Mitigating the Drawbacks of the L₀ Norm and the Total Variation Norm

Area of Science:

• Image reconstruction
• Compressed sensing
• Signal processing

Background:

• The L0 norm minimization is considered optimal for enforcing sparse solutions in compressed sensing.
• However, L0 norm minimization is computationally challenging for gradient-based iterative algorithms.
• Total Variation (TV) norm minimization is a common alternative but may not sufficiently enforce piecewise-constant images.

Purpose of the Study:

• To investigate the effectiveness of L0 norm minimization for enforcing piecewise-constant images in limited-angle tomography.
• To address the gradient issues of L0 norm minimization in iterative algorithms.
• To propose a novel approach that improves upon existing methods for sparse signal reconstruction.

Main Methods:

• Utilizing limited-angle tomography to demonstrate the application of L0 norm minimization.
• Introducing a novel technique by replacing the zero derivative of the L0 norm with a zero-mean random variable.
• Developing a gradient-based iterative image reconstruction algorithm incorporating this randomized L0 norm.

Main Results:

• Computer simulations show that the proposed L0 norm minimization method outperforms TV minimization.
• The novel approach successfully encourages piecewise-constant images.
• Quantitative evaluations demonstrate improvements in Structural Similarity (SSIM) and Peak Signal-to-Noise Ratio (PSNR).

Conclusions:

• The proposed randomized L0 norm minimization is a viable and effective alternative to TV norm minimization for image reconstruction.
• The introduction of randomness in the gradient of the objective function overcomes the limitations of L0 norm derivatives.
• The method shows significant potential for improving image quality in compressed sensing applications, particularly in limited-angle tomography.