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Abstract We introduce a lifted $$\ell _1$$ (LL1) regularization framework for the recovery of sparse signals. The proposed LL1 regularization is a generalization of several popular regularization methods in the field and is motivated by recent advancements in re-weighted $$\ell _1$$ approaches for sparse recovery. Through a comprehensive analysis of the relationships between existing methods, we identify two distinct types of lifting functions that guarantee equivalence to the $$\ell _0$$ minimization problem, which is a key objective in sparse signal recovery. To solve the LL1 regularization problem, we propose an algorithm based on the alternating direction method of multipliers and provide proof of convergence for the unconstrained formulation. Our experiments demonstrate the improved performance of the LL1 regularization compared with state-of-the-art methods, confirming the effectiveness of our proposed framework. In conclusion, the LL1 regularization presents a promising and flexible approach to sparse signal recovery and invites further research in this area. 

SUMMARY Repeatedly recording seismic data over a period of months or years is one way to identify trapped oil and gas and to monitor CO2 injection in underground storage reservoirs and saline aquifers. This process of recording data over time and then differencing the images assumes the recording of the data over a particular subsurface region is repeatable. In other words, the hope is that one can recover changes in the Earth when the survey parameters are held fixed between data collection times. Unfortunately, perfect experimental repeatability almost never occurs. Acquisition inconsistencies such as changes in weather (currents, wind) for marine seismic data are inevitable, resulting in source and receiver location differences between surveys at the very least. Thus, data processing aimed at improving repeatability between baseline and monitor surveys is extremely useful. One such processing tool is regularization (or binning) that aligns multiple surveys with different source or receiver configurations onto a common grid. Data binned onto a regular grid can be stored in a high-dimensional data structure called a tensor with, for example, x and y receiver coordinates and time as indices of the tensor. Such a higher-order data structure describing a subsection of the Earth often exhibits redundancies which one can exploit to fill in gaps caused by sampling the surveys onto the common grid. In fact, since data gaps and noise increase the rank of the tensor, seeking to recover the original data by reducing the rank (low-rank tensor-based completion) successfully fills in gaps caused by binning. The tensor nuclear norm (TNN) is defined by the tensor singular value decomposition (tSVD) which generalizes the matrix SVD. In this work we complete missing time-lapse data caused by binning using the alternating direction method of multipliers (or ADMM) to minimize the TNN. For a synthetic experiment with three parabolic events in which the time-lapse difference involves an amplitude increase in one of these events between baseline and monitor data sets, the binning and reconstruction algorithm (TNN-ADMM) correctly recovers this time-lapse change. We also apply this workflow of binning and TNN-ADMM reconstruction to a real marine survey from offshore Western Australia in which the binning onto a regular grid results in significant data gaps. The data after reconstruction varies continuously without the large gaps caused by the binning process. 

In this paper, we aim to segment an image degraded by blur and Poisson noise. We adopt a smoothing-and-thresholding (SaT) segmentation framework that finds a piecewise-smooth solution, followed by k -means clustering to segment the image. Specifically for the image smoothing step, we replace the least-squares fidelity for Gaussian noise in the Mumford-Shah model with a maximum posterior (MAP) term to deal with Poisson noise and we incorporate the weighted difference of anisotropic and isotropic total variation (AITV) as a regularization to promote the sparsity of image gradients. For such a nonconvex model, we develop a specific splitting scheme and utilize a proximal operator to apply the alternating direction method of multipliers (ADMM). Convergence analysis is provided to validate the efficacy of the ADMM scheme. Numerical experiments on various segmentation scenarios (grayscale/color and multiphase) showcase that our proposed method outperforms a number of segmentation methods, including the original SaT.