Denosing Using Wavelets and Projections onto the ` 1 - Ball

Both wavelet denoising and denoising methods using the concept of sparsity are based on softthresholding. In sparsity-based denoising methods, it is assumed that the original signal is sparse in some transform domains such as the Fourier, DCT, and/or wavelet domain. The transfer domain coefficients of the noisy signal are projected onto `1-balls to reduce noise. In this lecture note, we establish the relation between the standard soft-thresholding-based denoising methods and sparsity-based wavelet denoising. We introduce a new deterministic soft-threshold estimation method using the epigraph set of `1-ball cost function. It is shown that the size of the `1-ball determined using linear algebra. The size of the `1-ball in turn determines the soft threshold. The key step is an orthogonal projection onto the epigraph set of the `1-norm cost function.