On Manifold Learning and Mesh Editing

Raif M. Rustamov, accepted as a poster to SGP 2008

We notice a formal connection between two fields -- manifold  learning and mesh editing, and exploit this connection to introduce a generalization of the naïve Laplacian mesh editing that uses diffusion wavelets.

Technical Report

Boundary Element Formulation of Harmonic Coordinates

Raif M. Rustamov, Technical Report

We explain how Boundary Element Methods (BEM) can be used to speed up the computation and reduce the storage associated with Harmonic Coordinates. In addition, BEM formulation allows extending the harmonic coordinates to the exterior and makes possible to compare the transfinite harmonic coordinates with transfinite Shepard interpolation and Mean Value Coordinates. This comparison reveals that there are unifying formulas, yet harmonic coordinates belong to a fundamentally different end of the spectrum. This observation allows us to generalize harmonic coordinates by introducing a novel class of interpolates which we call weakly singular interpolates.

Technical Report

Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation

Raif M. Rustamov, SGP 07

A deformation invariant representation of surfaces, the GPS embedding, is introduced using the eigenvalues and eigenfunctions of the Laplace-Beltrami differential operator. Notably, since the definition of the GPS embedding completely avoids the use of geodesic distances, and is based on objects of global character, the obtained representation is robust to local topology changes. The GPS embedding captures enough information to handle various shape processing tasks as shape classification, segmentation, and correspondence. To demonstrate the practical relevance of the GPS embedding, we introduce a deformation invariant shape descriptor called G2-distributions, and demonstrate their discriminative power, invariance under natural deformations, and robustness.

Paper  Project Page  GPS Based Search Engine

Augmented Symmetry Transforms

Raif M. Rustamov, SMI 07

Symmetry has been playing an increasing role in 3D shape processing. Recently introduced Planar Reflective Symmetry Transform(PRST) has been found useful for canonical coordinate frame determination, shape matching, retrieval, and segmentation. Guided by the intuition that every imperfect symmetry is imperfect in its own way, we investigate the possibility of incorporating more information into symmetry transforms like PRST. As a step in this direction, the concept of Augmented Symmetry Transform is introduced; we obtain a family of symmetry transforms indexed by a parameter. While the original PRST measures how much the symmetry is broken, the Augmented PRST also gives some information about how it is broken. Several approaches to calculating the augmented transform are described. We demonstrate that the augmented transform is useful for shape retrieval.

Paper

Work in Topology

My PhD work concentrated on low-dimensional topology, specifically, Heegaard Floer homology and its interplay with singularity theory and symplectic geometry.

The Renormalized Euler Characteristic and L-space Surgeries.

On Plumbed L-spaces.

Surgery formula for the renormalized Euler characteristic of Heegaard Floer homology.

On Heegaard Floer homology of plumbed three-manifolds with b1 = 1.

Calculation of Heegaard Floer homology for a class of Brieskorn spheres.