Wavelet Transform for Compressive Sensing in Image Processing

Application of Wavelet Transform in Compressive Sensing for Image Processing

Wavelet transform has been widely adopted in signal and image processing due to its excellent time-frequency localization properties. Combined with compressive sensing theory, wavelet transform provides an efficient sparse representation method for images, enabling high-quality reconstruction even at sampling rates far below the Nyquist rate.

The core principle of compressive sensing lies in exploiting signal sparsity, and wavelet transform naturally represents images in a sparse or compressible form in the transform domain. By performing random measurement sampling on wavelet coefficients, data acquisition requirements can be significantly reduced. Subsequently, reconstruction algorithms recover the original image from limited measurements.

Three mainstream algorithms demonstrate distinct characteristics during reconstruction:

Optimization-based methods, such as L1-minimization algorithms, reconstruct sparse signals by solving convex optimization problems. These provide stable recovery performance but involve higher computational complexity. Common implementations use linear programming solvers or specialized optimization packages like CVX in MATLAB.

Iterative thresholding algorithms (e.g., ISTA - Iterative Soft Thresholding Algorithm) achieve reconstruction through alternating gradient descent and soft thresholding operations. They offer higher computational efficiency suitable for large-scale problems but exhibit slower convergence rates. Code implementation typically involves iterative loops with thresholding functions applied to wavelet coefficients.

Greedy pursuit algorithms (like OMP - Orthogonal Matching Pursuit) construct signal representations by progressively selecting the most correlated atoms. They feature simple implementation and fast execution but require measurement matrices with low coherence. Programming implementations often use iterative atom selection and residual update procedures.

Each algorithm has specific advantages, and practical selection depends on computational resources, accuracy requirements, and real-time constraints. The integration of wavelet transform with compressive sensing provides new technical pathways for image acquisition, transmission, and storage, showing significant potential in medical imaging, remote sensing, and related fields.