An efficient and effective l₀–l₂ minimisation algorithm for compressive imaging

Abstract

Compressive imaging has been intensively studied during the past few years, capable of reconstructing high-resolution images with sampling ratios far below the Nyquist rate. In contrast to previous works, a new l₀–l₂ minimisation approach is proposed for compressive imaging in this paper, regularised by sparsity constraints in three complementary frames. The new approach stems from the observation that images of practical interest may consist of different morphological components (e.g. point singularities, oscillating textures, curvilinear edges), and therefore, cannot be sparsely represented in one single frame. The alternating split Lagrangian method is further exploited to resolve the l₀–l₂ minimisation problem, leading to an efficient iteration scheme for compressive imaging from partial Fourier data. In addition, we analyse the convergence properties of the proposed algorithm and compare its performance against several recently proposed methods. Numerical simulations on natural and magnetic resonance images show that the proposed approach achieves state-of-the-art performance.

Keywords:

• Compressed sensing
• Compressive imaging
• Tight frame
• Splitting Lagrangian
• Iterative hard thresholding
• Texture preserving