An MM-Based Algorithm for -Regularized Least-Squares Estimation With an Application to Ground Penetrating Radar Image Reconstruction

Abstract

An estimation method known as least absolute shrinkage and selection operator (LASSO) or ℓ1-regularized LS estimation has been found to perform well in a number of applications. In this paper, we use the majorize-minimize method to develop an algorithm for minimizing the LASSO objective function, which is the sum of a linear LS objective function plus an ℓ1 penalty term. The proposed algorithm, which we call the LASSO estimation via majorization-minimization (LMM) algorithm, is straightforward to implement, parallelizable, and guaranteed to produce LASSO objective function values that monotonically decrease. In addition, we formulate an extension of the LMM algorithm for reconstructing ground penetrating radar (GPR) images, that is much faster than the standard LMM algorithm and utilizes significantly less memory. Thus, the GPR specific LMM (GPR-LMM) algorithm is able to accommodate the big data associated with GPR imaging. We compare our proposed algorithms to the state-of-the-art ℓ1-regularized LS algorithms using a time and space complexity analysis. The GPR-LMM greatly outperforms the competing algorithms in terms of the performance metrics we considered. In addition, the reconstruction results of the standard LMM and GPR-LMM algorithms are evaluated using both simulated and real GPR data.

Through-Wall UWB Radar Based on Sparse Deconvolution with Arctangent Regularization for Locating Human Subjects

“…Next, the calibrated signal was down-sampled by "downsample" command MATLAB, where the normalized frequency (ω 0 ) was evaluated in Equation (30), ω 0 = 2π3 × 10 9 × 3.125 × 10 −12 × 4 = 0.2356 rad/sample with f c = 3 GHz, ∆t = 3.125 ps, and r down = 4. The down-sampling directly reduced the calcaultion time of the inverse matrix of the sparse deconvolution, from (N × N) to (N/4 × N/4), dimension while maintaing the Nyquist sampling condition [37].…”

“…Nowadays, sparse deconvolution plays an important role in extracting the original data from the noisy received signal; it has been widely used in denoising, interpolation, super-resolution, and declipping [16][17][18][19][20][21][22][23][24][25][26]. Whereas linear time-invariant (LTI) filters, such as low-pass, band-pass and high pass, have amplitude distortions on the original signal resolution [27][28][29][30][31][32][33][34][35].…”

“…The sparse deconvolution algorithm is a numerical method to restore the original signal by optimization formulation with L1 norm regularization. L1 norm regularization could effectively remove noise and rectify parts of signal distortion [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. Meanwhile, L2 norm regularization fails to remove noise, while aggravating signal distortion, and the Lp pseudo-norm regularization (0 < p < 1) completely removes noise and corrects most of the signal distortion [20,21].…”

Weak Fault Detection for Gearboxes Using Majorization–Minimization and Asymmetric Convex Penalty Regularization

“…Before the derivation of the proposed algorithm, a review to the majorization-minimization (MM) algorithm is given. Usually, the MM algorithm is used to reduce the computational complexity-in other words, for a minimization problem that is too complex to solve directly, the MM method can simplify it into a series of simpler problems by constructing the majorization functions [37][38][39].…”

Fast image reconstruction for near-field terahertz imaging with multistatic non-uniform sparse arrays

“…However, the special multistatic structure of SPA is such that it is not possible to directly apply conventional fast Fourier transform (FT)-based image reconstruction techniques in the nearfield (NF) [19,20]. On the other hand, traditional non-Fourier techniques for scene image reconstruction, such as generalized synthetic aperture focusing technique (GSAFT) [17,21], least-squares [22,23] and matched filter [24,25], have a very high computational time. *a.molaei@qub.ac.uk; fax 44 28 9097 1702; pure.qub.ac.uk/en/persons/amir-masoud-molaei Aiming to address the above, our focus in this paper is on the development of a computationally efficient algorithm for image reconstruction compatible with SPAs in a NF multistatic THz imaging scenario.…”