Instead of L2-norm solutions obtained by the conventional LS solution, Lp-norm minimization solutions, with , are often tried. Iterative inversion algorithms called IRLS (Iteratively Reweighted Least Squares) algorithms have been developed to solve these problems, which lie between the least-absolute-values problem and the classical least-squares problem. The main advantage of IRLS is to provide an easy way to compute the approximate L1-norm solution. L1-norm solutions are known to be more robust than L2-norm solutions, being less sensitive to spiky, high-amplitude noise Claerbout and Muir (1973); Scales et al. (1988); Scales and Gersztenkorn (1987); Taylor et al. (1979).
The problem solved by IRLS is a minimization of the weighted residual in the least-squares sense :
(5) |
(6) |
(7) |
(8) |
where is a value that is used as a threshold between L1 and L2 -norms.
IRLS can be easily incorporated in CG algorithms by including a weight such that the operator has a postmultiplier and the the adjoint operator has a premultiplier Claerbout (2004). Even though we do not know the real Lp-norm residual vector at the beginning of the iteration, we can approximate the residual with a residual of the previous iteration step, and it will converge to a residual that is very close to the Lp-norm residual as the iteration step continues. This can be summarized as follows:
iterate { }.