Compressed Sensing Image Reconstruction Using Two-Step Iterative Shrinkage Algorithm and Complex Wavelets

Compressed sensing represents a revolutionary signal acquisition and reconstruction technique that breaks through the limitations of traditional Nyquist sampling theorem. It enables high-quality signal reconstruction at sampling rates significantly below the Nyquist rate. In the field of image processing, compressed sensing technology has gained particular attention due to its ability to substantially reduce data volume in image acquisition and transmission while maintaining high image quality. Code implementation typically involves designing measurement matrices and optimization algorithms to recover sparse signals from limited measurements.

The Two-Step Iterative Shrinkage/Thresholding (TwIST) algorithm is an efficient optimization method specifically designed for solving sparse signal reconstruction problems in compressed sensing. Compared to traditional iterative shrinkage algorithms, TwIST significantly improves convergence speed by introducing a two-step update mechanism, making large-scale image reconstruction problems feasible. The method combines solutions from the previous two iterations at each step, effectively reducing oscillations and achieving faster convergence toward the optimal solution. In MATLAB implementations, TwIST typically requires defining proper shrinkage operators and convergence criteria.

Complex wavelet transforms serve as excellent multi-scale analysis tools for image sparse representation. Compared to real-valued wavelets, complex wavelets demonstrate superior translation invariance and directional selectivity, enabling more effective capture of image edge and texture information. Within the compressed sensing framework, complex wavelets as sparse bases provide more compact signal representations, thereby enhancing reconstructed image quality. Implementation often involves using dual-tree complex wavelet transform (DTCWT) libraries that handle both magnitude and phase information.

The integration of Two-Step Iterative Shrinkage algorithm with complex wavelets creates an efficient compressed sensing image reconstruction methodology. This approach first employs complex wavelets for sparse image representation, then applies the TwIST algorithm in the measurement domain to progressively recover the original image. This combination not only inherits TwIST's rapid convergence characteristics but also fully leverages complex wavelets' advantages in image representation, achieving high-quality, high-efficiency image reconstruction. The algorithm workflow typically includes initialization, iterative updates with shrinkage operations, and convergence checking.

In practical applications, this method proves particularly suitable for medical imaging, remote sensing image processing, and video surveillance scenarios where data acquisition costs are high or transmission bandwidth is limited. By adjusting iteration counts and shrinkage thresholds, practitioners can achieve optimal balance between reconstruction quality and computational efficiency to meet diverse application requirements. Key parameters like regularization weights and stopping criteria can be fine-tuned based on specific image characteristics and noise levels.