Integrable Generalized Principal Chiral Models

We study 2D non-linear sigma models on a group manifold with a special form of the metric. We address the question of integrability for this special class of sigma models. We derive two algebraic conditions for the metric on the group manifold. Each solution of these conditions defines an integrable model. Although the algebraic system is overdetermined in general, we give two examples of solutions. We find the Lax field for these models and calculate their Poisson brackets. We also obtain the renormalization group (RG) equations, to first order, for the generic model. We solve the RG equations for the examples we have and show that they are integrable along the RG flow.Comment: 14 pages, harvmac (l

The finite harmonic oscillator and its associated sequences

A system of functions (signals) on the finite line, called the oscillator system, is described and studied. Applications of this system for discrete radar and digital communication theory are explained. Keywords: Weil representation, commutative subgroups, eigenfunctions, random behavior, deterministic constructionComment: Published in the Proceedings of the National Academy of Sciences of the United States of America (Communicated by Joseph Bernstein, Tel Aviv University, Tel Aviv, Israel

Level Set KSVD

We present a new algorithm for image segmentation - Level-set KSVD. Level-set KSVD merges the methods of sparse dictionary learning for feature extraction and variational level-set method for image segmentation. Specifically, we use a generalization of the Chan-Vese functional with features learned by KSVD. The motivation for this model is agriculture based. Aerial images are taken in order to detect the spread of fungi in various crops. Our model is tested on such images of cotton fields. The results are compared to other methods.Comment: 25 pages, 14 figures. Submitted to IJC

Learning Big (Image) Data via Coresets for Dictionaries

Signal and image processing have seen an explosion of interest in the last few years in a new form of signal/image characterization via the concept of sparsity with respect to a dictionary. An active field of research is dictionary learning: the representation of a given large set of vectors (e.g. signals or images) as linear combinations of only few vectors (patterns). To further reduce the size of the representation, the combinations are usually required to be sparse, i.e., each signal is a linear combination of only a small number of patterns. This paper suggests a new computational approach to the problem of dictionary learning, known in computational geometry as coresets. A coreset for dictionary learning is a small smart non-uniform sample from the input signals such that the quality of any given dictionary with respect to the input can be approximated via the coreset. In particular, the optimal dictionary for the input can be approximated by learning the coreset. Since the coreset is small, the learning is faster. Moreover, using merge-and-reduce, the coreset can be constructed for streaming signals that do not fit in memory and can also be computed in parallel. We apply our coresets for dictionary learning of images using the K-SVD algorithm and bound their size and approximation error analytically. Our simulations demonstrate gain factors of up to 60 in computational time with the same, and even better, performance. We also demonstrate our ability to perform computations on larger patches and high-definition images, where the traditional approach breaks down