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Dirichlet Energy for Analysis and Synthesis of Soft Maps
Author(s) -
Solomon Justin,
Guibas Leonidas,
Butscher Adrian
Publication year - 2013
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/cgf.12186
Subject(s) - dirichlet distribution , discretization , probabilistic logic , mathematics , computation , measure (data warehouse) , computer science , distortion (music) , algorithm , point (geometry) , mathematical optimization , mathematical analysis , geometry , boundary value problem , artificial intelligence , data mining , computer network , bandwidth (computing) , amplifier
Abstract Soft maps taking points on one surface to probability distributions on another are attractive for representing surface mappings in the presence of symmetry, ambiguity, and combinatorial complexity. Few techniques, however, are available to measure their continuity and other properties. To this end, we introduce a novel Dirichlet energy for soft maps generalizing the classical map Dirichlet energy, which measures distortion by computing how soft maps transport probabilistic mass from one distribution to another. We formulate the computation of the Dirichlet energy in terms of a differential equation and provide a finite elements discretization that enables all of the quantities introduced to be computed. We demonstrate the effectiveness of our framework for understanding soft maps arising from various sources. Furthermore, we suggest how these energies can be applied to generate continuous soft or point‐to‐point maps.