A generalised Lasso iteratively reweighted scheme is here introduced to perform spatially regularised Hurst estimation on semi-local, weakly self-similar processes. This is extended further to the robust, heavy-tailed case whereupon the generalised M-Lasso is proposed. The design successfully incorporates both a spatial derivative in the generalised Lasso regulariser operator and a weight matrix formulated in the wavelet domain. The result simultaneously spatially smooths the Hurst estimates and downweights outliers. Experiments using a Hampel score function confirm that the method yields superior Hurst estimates in the presence of strong outliers. Moreover, it is shown that the inferred weight matrix can be used to perform wavelet shrinkage and denoise fractional Brownian surfaces in the presence of strong, localised, band-limited noise.
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