Linear representations of images are commonly used in object recognition; however, frequently used ones (namely, PCA, ICA, and FDA) are generally far from optimal in terms of actual recognition performance. We propose a (Monte-Carlo) simulated annealing algorithm that leads to optimal linear representations by maximizing the performance over subspaces. We illustrate its effectiveness using recognition experiments.

Monte Carlo (MC) methods have become an important tool for inferences in non-Gaussian and non-Euclidean settings. We study their use in those signal/image processing scenarios where the parameter spaces are certain Riemannian manifolds (finite-dimensional Lie groups and their quotient sets). We investigate the estimation of means and variances of the manifold-valued parameters, using two popular sampling methods: independent and importance sampling. Using Euclidean embeddings, we specify a notion of extrinsic means, employ Monte Carlo methods to estimate these means, and utilize large-sample asymptotics to approximate the estimator covariances. Experimental results are presented for target pose estimation (orthogonal groups) and signal subspace estimation (Grassmann manifolds). Asymptotic covariances are utilized to construct confidence regions, to compare estimators, and to determine the sample size for MC sampling.

Linear representations and linear dimension reduction techniques are very common in signal and image processing. Many such applications reduce to solving problems of stochastic optimizations or statistical inferences on the set of all subspaces, i.e. a Grassmann manifold. Central to solving them is the computation of an "exponential" map (for constructing geodesics) and its inverse on a Grassmannian. Here we suggest efficient techniques for these two steps and illustrate two applications: (i) For image-based object recognition, we define and seek an optimal linear representation using a Metropolis-Hastings type, stochastic search algorithm on a Grassmann manifold. (ii) For statistical inferences, we illustrate computation of sample statistics, such as mean and variances, on a Grassmann manifold.

In the problem of recognizing targets from their observed images, the estimation of target orientations, as elements of the rotation group SO(3), plays an important role. For k- objects the unknown parameter is an element of SO(3) k . Since k may be unknown a-priori, the parameter space is extended to X = [ 1 k=0 SO(3) k . In this representation, both the target orientations and their numbers have to be estimated simultaneously. We present a Bayesian approach which builds a posterior probability measure on X . Then, utilizing a Markov jump-diffusion process X(t), we sample from this posterior to empirically generate the estimates. The two components of X(t), jumps and diffusions, are chosen in such a way that the resulting Markov process has the desired ergodic property: averages along its sample paths converge to the expectations under the posterior. Proper choice of the diffusion parameters and the jump intensities is demonstrated and the ergodic result associated wi...

We address the problem of tracking principal subspaces using ideas from nonlinear filtering. The subspaces are represented by their complex projection-matrices, and time-varying subspaces correspond to trajectories on the Grassmann manifold. Under a Bayesian approach, we impose a smooth prior on the velocities associated with the subspace motion. This prior combined with any standard likelihood function forms a posterior density on the Grassmannian, for filtering and estimation. Using a sequential Monte Carlo method, a recursive nonlinear tracking algorithm is derived and some implementation results are presented.

We address the problem of tracking the time-varying linear subspaces (of a larger system) under a Bayesian framework. Variations in subspaces are treated as a piecewise-geodesic process on a complex Grassmann manifold and a Markov prior is imposed on it. This prior model, together with an observation model, gives rise to a hidden Markov model on a Grassmann manifold, and admits Bayesian inferences. A sequential Monte Carlo method is used for sampling from the time-varying posterior and the samples are utilized to estimate the underlying process. Simulation results are presented for principal subspace tracking in array signal processing.

A variety of engineering problems can be studied as inferences on constrained sets, Lie groups in particular. Additionally, the number of parameters to be estimated, namely the model-order, may also be unknown a-priori. We present a Bayesian approach by building a posterior probability distribution on a countable unions of Lie groups and utilizing the jump-diffusion processes to generate optimal estimators empirically, under this posterior. This approach is presented in the context of two well-known problems: pose estimation in object recognition and subspace estimation in signal processing. 1. Introduction We present a Bayesian formulation and an inference technique to solve problems involving: (i) model-order variability in addition to the unknown parameters, and (ii) the parametric variations taking place on curved Lie groups. The inclusion of latter extends the approach introduced in Grenander-Miller [8] which focuses only on Euclidean parameters. An important aspect of this work...

This paper discusses non-parametric regression between Riemannian manifolds. This learning problem arises frequently in many application areas ranging from signal processing, computer vision, over robotics to computer graphics. We present a new algorithmic scheme for the solution of this general learning problem based on regularized empirical risk minimization. The regularization functional takes into account the geometry of input and output manifold, and we show that it implements a prior which is particularly natural. Moreover, we demonstrate that our algorithm performs well in a difficult surface registration problem. 1

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