Undecimated wavelet shrinkage estimate of the 1D and 2D spectra

We study the problem of estimating the log-spectrum of a stationary Gaussian time series by thresholding the wavelet coefficients. We propose the use of the undecimated wavelet transform to denoise the log-periodogram. For this, we review a denoising method based on undecimated wavelet transform, and we propose a level-dependent threshold which considers that one undecimated scale has N/b coefficients “repeating” b times. The result corresponds to the average of all log-peridogram circulant shifts denoised by a decimated wavelet transform. The purpose of this undecimated thresholding is to make the reconstructed log-spectrum as nearly noise-free as possible, but with a keep of all small frequential components. Since the wavelet denoising method can be generalized to images, we develop an estimation technique of the 2D log-spectrum based on 2D undecimated wavelet. We derive a new technique, easy to apply, which gives information about the 2D frequential components of an image