Abstract
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.
Highlights
Over the past couple of decades, l1 regularization techniques such as total variation have become increasingly popular methods for image and signal denoising and reconstruction problems
Synthetic aperture radar (SAR) [2, 3], magnetic resonance imaging (MRI) [4,5,6], electron tomography [7, 8], and inpainting [9, 10] are all image recovery applications that have advanced in part due to l1 regularization methods, and in each case the approach can be tailored to the challenges that the particular application poses
higher-order TV (HOTV) circumvents the staircasing often observed in total variation (TV) solutions and has been shown to be more effective for problems with fine features, where resolution can be improved by increasing the order of derivatives in the regularization term [8]
Summary
Over the past couple of decades, l1 regularization techniques such as total variation have become increasingly popular methods for image and signal denoising and reconstruction problems. Synthetic aperture radar (SAR) [2, 3], magnetic resonance imaging (MRI) [4,5,6], electron tomography [7, 8], and inpainting [9, 10] are all image recovery applications that have advanced in part due to l1 regularization methods, and in each case the approach can be tailored to the challenges that the particular application poses With many problems such as two- and three-dimensional electron microscopy imaging, the challenge is often to acquire as little data as. For more general piecewise smooth functions higher-order TV (HOTV) regularization methods are effective [14, 16, 17], and they do not suffer from the staircasing effects In this case, the transform maps f to approximations of discrete derivatives of f, e.g., higher-order finite differences of f. Most of the coefficients can be neglected, and a sparse approximation of f exists with respect to the basis
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