]> Regularized Least-Squares

Regularized Least-Squares

Invert the forward problem by minimizing the cost function in a least-squared sense. Either the functional form of the cost function itself or halting after a fixed number of iterations can provide the regularization.

Function Summary

Syntax:	X = art(A, Y, X0, nIter, W, iMeas);
Inputs:	A	Forward matrix
	Y	The residue appropriate to A
	lambda	Regularization parameter
	X0	Initial guess or empty matrix [] for zeros
	nIter	Number of iterations to calculate
	tol	Stepping tolerance. Optional: 1.0e-6 if not specified
Outputs:	X	The reconstructed image

Detailed Descriptions

If the minimization is being regularized by limiting the number of iterations, then the cost function being minimized is the usual $L_{2}$ -norm

C = {|Y - A X|}^{2} .

Embedding the regularization inside the cost function, on the other hand, amounts to minimizing the modified cost function

C = {|[\begin{matrix} Y \\ 0 \end{matrix}] - [\begin{matrix} A \\ α I \end{matrix}] X|}^{2},

which can be derived from the usual Tikhonov cost function.