Shape-from-Operator: Recovering Shapes from Intrinsic Operators

• Published in: Computer graphics forum Vol. 34; no. 2; pp. 265 - 274
• Main Authors: Boscaini, Davide, Eynard, Davide, Kourounis, Drosos, Bronstein, Michael M.
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.05.2015
• Subjects: Analogies; Analysis; Categories and Subject Descriptors (according to ACM CCS); Collection; Computer graphics; Eigenvalues; I.3 [Computer graphics]: Shape modeling-Shape analysis; Image processing systems; Laplace transforms; Operators; Optimization; Reconstruction; Recovering; Splitting; Studies
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.12558

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.