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    Surface Map Homology Inference
    (The Eurographics Association and John Wiley & Sons Ltd., 2021) Born, Janis; Schmidt, Patrick; Campen, Marcel; Kobbelt, Leif; Digne, Julie and Crane, Keenan
    A homeomorphism between two surfaces not only defines a (continuous and bijective) geometric correspondence of points but also (by implication) an identification of topological features, i.e. handles and tunnels, and how the map twists around them. However, in practice, surface maps are often encoded via sparse correspondences or fuzzy representations that merely approximate a homeomorphism and are therefore inherently ambiguous about map topology. In this work, we show a way to infer topological information from an imperfect input map between two shapes. In particular, we compute a homology map, a linear map that transports homology classes of cycles from one surface to the other, subject to a global consistency constraint. Our inference robustly handles imperfect (e.g., partial, sparse, fuzzy, noisy, outlier-ridden, non-injective) input maps and is guaranteed to produce homology maps that are compatible with true homeomorphisms between the input shapes. Homology maps inferred by our method can be directly used to transfer homological information between shapes, or serve as foundation for the construction of a proper homeomorphism guided by the input map, e.g., via compatible surface decomposition.
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    TinyAD: Automatic Differentiation in Geometry Processing Made Simple
    (The Eurographics Association and John Wiley & Sons Ltd., 2022) Schmidt, Patrick; Born, Janis; Bommes, David; Campen, Marcel; Kobbelt, Leif; Campen, Marcel; Spagnuolo, Michela
    Non-linear optimization is essential to many areas of geometry processing research. However, when experimenting with different problem formulations or when prototyping new algorithms, a major practical obstacle is the need to figure out derivatives of objective functions, especially when second-order derivatives are required. Deriving and manually implementing gradients and Hessians is both time-consuming and error-prone. Automatic differentiation techniques address this problem, but can introduce a diverse set of obstacles themselves, e.g. limiting the set of supported language features, imposing restrictions on a program's control flow, incurring a significant run time overhead, or making it hard to exploit sparsity patterns common in geometry processing. We show that for many geometric problems, in particular on meshes, the simplest form of forward-mode automatic differentiation is not only the most flexible, but also actually the most efficient choice. We introduce TinyAD: a lightweight C++ library that automatically computes gradients and Hessians, in particular of sparse problems, by differentiating small (tiny) sub-problems. Its simplicity enables easy integration; no restrictions on, e.g., looping and branching are imposed. TinyAD provides the basic ingredients to quickly implement first and second order Newton-style solvers, allowing for flexible adjustment of both problem formulations and solver details. By showcasing compact implementations of methods from parametrization, deformation, and direction field design, we demonstrate how TinyAD lowers the barrier to exploring non-linear optimization techniques. This enables not only fast prototyping of new research ideas, but also improves replicability of existing algorithms in geometry processing. TinyAD is available to the community as an open source library.