Multiscale Representations of Markov Random Fields

Abstract

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Abstract 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 Ma.rkov random fields (MRF's) can be represented within this framework. Markov processes in one-dimension and Markov randoml fields in two-dimensions are widely used classes of models for analysis, design and statistical inference. 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 MIR,F's are well known to be very difficult to analyze due to their non-causal structure, and thus their use typically leads to colmputationally intensive algoritlllhms for smoothing and parameter idlentification. Our multiscale representations are based on scale-recursive models, thus providing a framework for the development of new and efficient algorithms for Markov processes and MRF's. In 1-D, the representation generalizes the mid-point deflection construction of Brownian motion. In 2-D, we use a further generalization to a. "mid-line" deflection construction. Our exact representations of 2-D MRF's are of potentially high dimension, and this motivates a class of approximate models ba.sed on one-dim7ensional wavelet transforms. We demonstrate the use of these 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 illustrate how the quality of the representations varies a.s a. function of the underlying MRF and the complexity of the wavelet-based approximate representation.

Additive noise reduction in natural images using second‐generation wavelet transform hidden Markov models

“…The main aim of statistical image denoising is to figure out a realistic paradigm that approximates f ( X ) and allows obtaining an effective processing algorithm. Thus, several approaches are considered to model the local joint statistics of the image pixels in the spatial domain, with the most predominant being the Markov random field model . In our study, the model of wavelet transform domain is based on the idea that often the linear, invertible transformation will probably ‘restructure’ the original image, and, on the other hand, leave the transform coefficients whose its structure is ‘simpler’ to process.…”

Adaptive fusion of multisensor precipitation using Gaussian-scale mixtures in the wavelet domain

“…Minimizing the trace of the covariance matrix of the estimates, this formalism allows us to sequentially obtain the conditional expectation of the system state variables t = [ x t ∣ y 0 , y 1 ,…, y t ] in time, given the noisy observations in the framework of an affine measurement equation y t = C t x t + v t , where C t relates the system state to the measurements and v t ∼ (0, Obviously, for such a Gaussian dynamic system, Kalman filter (KF) is an optimal estimator as the conditional expectation, and the associated covariance can fully explain the entire probabilistic structure of the system. Replacing the notion of time with scale, the original idea of the linear estimation of temporal Gauss‐Markov systems was further expanded by Chou et al [1994] to the optimal estimation of multiresolution auto‐regressive (MAR) Gaussian processes [e.g., Luettgen et al , 1993; Daniel and Willsky , 1999; Willsky , 2002]. In MAR representation, a multiresolution process is naturally defined on a treelike graph structure (see Figure 4), where each node s ∈ on the tree is a 3‐tuple which indicates the signal quantity x ( s ) in a specific translational offset and scale level: In this coarse‐to‐fine multiresolution dynamics, s denotes the parent node of s , A ( s ) is the transition matrix and w ( s ) ∼ (0, Σ w ( s ) ) is a Gaussian white noise.…”

Multiresolution optimal interpolation and statistical analysis of TOPEX/POSEIDON satellite altimetry

“…Furthermore, since we have a model (12) for the estimation errors in this time-recursive statistical estimation problem, we can use the measurement residuals…”

“…In addition to the computational efficiencies admitted by these multiscale models, they also can be used to capture the statistical structure of rich classes of phenomena. For example, in [12] it is shown that multiscale models as in (14),(15) can be constructed to represent phenomena frequently modeled using MRF's. Of more direct importance here is the fact that this multiscale framework is directly suited to capturing phenomena that display a multitude of correlation scales.…”