GENERALIZED CROSS-VALIDATION FOR WAVELET THRESHOLDING

Noisy data are often fitted using a smoothing parameter, controlling t he importance of two objectives that are opposite to a certain extent. One of these two is smoothness and the other is closeness to the inpu t data. The optimal value of this paramater minimizes the error of the result (as compared to the unknown, exact data), usually expressed in the L(2) norm. This optimum cannot be found exactly, simply because t he exact data are unknown. In spline theory, the generalized cross val idation (GCV) technique has proved to be an effective (though rather s low) statistical way for estimating this optimum. On the other hand, w avelet theory is well suited for signal and image processing. This pap er investigates the possibility of using GCV in a noise reduction algo rithm, based on wavelet-thresholding, where the threshold can be seen as a kind of smoothing parameter. The GCV method thus allows choosing the (nearly) optimal threshold, without knowing the noise variance. Bo th an original theoretical argument and practical experiments are used to show this successful combination. (C) 1997 Elsevier Science B.V.