  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. A key component for their success is that under certain
assumptions, the solution of minimum $\ell_1$ norm is a good approximation to
the solution of minimum $\ell_0$ norm. In this work, we demonstrate that this
approximation can result in artifacts that are inconsistent with desired
sparsity promoting $\ell_0$ properties, resulting in subpar results in {some}
instances. 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. We also develop the tools necessary for MHOTV
computations to be performed efficiently, via operator decomposition and
alternatively converting the problem into Fourier space. 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.
