Inter-Surface Maps via Constant-Curvature Metrics

Patrick Schmidt, Marcel Campen, Janis Born, Leif Kobbelt

We propose a novel approach to represent maps between two discrete surfaces of the same genus and to minimize intrinsic mapping distortion. Our maps are well-defined at every surface point and are guaranteed to be continuous bijections (surface homeomorphisms). As a key feature of our approach, only the images of vertices need to be represented explicitly, since the images of all other points (on edges or in faces) are properly defined implicitly. This definition is via unique geodesics in metrics of constant Gaussian curvature. Our method is built upon the fact that such metrics exist on surfaces of arbitrary topology, without the need for any cuts or cones (as asserted by the uniformization theorem). Depending on the surfaces' genus, these metrics exhibit one of the three classical geometries: Euclidean, spherical or hyperbolic. Our formulation handles constructions in all three geometries in a unified way. In addition, by considering not only the vertex images but also the discrete metric as degrees of freedom, our formulation enables us to simultaneously optimize the images of these vertices and images of all other points.

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author = {Schmidt, Patrick and Campen, Marcel and Born, Janis and Kobbelt, Leif},
title = {Inter-Surface Maps via Constant-Curvature Metrics},
journal = {ACM Transactions on Graphics},
issue_date = {July 2020},
volume = {39},
number = {4},
month = jul,
year = {2020},
articleno = {119},
url = {https://doi.org/10.1145/3386569.3392399},
doi = {10.1145/3386569.3392399},
publisher = {ACM},
address = {New York, NY, USA},

Fast and Robust QEF Minimization using Probabilistic Quadrics

Philip Trettner, Leif Kobbelt
Computer Graphics Forum (Proc. EUROGRAPHICS 2020)

Error quadrics are a fundamental and powerful building block in many geometry processing algorithms. However, finding the minimizer of a given quadric is in many cases not robust and requires a singular value decomposition or some ad-hoc regularization. While classical error quadrics measure the squared deviation from a set of ground truth planes or polygons, we treat the input data as genuinely uncertain information and embed error quadrics in a probabilistic setting ("probabilistic quadrics") where the optimal point minimizes the expected squared error. We derive closed form solutions for the popular plane and triangle quadrics subject to (spatially varying, anisotropic) Gaussian noise. Probabilistic quadrics can be minimized robustly by solving a simple linear system - 50x faster than SVD. We show that probabilistic quadrics have superior properties in tasks like decimation and isosurface extraction since they favor more uniform triangulations and are more tolerant to noise while still maintaining feature sensitivity. A broad spectrum of applications can directly benefit from our new quadrics as a drop-in replacement which we demonstrate with mesh smoothing via filtered quadrics and non-linear subdivision surfaces.

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@article {10.1111:cgf.13933,
journal = {Computer Graphics Forum},
title = {{Fast and Robust QEF Minimization using Probabilistic Quadrics}},
author = {Trettner, Philip and Kobbelt, Leif},
year = {2020},
publisher = {The Eurographics Association and John Wiley & Sons Ltd.},
ISSN = {1467-8659},
DOI = {10.1111/cgf.13933}

High-Fidelity Point-Based Rendering of Large-Scale 3D Scan Datasets

Patric Schmitz, Timothy Blut, Christian Mattes, Leif Kobbelt
IEEE Computer Graphics and Applications

Digitalization of 3D objects and scenes using modern depth sensors and high-resolution RGB cameras enables the preservation of human cultural artifacts at an unprecedented level of detail. Interactive visualization of these large datasets, however, is challenging without degradation in visual fidelity. A common solution is to fit the dataset into available video memory by downsampling and compression. The achievable reproduction accuracy is thereby limited for interactive scenarios, such as immersive exploration in Virtual Reality (VR). This degradation in visual realism ultimately hinders the effective communication of human cultural knowledge. This article presents a method to render 3D scan datasets with minimal loss of visual fidelity. A point-based rendering approach visualizes scan data as a dense splat cloud. For improved surface approximation of thin and sparsely sampled objects, we propose oriented 3D ellipsoids as rendering primitives. To render massive texture datasets, we present a virtual texturing system that dynamically loads required image data. It is paired with a single-pass page prediction method that minimizes visible texturing artifacts. Our system renders a challenging dataset in the order of 70 million points and a texture size of 1.2 terabytes consistently at 90 frames per second in stereoscopic VR.

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High-Performance Image Filters via Sparse Approximations

Kersten Schuster, Philip Trettner, Leif Kobbelt
Proceedings of the ACM on Computer Graphics and Interactive Techniques, Vol. 3, No. 2, 2020

We present a numerical optimization method to find highly efficient (sparse) approximations for convolutional image filters. Using a modified parallel tempering approach, we solve a constrained optimization that maximizes approximation quality while strictly staying within a user-prescribed performance budget. The results are multi-pass filters where each pass computes a weighted sum of bilinearly interpolated sparse image samples, exploiting hardware acceleration on the GPU. We systematically decompose the target filter into a series of sparse convolutions, trying to find good trade-offs between approximation quality and performance. Since our sparse filters are linear and translation-invariant, they do not exhibit the aliasing and temporal coherence issues that often appear in filters working on image pyramids. We show several applications, ranging from simple Gaussian or box blurs to the emulation of sophisticated Bokeh effects with user-provided masks. Our filters achieve high performance as well as high quality, often providing significant speed-up at acceptable quality even for separable filters. The optimized filters can be baked into shaders and used as a drop-in replacement for filtering tasks in image processing or rendering pipelines.

Cost Minimizing Local Anisotropic Quad Mesh Refinement

Max Lyon, David Bommes, Leif Kobbelt
Eurographics Symposium on Geometry Processing 2020

Quad meshes as a surface representation have many conceptual advantages over triangle meshes. Their edges can naturally be aligned to principal curvatures of the underlying surface and they have the flexibility to create strongly anisotropic cells without causing excessively small inner angles. While in recent years a lot of progress has been made towards generating high quality uniform quad meshes for arbitrary shapes, their adaptive and anisotropic refinement remains difficult since a single edge split might propagate across the entire surface in order to maintain consistency. In this paper we present a novel refinement technique which finds the optimal trade-off between number of resulting elements and inserted singularities according to a user prescribed weighting. Our algorithm takes as input a quad mesh with those edges tagged that are prescribed to be refined. It then formulates a binary optimization problem that minimizes the number of additional edges which need to be split in order to maintain consistency. Valence 3 and 5 singularities have to be introduced in the transition region between refined and unrefined regions of the mesh. The optimization hence computes the optimal trade-off and places singularities strategically in order to minimize the number of consistency splits — or avoids singularities where this causes only a small number of additional splits. When applying the refinement scheme iteratively, we extend our binary optimization formulation such that previous splits can be undone if this prevents degenerate cells with small inner angles that otherwise might occur in anisotropic regions or in the vicinity of singularities. We demonstrate on a number of challenging examples that the algorithm performs well in practice.

A Three-Level Approach to Texture Mapping and Synthesis on 3D Surfaces

Kersten Schuster, Philip Trettner, Patric Schmitz, Leif Kobbelt
Proceedings of the ACM on Computer Graphics and Interactive Techniques, Vol. 3, No. 1, 2020

We present a method for example-based texturing of triangular 3D meshes. Our algorithm maps a small 2D texture sample onto objects of arbitrary size in a seamless fashion, with no visible repetitions and low overall distortion. It requires minimal user interaction and can be applied to complex, multi-layered input materials that are not required to be tileable. Our framework integrates a patch-based approach with per-pixel compositing. To minimize visual artifacts, we run a three-level optimization that starts with a rigid alignment of texture patches (macro scale), then continues with non-rigid adjustments (meso scale) and finally performs pixel-level texture blending (micro scale). We demonstrate that the relevance of the three levels depends on the texture content and type (stochastic, structured, or anisotropic textures).

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author = {Schuster, Kersten and Trettner, Philip and Schmitz, Patric and Kobbelt, Leif},
title = {A Three-Level Approach to Texture Mapping and Synthesis on 3D Surfaces},
year = {2020},
issue_date = {Apr 2020},
publisher = {The Association for Computers in Mathematics and Science Teaching},
address = {USA},
volume = {3},
number = {1},
url = {https://doi.org/10.1145/3384542},
doi = {10.1145/3384542},
journal = {Proc. ACM Comput. Graph. Interact. Tech.},
month = apr,
articleno = {1},
numpages = {19},
keywords = {material blending, surface texture synthesis, texture mapping}

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