Optimizing the resolution level in wavelet transform-applied to image restoration

We empirically show that there exists a better threshold than the minimax or universal estimators for resolution cutoff J in the wavelet transform. Tests are conducted on de-noising both 1-D and 2-D signals to which Gaussian white noise is added. Donoho (1992) and Donoho and Johnstone (1994, 1995) proposed the Waveshrink method for reconstructing an unknown function from input data corrupted by Gaussian noise. The Waveshrink has three steps: (i) transform input noisy data into empirical wavelet coefficients wavelet domain; (ii) threshold the results from (i); and (iii) inverse transform the shrunk coefficients. The threshold in step (ii) is estimated from the wavelet coefficients of a given low-resolution cutoff J₀. In Waveshrink, such obtained threshold is applicable only to the empirical wavelet coefficients of levels J, J>J₀. It is unknown, however, what value of J, produces the optimal threshold. Furthermore, our results show that the threshold value either estimated by minimax or universal estimator is often located too far away from the histogram edge of the wavelet coefficient, as a result, optimal performance of de-noising cannot be achieved. Wavelet coefficients of 1-D and 2-D signals are compared too