Denoising by singularity detection

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 denoising 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, and therefore, complicated reconstruction is avoided. Instead, we propose to estimate the regularity of a signal by computing the sum of the modulus of its wavelet coefficients inside the corresponding "cone of influence" and select the coefficients that correspond to the regular part of the signal for reconstruction. In the selection procedure, we propose to use both the techniques of "interscale ratio" and "interscale difference" to obtain the required wavelet coefficients. The algorithm gives an improved denoising result as compared with the previous approaches in terms of mean squared error and visual quality. The new denoising algorithm is also invariant to translation. It does not introduce spurious oscillations and requires very little a priori information of the signal or noise. Besides, we extend the method to two-dimensions to estimate the regularity of an image by computing the sum of the modulus of its wavelet coeflicients inside the so-called "directional cone of influence." The denoising technique is applied to tomographic image reconstruction, where the improved performance of the new approach can clearly be observed.