# Threshold both detail and analysis coefficients?

Suppose I perform a discrete wavelet transform on some noisy signal $$x$$ and obtain $${a_1, d_1, d_2, \ldots, d_n}$$ where $$a_1$$ is the set of analysis coefficients and $$\{d_j\}$$ are the detail coefficients.

Suppose I wish to denoise the signal by thresholding of some sort. Many authors recommend only thresholding the detail coefficients $$d_{j}$$, and leaving the analysis coefficients alone. The reasoning is that the analysis coefficients are "low frequency", and that noise is "high frequency".

However, in the case of AWGN, noise is contained in every frequency, and since the DWT is orthogonal, noise is just as likely to end up in the analysis as the detail coefficients. In addition, I was under the impression that the thoughts one should have in the wavelet domain were about sparsity, not frequency. So I am confused as to whether I should threshold the whole vector or preserve the analysis coefficients.

Has a comparison of thresholding analysis coefficients vs leaving them alone been performed? Is there a consensus among experts on which way it should be done?

• In denosing task, you want to preserve the shape of the signal and remove the high frequency noise. If you remove the analysis coefficients, then automatically the denoised signal does not preserve the shape of the original signal. The main issue with wavelet denoising technique is that you observe Gibbs phenomenon at the edges. I'd recommend you to look at the following example – Maxtron Dec 20 '18 at 5:44
• I want to remove low frequency noise too. I don't care what frequency the noise is. I don't want to remove the analysis coefficients, I want to threshold them. – user14717 Dec 20 '18 at 17:44