Download Signal and Image Restoration: Information-Theoretic by Joseph Noonan PDF
By Joseph Noonan
The objective of this ebook is to give a unified information-theoretic method of remedy estimation difficulties in general confronted in sign processing purposes. The authors supply new methods to this challenge, in addition to new interpretations for present suggestions. significant purposes of this paintings comprise snapshot recovery, conversation channel estimation, textual content recovery, and method modeling. A basic method of fixing a couple of detection and estimation difficulties using recommendations from details concept is constructed, and the theoretical improvement of the technique - in addition to vital purposes - is given.
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We note here that the constraint presented in this section is of importance to our work since the same constraint is used in deriving the GMF. 4 Bayesian Restoration It is possible to formulate the problem of image restoration as an estimation problem. Using this formulation one may then bring to the problem established techniques from estimation theory such as ML estimation and MAP estimation. Assume that the p (x) and p (y) are the probability distribution functions of the object and its estimate.
15) Again, define τ = Ef (T) and u = Ef (U). 16) where α0 maximizes L(α, τ 0 ). 17) Error Bounds for Maximum-Entropy Estimates 43 where α ¯ satisfies the gradient equation τ 0 = ∇α ψ(α, β). Note that φ(τ 0 , β) is ¯ and a concave function of β. Thus, φ(τ 0 , β) has a unique maximum at (α, ¯ β) 0 0 0 ¯ ¯ φ(τ , β) > φ(τ , β) for any other β. 18) from an n-parameter canonical exponential family has the maximum divergence distance from the estimate fn (x, α0 ). 2. Of all PDFs from an n-parameter canonical exponential family, ¯ has the maximum divergence distance from the ME estimate fm (x, α0 ) fn (x, α, ¯ β) computed from the information τ 0 = Ef (T).
An operator R(y, α) depending on the regularization parameter α is called a regularizing operator for Eq. 1) in a neighborhood of yT , the noiseless observations, if 1. there exists a δ1 > 0 such that the operator R(y, α) is defined for every α > 0 and every y ∈ Y for which ρy (y, yT ) ≤ δ ≤ δ1 , 2. there exists a function α = α(δ) such that for every ε > 0, there is a δ(ε) ≤ δ1 such that y ∈ Y and ρy (yT , yδ ) ≤ δ(ε) ⇒ ρx (xT , xα ) ≤ ε, where xα = R(yδ , α(δ)) is called the regularized solution, and the regularizing operator is not necessarily unique.