Denosing Using Wavelets and Projections onto the L1-Ball

Both wavelet denoising and denosing 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 wavelet domain and the wavelet subsignals of the noisy signal are projected onto `1-balls to reduce noise. In this lecture note, it is shown that the size of the `1-ball or equivalently the soft threshold value can be determined using linear algebra. The key step is an orthogonal projection onto the epigraph set of the `1 norm cost function. In standard wavelet denoising, a signal corrupted by additive noise is wavelet transformed and resulting wavelet subsignals are soft and/or hard thresholded. After this step the denoised signal is reconstructed from the thresholded wavelet subsignals [1, 2]. Thresholding the wavelet coefficients intuitively makes sense because wavelet subsignals obtained from an orthogonal or biorgthogonal wavelet filterbank exhibit large amplitude coefficients only around edges or change locations of the original signal. Other small amplitude coefficients should be due to noise. Many other related wavelet denoising methods are developed based on Donoho and Johnstone’s idea, see e.g. [1–6]. Most denoising methods take advantage of sparse nature of practical signals in wavelet domain to reduce the noise [7–12]. Consider the following basic denoising framework. Let v[n] be a discrete-time signal and x[n] be a noisy version of v[n]: x[n] = v[n] + ξ[n], n = 0, 1, 2, . . . , N − 1. (1)