As-Killing-As-Possible Vector Fields for Planar Deformation
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Date
2011
Journal Title
Journal ISSN
Volume Title
Publisher
The Eurographics Association and Blackwell Publishing Ltd.
Abstract
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.
Description
@article{:10.1111/j.1467-8659.2011.02028.x,
journal = {Computer Graphics Forum},
title = {{As-Killing-As-Possible Vector Fields for Planar Deformation}},
author = {Solomon, Justin and Ben-Chen, Mirela and Butscher, Adrian and Guibas, Leonidas},
year = {2011},
publisher = {The Eurographics Association and Blackwell Publishing Ltd.},
ISSN = {1467-8659},
DOI = {/10.1111/j.1467-8659.2011.02028.x}
}