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In the realm of signal and image denoising and reconstruction, $\ell_1$regularization techniques have generated a great deal of attention with amultitude of variants. A key component for their success is that under certainassumptions, the solution of minimum $\ell_1$ norm is a good approximation tothe solution of minimum $\ell_0$ norm. In this work, we demonstrate that thisapproximation can result in artifacts that are inconsistent with desiredsparsity promoting $\ell_0$ properties, resulting in subpar results in {some}instances. With this as our motivation, we develop a multiscale higher ordertotal variation (MHOTV) approach, which we show is related to the use ofmultiscale Daubechies wavelets. We also develop the tools necessary for MHOTVcomputations to be performed efficiently, via operator decomposition andalternatively converting the problem into Fourier space. The relationship ofhigher order regularization methods with wavelets, which we believe hasgenerally gone unrecognized, is shown to hold in several numerical results,although notable improvements are seen with our approach over both wavelets andclassical HOTV.

Multiscale Higher Order TV Operators for L1 Regularization