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Stein's unbiased risk estimator (SURE) has been shown to be an effective metric for determining optimal parameters for many applications. The topic of this article is focused on the use of SURE for determining parameters for blind deconvolution. The parameters include those that define the shape of the point spread function (PSF), as well as regularization parameters in the deconvolution formulas. Within this context, the optimal parameters are typically determined via a brute for search over the feasible parameter space. When multiple parameters are involved, this parameter search is prohibitively costly due to the curse of dimensionality. In this work, novel fixed point iterations are proposed for optimizing these parameters, which allows for rapid estimation of a relatively large number of parameters. We demonstrate that with some mild tuning of the optimization parameters, these fixed point methods typically converge to the ideal PSF parameters in relatively few iterations, e.g. 50-100, with each iteration requiring very low computational cost.

BM3D has been considered the standard for comparison in the image denoising literature for the last decade. Though it has been shown to be surpassed numerous times by alternative algorithms in terms of PSNR, the margins are very thin, and denoising is approaching a limiting point. The reason for the continued use of BM3D within the literature is due to its off-the-shelf ease-of-use in any application, which alternative improved denoising algorithms sometimes fail to match. This article proposes a new variation of BM3D, which maintains its ease of use but is notably faster. This development brings us closer to real-time ease-of-use application of new state-of-the-art image reconstruction algorithms such as plug-and-play priors. We refer to our variation of BM3D as G-BM3D. In terms of image quality, our algorithm attains very similar denoising performance to the original algorithm. Though our algorithm is written completely in MATLAB software, it is already between 5-20 times faster than the original algorithm, and the modifications to the algorithm are such that it is expected to be significantly faster when ported to CUDA language and with more powerful GPUs. The improved processing time is achieved by two main components. The first component is a new computational strategy that achieves faster block matching, and the second is a new global approach to the 3D wavelet filtering step that allows for significantly improved processing times on GPUs. The fast block matching strategy could also be applied to any of the vast number of nonlocal self-similarity (NSS) denoisers to improve processing times.

This article is written to serve as an introduction and survey of imaging with synthetic aperture radar (SAR). The reader will benefit from having some familiarity with harmonic analysis, electromagnetic radiation, and inverse problems. After an overview of the SAR problem and some main concepts, the SAR imaging problem is contextualized in terms of classical harmonic analysis. Within this context, we consider partial Fourier sums of off-centered Fourier data and correspondingly the convolutional kernels resulting from conventional SAR image formation techniques. Following this, we revisit imaging of random complex signals from frequency data as in SAR, providing simpler derivations of some previous results and extending these ideas to the continuous setting. These concepts are tied in with the derived convolutional kernels, and it is deduced how good an image approximation is when it is obtained from only a small band of high frequency Fourier coefficients. Finally, regularization methods are presented to improve the quality of SAR images. Corresponding MATLAB software is made available for reproducibility of most figures and to facilitate further exploration of the methods presented here.

The choice of the parameter value for regularized inverse problems is critical to the results and remains a topic of interest. This article explores a criterion for selecting a good parameter value by maximizing the probability of the data, {{with no prior knowledge of the noise variance}}. These concepts are developed for $\ell_2$ and consequently $\ell_1$ regularization models by way of their Bayesian interpretations. Based on these concepts, an iterative scheme is proposed and demonstrated to converge accurately, and analytical convergence results are provided that substantiate these empirical observations. For some of the most common inverse problems, including MRI, SAR, denoising, and deconvolution, an extremely efficient algorithm is derived, making the iterative scheme very attractive for real case use. The computational concerns associated with the general case for any inverse problem are also carefully addressed. A robust set of 1D and 2D numerical simulations confirm the effectiveness of the proposed approach.

Tomography is an imaging technique that works by reconstructing a scene from acquired data in the form of line integrals of the imaging domain. A fundamental underlying assumption in the reconstruction procedure is the precise alignment of the data values, i.e. the relationship between the data values and the paths of the lines of integration is accurately known. In many applications, e.g. electron and X-ray tomography, it is necessary to establish this relationship using software alignment techniques or image registration due to misalignment when rotating the physical specimen. Unfortunately, highly accurate software alignment is still a challenge to achieve in many cases, and improper alignment results in severe loss in the imaging resolution. In this article, we develop a new approach that considers the alignment problem through a completely different lens, as a problem of recovering phase shifts in Fourier domain {within} the reconstruction algorithm. The recovery of these phase shifts serves as the data alignment, which is done by calculating discrepancies between the misaligned data and the current reconstruction. In the development of the approach, we investigate proper selection of parameters, and we show that it is fairly flexible and surprisingly accurate. Finally, the analysis of our approach provides insight into why projection matching alignment by cross-correlation can be improved through low pass filtering, which we demonstrate. Our methods are validated in a wide range of examples and settings.

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.

Popular methods for finding regularized solutions to inverse problems include sparsity promoting $\ell_1$ regularization techniques, one in particular which is the well known total variation (TV) regularization. More recently, several higher order (HO) methods similar to TV have been proposed, which we generally refer to as HOTV methods. In this letter, we investigate problem of the often debated selection of $\lambda$, the parameter used to carefully balance the interplay between data fitting and regularization terms. We theoretically argue for a scaling of the parameter that works for all orders for HOTV methods, based off of a single selection of the parameter for any one of the orders. We also provide numerical results which justify our theoretical findings.

Over the last decade or so, reconstruction methods using $\ell_1$ regularization, often categorized as compressed sensing (CS) algorithms, have significantly improved the capabilities of high fidelity imaging in electron tomography. The most popular $\ell_1$ regularization approach within electron tomography has been total variation (TV) regularization. In addition to reducing unwanted noise, TV regularization encourages a piecewise constant solution with sparse boundary regions. In this paper we propose an alternative $\ell_1$ regularization approach for electron tomography based on higher order total variation (HOTV). Like TV, the HOTV approach promotes solutions with sparse boundary regions. In smooth regions however, the solution is not limited to piecewise constant behavior. We demonstrate that this allows for more accurate reconstruction of a broader class of images -- even those for which TV was designed for -- particularly when dealing with pragmatic tomographic sampling patterns and very fine image features. We develop results for an electron tomography data set as well as a phantom example, and we also make comparisons with discrete tomography approaches.