  The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer
graphics and geometry processing applications. In particular, Laplacian
eigenbases allow generalizing the classical Fourier analysis to manifolds. A
key drawback of such bases is their inherently global nature, as the Laplacian
eigenfunctions carry geometric and topological structure of the entire
manifold. In this paper, we introduce a new framework for local spectral shape
analysis. We show how to efficiently construct localized orthogonal bases by
solving an optimization problem that in turn can be posed as the
eigendecomposition of a new operator obtained by a modification of the standard
Laplacian. We study the theoretical and computational aspects of the proposed
framework and showcase our new construction on the classical problems of shape
approximation and correspondence. We obtain significant improvement compared to
classical Laplacian eigenbases as well as other alternatives for constructing
localized bases.
