Image Compressive Sensing with Wavelet Transform

In the field of image processing, the integration of wavelet transform with compressive sensing provides an innovative approach for efficient data compression and reconstruction. Wavelet transform decomposes images into subbands of different frequencies, while compressive sensing leverages signal sparsity to achieve efficient data acquisition at sampling rates significantly below the Nyquist rate. The implementation typically involves applying discrete wavelet transform (DWT) functions like wavedec2() in MATLAB to obtain sparse representations.

The application of wavelet transform in compressive sensing primarily manifests in the sparse representation stage. After undergoing wavelet transformation, image energy concentrates predominantly on a small number of coefficients, demonstrating excellent sparsity in the wavelet domain. This sparsity is precisely the key characteristic required by compressive sensing theory, enabling reconstruction of original signals using far fewer measurements than traditional methods. Code implementation often involves thresholding operations to enhance sparsity using functions like wthresh() for coefficient shrinkage.

The OMP (Orthogonal Matching Pursuit) algorithm serves as a commonly used reconstruction algorithm suitable for compressive sensing frameworks. OMP iteratively selects the most correlated atoms to approximate signals, progressively reconstructing the original image. Built upon wavelet transform, OMP can more effectively utilize signal sparsity to achieve high-quality reconstruction. A typical Python implementation would involve creating a measurement matrix and iteratively solving the optimization problem using orthogonal projection.

The relationship between performance analysis and sampling points constitutes a key research focus. Experimental results demonstrate that as sampling rates increase, reconstructed image quality improves significantly, though this improvement is nonlinear. Within a certain range, increasing sampling points can substantially enhance reconstruction quality metrics like Peak Signal-to-Noise Ratio (PSNR) or Structural Similarity Index (SSIM), but beyond a specific threshold, improvement effects plateau. Therefore, practical applications require balancing sampling rates against reconstruction quality to identify optimal sampling points. Performance evaluation code typically involves calculating PSNR using psnr() functions and SSIM using dedicated image quality assessment toolboxes.

Furthermore, wavelet basis selection significantly impacts compressive sensing performance. Different wavelet bases (such as Haar, Daubechies, etc.) possess varying sparse representation capabilities, further influencing reconstruction outcomes. Thus, practical implementations require selecting appropriate wavelet bases and sampling strategies based on specific requirements to achieve optimal compression and reconstruction results. Code implementations often include comparative analyses using different wavelet families through functions like dwt() with varying wavelet type parameters.