Abstract
The Split Bregman method (SBM), a popular and universal CS reconstruction algorithm for inverse problems with both l1-norm and TV-norm regularization, has been extensively applied in complex domains through the complex-to-real transforming technique, e.g., MRI imaging and radar. However, SBM still has great potential in complex applications due to the following two points; Bregman Iteration (BI), employed in SBM, may not make good use of the phase information for complex variables. In addition, the converting technique may consume more time. To address that, this paper presents the complex-valued Split Bregman method (CV-SBM), which theoretically generalizes the original SBM into the complex domain. The complex-valued Bregman distance (CV-BD) is first defined by replacing the corresponding regularization in the inverse problem. Then, we propose the complex-valued Bregman Iteration (CV-BI) to solve this new problem. How well-defined and the convergence of CV-BI are analyzed in detail according to the complex-valued calculation rules and optimization theory. These properties prove that CV-BI is able to solve inverse problems if the regularization is convex. Nevertheless, CV-BI needs the help of other algorithms for various kinds of regularization. To avoid the dependence on extra algorithms and simplify the iteration process simultaneously, we adopt the variable separation technique and propose CV-SBM for resolving convex inverse problems. Simulation results on complex-valued l1-norm problems illustrate the effectiveness of the proposed CV-SBM. CV-SBM exhibits remarkable superiority compared with SBM in the complex-to-real transforming technique. Specifically, in the case of large signal scale n = 512, CV-SBM yields 18.2%, 17.6%, and 26.7% lower mean square error (MSE) as well as takes 28.8%, 25.6%, and 23.6% less time cost than the original SBM in 10 dB, 15 dB, and 20 dB SNR situations, respectively.
Highlights
Compressed sensing (CS) theory has been thoroughly analyzed and extensively applied in the signal processing [1,2] and image processing community [3,4,5] during the past decades
Split Bregman method (SBM) still has great potential in terms of both reconstruction performance and time cost considering the following two points: The original Bregman Iteration (BI) defined in the real domain may not make good use of the phase information for complex variables, which degrades the recovery accuracy; secondly, the converting technique quadruples the elements of the sensing matrix A to 2m × 2n, which consumes more memory and time within the iteration process
The tests mentioned above have shown that the proposed complex-valued Split Bregman method (CV-SBM) presents remarkable performance compared with RV-SBM in the same experimental environment
Summary
Compressed sensing (CS) theory has been thoroughly analyzed and extensively applied in the signal processing [1,2] and image processing community [3,4,5] during the past decades. The Split Bregman method (SBM) proposed in [28] is a universal convex optimization algorithm for both l1 -norm and TV-norm regularization problems. SBM still has great potential in terms of both reconstruction performance and time cost considering the following two points: The original BI defined in the real domain may not make good use of the phase information for complex variables, which degrades the recovery accuracy; secondly, the converting technique quadruples the elements of the sensing matrix A to 2m × 2n, which consumes more memory and time within the iteration process. Inspired by SBM, we adopt the variable separation technique to avoid the requirement of other optimization algorithms and present CV-SBM to settle the convex inverse problems with the simplified solution.
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