Denoising by singularity rejection

In this paper, a new algorithm for noise reduction using the wavelet transform is proposed. The new approach can be viewed as a combination of Mallat and Donoho's de-noising methods. Similar to Mallat's approach, we estimate the regularity of a signal from the evolution of its wavelet transform coefficients across scales. However, we do not perform maxima detection and processing whence complicated reconstruction is avoided. Instead, we propose to estimate the regularity of a signal by computing the sum of the "cone of influence" and select the coefficients which correspond to the regular part of the signal for reconstruction. Using a non-decimated wavelet with only one vanishing moment, the method can give an improved de-noising result as compared with the previous methods in terms of mean squared error and visual quality. The new de-noising method is also invariant to translation. It does not introduce spurious oscillations and requires very few parameters to be specified. Besides, we extend the method to two-dimensions using a "directional sum of cone" method. The denoising technique is applied to tomographic image reconstruction where the improved performance of the new approach can clearly be observed.