]> Tikhonov Regularization

Tikhonov Regularization

Invert the forward problem using Tikhonov regularization and a user-supplied regularization parameter.

Function Summary

Syntax:	X = tik(A, Y, alpha[, CD[, CI]]);
Inputs:	A	Forward matrix
	Y	The residue appropriate to A
	alpha	Regularization parameter, multiplies CI
	CD	Covariance of the data. Optional: identity of not specified
	CI	Covariance of the image. Optional: identity of not specified
Outputs:	X	The reconstructed image

Detailed Descriptions

Tikhonov regularizes the matrix inverse by adding an extra term to the $L_{2}$ -norm of the residue. Common choices are the 2-norm of the magnitude of the reconstructed image (weighted by an a priori image covariance where possible) ${|σ^{-1} X|}^{2}$ and the curvature (the elastic cost) ${|σ^{-1} \nabla X|}^{2} .$ The PMI toolbox, however, implements the only former. Thus, the reconstructed image can be written in terms of the forward matrix $A$ , the data covariance $σ_{y}^{2},$ the a priori image covariance $σ_{x}^{2},$ and the regularization parameter $α$ (a non-negative constant parameter that controls the strength of the regularization) as

Δ X = {(A^{†} σ_{y}^{-2} A + α σ_{x}^{-2})}^{-1} A^{†} σ_{y}^{-2} Y