Generalized cross validation for wavelet thresholding

Noisy data are often fitted using a smoothing parameter, controling the importance of two objectives that are opposite to a certain extent. One of these two is smoothness and the other is closeness to the input data. The optimal value of this paramater minimizes the error of the result (as compared to the unknown, exact data), usually expressed in the L2 norm. This optimum cannot be found exactly, simply because the exact data are unknown. In spline theory, the Generalized Cross Validation (GCV) technique has proven to be an effective (though rather slow) statistical way for estimating this optimum. On the other hand, wavelet theory is well suited for signal and image processing. This paper investigates the possibility of using GCV in a noise reduction algorithm, 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. Both an original theoretical argument and practical experiments are used to show this successful combination.