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Shape‐from‐Operator: Recovering Shapes from Intrinsic Operators
Author(s) -
Boscaini Davide,
Eynard Davide,
Kourounis Drosos,
Bronstein Michael M.
Publication year - 2015
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.12558
Subject(s) - embedding , operator (biology) , metric (unit) , laplace operator , mathematics , eigenvalues and eigenvectors , scaling , intrinsic metric , computer science , algorithm , mathematical analysis , metric space , injective metric space , geometry , artificial intelligence , physics , biochemistry , chemistry , operations management , repressor , quantum mechanics , transcription factor , economics , gene
We formulate the problem of shape‐from‐operator (SfO), recovering an embedding of a mesh from intrinsic operators defined through the discrete metric (edge lengths). Particularly interesting instances of our SfO problem include: shape‐from‐Laplacian, allowing to transfer style between shapes; shape‐from‐difference operator, used to synthesize shape analogies; and shape‐from‐eigenvectors, allowing to generate ‘intrinsic averages’ of shape collections. Numerically, we approach the SfO problem by splitting it into two optimization sub‐problems: metric‐from‐operator (reconstruction of the discrete metric from the intrinsic operator) and embedding‐from‐metric (finding a shape embedding that would realize a given metric, a setting of the multidimensional scaling problem). We study numerical properties of our problem, exemplify it on several applications, and discuss its imitations.