Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications

We propose an affine framework for perspective views, captured by a single extremely simple equation based on a viewer-centered invariant we call "relative affine structure". Via a number of corollaries of our main results we show that our framework unifies previous work --- including Euclidean, projective and affine --- in a natural and simple way, and introduces new, extremely simple, algorithms for the tasks of reconstruction from multiple views, recognition by alignment, and certain image coding applications.

Support Vector Machines: Training and Applications

The Support Vector Machine (SVM) is a new and very promising classification technique developed by Vapnik and his group at AT&T Bell Labs. This new learning algorithm can be seen as an alternative training technique for Polynomial, Radial Basis Function and Multi-Layer Perceptron classifiers. An interesting property of this approach is that it is an approximate implementation of the Structural Risk Minimization (SRM) induction principle. The derivation of Support Vector Machines, its relationship with SRM, and its geometrical insight, are discussed in this paper. Training a SVM is equivalent to solve a quadratic programming problem with linear and box constraints in a number of variables equal to the number of data points. When the number of data points exceeds few thousands the problem is very challenging, because the quadratic form is completely dense, so the memory needed to store the problem grows with the square of the number of data points. Therefore, training problems arising in some real applications with large data sets are impossible to load into memory, and cannot be solved using standard non-linear constrained optimization algorithms. We present a decomposition algorithm that can be used to train SVM's over large data sets. The main idea behind the decomposition is the iterative solution of sub-problems and the evaluation of, and also establish the stopping criteria for the algorithm. We present previous approaches, as well as results and important details of our implementation of the algorithm using a second-order variant of the Reduced Gradient Method as the solver of the sub-problems. As an application of SVM's, we present preliminary results we obtained applying SVM to the problem of detecting frontal human faces in real images.

Molecular computations for reactions and phase transitions: applications to protein stabilization, hydrates and catalysis

In this work we have made significant contributions in three different areas of interest: therapeutic protein stabilization, thermodynamics of natural gas clathrate-hydrates, and zeolite catalysis. In all three fields, using our various computational techniques, we have been able to elucidate phenomena that are difficult or impossible to explain experimentally. More specifically, in mixed solvent systems for proteins we developed a statistical-mechanical method to model the thermodynamic effects of additives in molecular-level detail. It was the first method demonstrated to have truly predictive (no adjustable parameters) capability for real protein systems. We also describe a novel mechanism that slows protein association reactions, called the “gap effect.” We developed a comprehensive picture of methioine oxidation by hydrogen peroxide that allows for accurate prediction of protein oxidation and provides a rationale for developing strategies to control oxidation. The method of solvent accessible area (SAA) was shown not to correlate well with oxidation rates. A new property, averaged two-shell water coordination number (2SWCN) was identified and shown to correlate well with oxidation rates. Reference parameters for the van der Waals Platteeuw model of clathrate-hydrates were found for structure I and structure II. These reference parameters are independent of the potential form (unlike the commonly used parameters) and have been validated by calculating phase behavior and structural transitions for mixed hydrate systems. These calculations are validated with experimental data for both structures and for systems that undergo transitions from one structure to another. This is the first method of calculating hydrate thermodynamics to demonstrate predictive capability for phase equilibria, structural changes, and occupancy in pure and mixed hydrate systems. We have computed a new mechanism for the methanol coupling reaction to form ethanol and water in the zeolite chabazite. The mechanism at 400°C proceeds via stable intermediates of water, methane, and protonated formaldehyde.