Wavelet-based statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly Gaussian. These models are unrealistic for many real-world signals. In this paper, we develop a new framework based on wavelet-domain hidden Markov models (HMMs). The framework enables us to concisely model the statistical dependencies and nonGaussian statistics often encountered in practice. Wavelet-domain HMMs are designed with the intrinsic properties of the wavelet transform in mind and provide powerful yet tractable probabilistic signal models. Efficient Expectation Maximization algorithms are developed for fitting the HMMs to observational signal data. The new framework is suitable for a wide range of applications, including signal estimation, detection, classification, prediction, and even synthesis. To demonstrate the utility of wavelet-domain HMMs, we develop novel algorithms for signal denoising, classification, and detectio...

Many approaches to Bayesian image segmentation have used maximum a posteriori (MAP) estimation in conjunction with Markov random fields (MRF). While this approach performs well, it has a number of disadvantages. In particular, exact MAP estimates cannot be computed, approximate MAP estimates are computationally expensive to compute, and unsupervised parameter estimation of the MRF is difficult. In this paper, we propose a new approach to Bayesian image segmentation which directly addresses these problems. The new method replaces the MRF model with a novel multiscale random field (MSRF), and replaces the MAP estimator with a sequential MAP (SMAP) estimator derived from a novel estimation criteria. Together, the proposed estimator and model result in a segmentation algorithm which is not iterative and can be computed in time proportional to MN where M is the number of classes and N is the number of pixels. We also develop a computationally effcient method for unsupervised estimation of m...

We present an algorithm based on statistical learning for synthesizing static and time-varying textures matching the appearance of an input texture. Our algorithm is general and automatic, and it works well on various types of textures including 1D sound textures, 2D texture images and 3D texture movies. The same method is also used to generate 2D texture mixtures that simultaneously capture the appearance of a number of different input textures. In our approach, input textures are treated as sample signals generated by a stochastic process. We first construct a tree representing a hierarchical multi-scale transform of the signal using wavelets. From this tree, new random trees are generated by learning and sampling the conditional probabilities of the paths in the original tree. Transformation of these random trees back into signals results in new random textures. In the case of 2D texture synthesis our algorithm produces results that are generally as good or better than those produce...

This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden

We present a tree-based reparameterization framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles. This class includes the belief propagation or sum-product algorithm [39, 36], as well as a rich set of variations and extensions of belief propagation. Algorithms in this class can be formulated as a sequence of reparameterization updates, each of which entails re-factorizing a portion of the distribution corresponding to an acyclic subgraph (i.e., a tree). The ultimate goal is to obtain an alternative but equivalent factorization using functions that represent (exact or approximate) marginal distributions on cliques of the graph. Our framework highlights an important property of BP and the entire class of reparameterization algorithms: the distribution on the full graph is not changed. The perspective of tree-based updates gives rise to a simple and intuitive characterization of the fixed points in terms of tree consistency. We develop interpretations of these results in terms of information geometry. The invariance of the distribution, in conjunction with the fixed point characterization, enables us to derive an exact relation between the exact marginals on an arbitrary graph with cycles, and the approximations provided by belief propagation, and more broadly, any algorithm that minimizes the Bethe free energy. We also develop bounds on this approximation error, which illuminate the conditions that govern their accuracy. Finally, we show how the reparameterization perspective extends naturally to more structured approximations (e.g., Kikuchi and variants [52, 37]) that operate over higher order cliques.

Recently, a framework for multiscale stochastic modeling was introduced based on coarse-to-fine scale-recursive dynamics defined on trees. This model class has some attractive characteristics which lead to extremely efficient, statistically optimal signal and image processing algorithms. In this paper, we show that this model class is also quite rich. In particular, we describe how 1-D Markov processes and 2-D Markov random fields (MRF’s) can be represented within this framework. The recursive structure of 1-D Markov processes makes them simple to analyze, and generally leads to computationally efficient algorithms for statistical inference. On the other hand, 2-D MRF’s are well known to be very difficult to analyze due to their noncausal structure, and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification. In contrast, our multiscale representations are based on scale-recursive models and thus lead naturally to scale-recursive algorithms, which can be substantially more efficient computationally than those associated with MRF models. In 1-D, the multiscale representation is a generalization of the midpoint deflection construction of Brownian motion. The representation of 2-D MRF’s is based on a further generalization to a “midline ” deflection construction. The exact representations of 2-D MRF’s are used to motivate a class of multiscale approximate MRF models based on one-dimensional wavelet transforms. We demonstrate the use of these latter models in the context of texture representation and, in particular, we show how they can be used as approximations for or alternatives to well-known MRF texture models.

We introduce a new image texture segmentation algorithm, HMTseg, based on wavelets and the hidden Markov tree (HMT) model. The HMT is a tree-structured probabilistic graph that captures the statistical properties of the coefficients of the wavelet transform. Since the HMT is particularly well suited to images containing singularities (edges and ridges), it provides a good classifier for distinguishing between textures. Utilizing the inherent tree structure of the wavelet HMT and its fast training and likelihood computation algorithms, we perform multiscale texture classification at a range of different scales. We then fuse these multiscale classifications using a Bayesian probabilistic graph to obtain reliable final segmentations. Since HMTseg works on the wavelet transform of the image, it can directly segment wavelet-compressed images without the need for decompression into the space domain. We demonstrate the performance of HMTseg with synthetic, aerial photo, and document image seg...

This thesis describes some new approaches to the representation and analysis of visual motion, as perceived by a biological or machine visual system. We begin by discussing the computation of image motion fields, the projection of motion in the three-dimensional world onto the two-dimensional image plane. This computation is notoriously difficult, and there are a wide variety of approaches that have been developed for use in image processing, machine vision, and biological modeling. We show that a large number of the basic techniques are quite similar in nature, differing primarily in conceptual motivation, and that they each fail to handle a set of situations that occur commonly in natural scenery. The central theme of the thesis is that the failure of these algorithms is due primarily to the use of vector fields as a representation for visual motion. We argue that the translational vector field representation is inherently impoverished and error-prone. Furthermore, there is evidence that a ...

We describe a technique for using the joint occurrence of local features at multiple resolutions to measure the similarity between texture images. Though superficially similar to a number of "Gabor" style techniques, which recognize textures through the extraction of multi-scale feature vectors, our approach is derived from an accurate generative model of texture, which is explicitly multiscale and non-parametric. The resulting recognition procedure is similarly non-parametric, and can model complex non-homogeneous textures. We report results on publicly available texture databases. In addition, experiments indicate that this approach may have sufficient discrimination power to perform target detection in synthetic aperture radar images (SAR). 1 Introduction The notion of texture is difficult to capture formally. Textures usually can be described informally as the output of some physical process wherein local structure is repeated seemingly at random. Two texture patches are consider...

This paper treats a multiresolution hidden Markov model for classifying images. Each image is represented by feature vectors at several resolutions, which are statistically dependent as modeled by the underlying state process, a multiscale Markov mesh. Unknowns in the model are estimated by maximum likelihood, in particular by employing the expectation-maximization algorithm. An image is classified by finding the optimal set of states with maximum a posteriori probability. States are then mapped into classes. The multiresolution model enables multiscale information about context to be incorporated into classification. Suboptimal algorithms based on the model provide progressive classification that is much faster than the algorithm based on single-resolution hidden Markov models.