A Convex Optimization Algorithm for Compressed Sensing in a Complex Domain: The Complex-Valued Split Bregman Method.

Area of Science:

• Computational Imaging
• Signal Processing
• Optimization Theory

Background:

• The Split Bregman method (SBM) is widely used for inverse problems with l1-norm and TV-norm regularization, often applied to complex domains via complex-to-real transformation.
• Existing SBM methods may not fully leverage phase information in complex variables and can be computationally intensive due to transformation techniques.

Purpose of the Study:

• To develop a generalized Split Bregman method (CV-SBM) that operates directly in the complex domain, enhancing reconstruction for complex-valued inverse problems.
• To improve the utilization of phase information and reduce computational overhead compared to existing SBM techniques.

Main Methods:

• Definition of complex-valued Bregman distance (CV-BD) by adapting regularization terms for complex variables.
• Proposal of complex-valued Bregman Iteration (CV-BI) for solving the new complex-valued optimization problem.
• Adoption of variable separation technique to create the final complex-valued Split Bregman method (CV-SBM) for efficient convex inverse problem resolution.

Main Results:

• Theoretical analysis confirms the well-definedness and convergence properties of CV-BI under convex regularization.
• Simulations on complex-valued l1-norm problems demonstrate the effectiveness of CV-SBM.
• CV-SBM achieved significant improvements over traditional SBM, showing 17.6-26.7% lower mean square error and 23.6-28.8% less time cost in various SNR conditions for n=512.

Conclusions:

• The proposed complex-valued Split Bregman method (CV-SBM) offers a theoretically sound and practically superior alternative for complex-valued inverse problems.
• CV-SBM effectively utilizes complex-valued information, leading to enhanced accuracy and computational efficiency, particularly beneficial for large-scale imaging and radar applications.