Multiscale higher-order TV operators for L1 regularization

In the realm of signal and image denoising and reconstruction, \(\ell _1\) regularization techniques have generated a great deal of attention with a multitude of variants. In this work, we demonstrate that the \(\ell _1\) formulation can sometimes result in undesirable artifacts that are inconsistent with desired sparsity promoting \(\ell _0\) properties that the \(\ell _1\) formulation is intended to approximate. With this as our motivation, we develop a multiscale higher-order total variation (MHOTV) approach, which we show is related to the use of multiscale Daubechies wavelets. The relationship of higher-order regularization methods with wavelets, which we believe has generally gone unrecognized, is shown to hold in several numerical results, although notable improvements are seen with our approach over both wavelets and classical HOTV. These results are presented for 1D signals and 2D images, and we include several examples that highlight the potential of our approach for improving two- and three-dimensional electron microscopy imaging. In the development approach, we construct the tools necessary for MHOTV computations to be performed efficiently, via operator decomposition and alternatively converting the problem into Fourier space.