SGP: Eurographics Symposium on Geometry Processing

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As-Killing-As-Possible Vector Fields for Planar Deformation
(The Eurographics Association and Blackwell Publishing Ltd., 2011) Solomon, Justin; Ben-Chen, Mirela; Butscher, Adrian; Guibas, Leonidas; Mario Botsch and Scott Schaefer
Cartoon animation, image warping, and several other tasks in two-dimensional computer graphics reduce to the formulation of a reasonable model for planar deformation. A deformation is a map from a given shape to a new one, and its quality is determined by the type of distortion it introduces. In many applications, a desirable map is as isometric as possible. Finding such deformations, however, is a nonlinear problem, and most of the existing solutions approach it by minimizing a nonlinear energy. Such methods are not guaranteed to converge to a global optimum and often suffer from robustness issues. We propose a new approach based on approximate Killing vector fields (AKVFs), first introduced in shape processing. AKVFs generate near-isometric deformations, which can be motivated as direction fields minimizing an as-rigid-as-possible (ARAP) energy to first order. We first solve for an AKVF on the domain given user constraints via a linear optimization problem and then use this AKVF as the initial velocity field of the deformation. In this way, we transfer the inherent nonlinearity of the deformation problem to finding trajectories for each point of the domain having the given initial velocities. We show that a specific class of trajectories - the set of logarithmic spirals - is especially suited for this task both in practice and through its relationship to linear holomorphic vector fields. We demonstrate the effectiveness of our method for planar deformation by comparing it with existing state-of-the-art deformation methods.
Blending of Hyperbolic Closed Curves
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Ikemakhen, Aziz; Ahanchaou, Taoufik; Digne, Julie and Crane, Keenan
In recent years, game developers are interested in developing games in the hyperbolic space. Shape blending is one of the fundamental techniques to produce animation and videos games. This paper presents two algorithms for blending between two closed curves in the hyperbolic plane in a manner that guarantees that the intermediate curves are closed. We deal with hyperbolic discrete curves on Poincaré disc which is a famous model of the hyperbolic plane. We use the linear interpolation approach of the geometric invariants of hyperbolic polygons namely hyperbolic side lengths, exterior angles and geodesic discrete curvature. We formulate the closing condition of a hyperbolic polygon in terms of its geodesic side lengths and exterior angles. This is to be able to generate closed intermediate curves. Finally, some experimental results are given to illustrate that the proposed methods generate aesthetic blending of closed hyperbolic curves.
Error Propagation Control in Laplacian Mesh Compression
(The Eurographics Association and John Wiley & Sons Ltd., 2018) Vasa, Libor; Dvořák, Jan; Ju, Tao and Vaxman, Amir
Laplacian mesh compression, also known as high-pass mesh coding, is a popular technique for efficiently storing both static and dynamic triangle meshes that gained further recognition with the advent of perceptual mesh distortion evaluation metrics. Currently, the usual rule of thumb that drives the decision for a mesh compression algorithm is whether or not accuracy in absolute scale is required: Laplacian mesh encoding is chosen when perceptual quality is the main objective, while other techniques provide better results in terms of mechanistic error measures such as mean squared error. In this work, we present a modification of the Laplacian mesh encoding algorithm that preserves its benefits while it substantially reduces the resulting absolute error. Our approach is based on analyzing the reconstruction stage and modifying the quantization of differential coordinates, so that the decoded result stays close to the input even in areas that are distant from anchor points. In our approach, we avoid solving an overdetermined system of linear equations and thus reduce data redundancy, improve conditioning and achieve faster processing. Our approach can be directly applied to both static and dynamic mesh compression and we provide quantitative results comparing our approach with the state of the art methods.
An Explicit Structure-preserving Numerical Scheme for EPDiff
(The Eurographics Association and John Wiley & Sons Ltd., 2018) Azencot, Omri; Vantzos, Orestis; Ben-Chen, Mirela; Ju, Tao and Vaxman, Amir
We present a new structure-preserving numerical scheme for solving the Euler-Poincaré Differential (EPDiff) equation on arbitrary triangle meshes. Unlike existing techniques, our method solves the difficult non-linear EPDiff equation by constructing energy preserving, yet fully explicit, update rules. Our approach uses standard differential operators on triangle meshes, allowing for a simple and efficient implementation. Key to the structure-preserving features that our method exhibits is a novel numerical splitting scheme. Namely, we break the integration into three steps which rely on linear solves with a fixed sparse matrix that is independent of the simulation and thus can be pre-factored. We test our method in the context of simulating concentrated reconnecting wavefronts on flat and curved domains. In particular, EPDiff is known to generate geometrical fronts which exhibit wave-like behavior when they interact with each other. In addition, we also show that at a small additional cost, we can produce globally-supported periodic waves by using our simulated fronts with wavefronts tracking techniques. We provide quantitative graphs showing that our method exactly preserves the energy in practice. In addition, we demonstrate various interesting results including annihilation and recreation of a circular front, a wave splitting and merging when hitting an obstacle and two separate fronts propagating and bending due to the curvature of the domain.
Harmonic Shape Interpolation on Multiply-connected Planar Domains
(The Eurographics Association and John Wiley & Sons Ltd., 2022) Shi, Dongbo; Chen, Renjie; Campen, Marcel; Spagnuolo, Michela
Shape interpolation is a fundamental problem in computer graphics. Recently, there have been some interpolation methods developed which guarantee that the results are of bounded amount of geometric distortion, hence ensure high quality interpolation. However, none of these methods is applicable to shapes within the multiply-connected domains. In this work, we develop an interpolation scheme for harmonic mappings, that specifically addresses this limitation. We opt to interpolate the pullback metric of the input harmonic maps as proposed by Chen et al. [CWKBC13]. However, the interpolated metric does not correspond to any planar mapping, which is the main challenge in the interpolation problem for multiply-connected domains. We propose to solve this by projecting the interpolated metric into the planar harmonic mapping space. Specifically, we develop a Newton iteration to minimize the isometric distortion of the intermediate mapping, with respect to the interpolated metric. For more efficient Newton iteration, we further derived a simple analytic formula for the positive semidefinite (PSD) projection of the Hessian matrix of our distortion energy. Through extensive experiments and comparisons with the state-of-the-art, we demonstrate the efficacy and robustness of our method for various inputs.
Optimising Perceived Distortion in Lossy Encoding of Dynamic Meshes
(The Eurographics Association and Blackwell Publishing Ltd., 2011) Vá a, L.; Petrík, O.; Mario Botsch and Scott Schaefer
Development of geometry data compression techniques in the past years has been limited by the lack of a metric with proven correlation with human perception of mesh distortion. Many algorithms have been proposed, but usually the aim has been to minimise mean squared error, or some of its derivatives. In the field of dynamic mesh compression, the situation has changed with the recent proposal of the STED metric, which has been shown to capture the human perception of mesh distortion much better than previous metrics. In this paper we show how existing algorithms can be steered to provide optimal results with respect to this metric, and we propose a novel dynamic mesh compression algorithm, based on trajectory space PCA and Laplacian coordinates, specifically designed to minimise the newly proposed STED error. Our experiments show that using the proposed algorithm, we were able to reduce the required data rate by up to 50% while preserving the introduced STED error.

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Browsing SGP: Eurographics Symposium on Geometry Processing by Subject "Animation"

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