On the problem of restoring original structure of signals (images) corrupted by noise

In the first part of this paper the problem of detecting piece-wise-linear structure of ID and 2D signals (images) corrupted by heavy Gaussian noise, is considered. In many cases the attribute function (response function) is continuous at the boundary points (lines) and only change of its derivatives indicates the transition from one region of smoothness to another. In the presence of heavy noise it is difficult to detect these changes with the help of a local operator and an approach taking into account the global behavior of signal should be introduced. The suggested approach is based on a combination of a least square (LS) criterion, dynamic programming and a probabilistic estimate for model selection. In the second part of the paper the signals corrupted by spikes are considered. In these cases the approaches based on the use of LS estimators are not efficient. It is shown that in the case of piece-wise-linear signals corrupted with such a noise the potentialities of known methods of Robust Regression are restricted. A modification of the Hough Transform is introduced as robust method for outliers detection. Following this procedure the outliers may be detected and excluded. The piece-wise-linear structure of 1D signals may be detected either by associating different linear pieces of the signal with the corresponding maxima in an accumulator array or (if the level of additional background noise is relatively high and the maxima are blurred) with the help of the above discribed methods applied to the signal cleaned of outliers. Results of experiments with 1D signals and 2D images are presented.