Orthogonally Constrained Sparse Approximations with Applications to Geometry Processing

Compressed manifold modes are solutions to an optimisation problem involving the $\ell_1$ norm and the orthogonality condition $X^TMX=I$. Such functions can be used in geometry processing as a basis for the function space of a mesh and are related to the Laplacian eigenfunctions. Compressed manifold modes and other alternatives to the Laplacian eigenfunctions are all special cases of generalised manifold harmonics, introduced here as solutions to a more general problem. An important property of the Laplacian eigenfunctions is that they commute with isometry. A definition for isometry between meshes is given and it is proved that compressed manifold modes also commute with isometry. The requirements for generalised manifold harmonics to commute with isometry are explored. A variety of alternative basis functions are tested for their ability to reconstruct specific functions -- it is observed that the function type has more impact than the basis type. The bases are also tested for their ability to reconstruct functions transformed by functional map -- it is observed that some bases work better for different shape collections. The Stiefel manifold is given by the set of matrices $X \in \mathbb{R}^{n \times k}$ such that $X^TMX = I$, with $M=I$. Properties and results are generalised for the $M \neq I$ case. A sequential algorithm for optimisation on the generalised Stiefel manifold is given and applied to the calculation of compressed manifold modes. This involves a smoothing of the $\ell_1$ norm. Laplacian eigenfunctions can be approximated by solving an eigenproblem restricted to a subspace. It is proved that these restricted eigenfunctions also commute with isometry. Finally, a method for the approximation of compressed manifold modes is given. This combines the method of fast approximation of Laplacian eigenfunctions with the ADMM solution to the compressed manifold mode problem. A significant improvement is made to the speed of calculation.