Structural Adaptive Smoothing Procedures

Abstract

An important problem in image and signal analysis is denoising. Given data y _j at locations x _j , j = 1, ..., N, in space or time, the goal is to recover the original image or signal m _j , j = 1, ..., N, from the noisy observations y _j , j = 1, ..., N. Denoising is a special case of a function estimation problem: If m _j = m(x _j ) for some function m(x), we may model the data y _j as real-valued random variables Y _j satisfying the regression relation

$$ Y_j = m\left( {x_j } \right) + \varepsilon _j , j = 1,...,N, $$

(6.1)

where the additive noise ε _j , j = 1, ..., N, is independent, identically distributed (i.i.d.) with mean $\mathbb{E}$ ε_j = 0. The original denoising problem is solved by finding an estimate $\hat m\left( x \right)$ of the regression function m(x) on some subset containing all the x _j . More generally, we may allow the function arguments to be random variables X _j ε ℝ^d themselves ending up with a regression model with stochastic design

$$ Y_j = m\left( {X_j } \right) + \varepsilon _j , j = 1,...,N, $$

(6.2)

where X _j ,Y _j are identically distributed, and $\mathbb{E}\left\{ {\varepsilon _j |X_j } \right\} = 0$ . In this case, the function m(x) to be estimated is the conditional expectation of Y _j given X _j = x.