In exterior calculus on smooth manifolds, the exterior derivative and wedge products are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and an antisymmetrized cup-like product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson’s cochain product defined using Whitney and de Rham maps.
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Functional Maps Representation On Product Manifolds
We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.
more » « less- Award ID(s):
- 1838071
- PAR ID:
- 10461152
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Computer Graphics Forum
- Volume:
- 38
- Issue:
- 1
- ISSN:
- 0167-7055
- Page Range / eLocation ID:
- p. 678-689
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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