Sparse Reconstruction Using Hyperbolic Tangent as Smooth l1-Norm Approximation

In the Compressed Sensing (CS) framework, underdetermined system of linear equation (USLE) can have infinitely many possible solutions. However, we intend to find sparsest solution, which is l0-norm minimization. finding an l0 norm solution out solutions NP-hard problem that becomes non-convex optimization problem. It has been a practically proven fact penalty be adequately estimated by l1 norm, recasts minimization convex non-differentiable and gradient-based algorithms are not applicable, due this very reason there need approximate its smooth approximation. Iterative shrinkage provide efficient method numerically minimize l1-regularized least square These required induce sparsity in their meet CS recovery requirement. research article, developed novel uses hyperbolic tangent function recover undersampled signal/images framework. our work, soft thresholding both approximated with functions. We also proposed criteria tune parameters get optimal results. The error bounds for approximation evaluated. To evaluate performance method, utilized dataset comprised 1-D sparse signal, compressively sampled MR image cardiac cine MRI. MRI important imaging modality assessing vascular function. provides ejection fraction output heart. advantage comes at cost slow acquisition process. Hence, it essential speed up process take full benefits Numerical results based on metrics, such as Structural Similarity (SSIM), Peak Signal Noise Ratio (PSNR) Root Mean Square Error (RMSE) show offers much better compared traditional Soft Thresholding (IST) methods.