Statistics for sparse transforms: some recent results with application to speech and image processing
Résumé
A transform is said to be sparse (or to achieve a sparse representation) if it can represent a signal by a relatively small number of large coefficients. In this qualitative sense, the wavelet transform is often regarded as sparse [1]. Under this sparsity model, [1] presents thresholding and shrinkage methods aimed at estimating a signal corrupted by independent and additive white Gaussian noise. These methods, called WaveShrink estimators, involve forcing to zero, the wavelet coefficients with amplitudes less than a certain threshold. For the threshold, [1] recommends the universal or minimax thresholds. On the other hand, a non-parametric approach is proposed in [2] for the detection of any random variable that has unknown distribution, amplitude larger than or equal to some known value A and probability of presence less than or equal to one half in additive white Gaussian noise with standard deviation .. The test proposed in [2] for detecting the realization of this random variable is a thresholding test with a specific threshold height that depends on A and .: the presence of the random variable (wich correspond to presence of usefull information in an observed wavelet coefficient) is accepted if the amplitude of the observation is above the threshold height and rejected otherwise. In a first part of the presentation, we describe how the non-parametric statistical decision test introduced in [2] can be used to estimate significant information in noisy wavelet coefficients via the most popular WaveShrink estimator, namely the soft thresholding, which is one of the approaches proposed in [1]. As far as the threshold height required by soft thresholding is concerned, theoretical results as well as tests on the signals of the WaveLab Toolbox suggest using the threshold height derived in [2, Theorem VII.1] instead of the standard universal and minimax thresholds. Experimental results in image denoising will also be presented so as to illustrate the conclusions of this first part. In a second part, we address the following problem. The results presented above are derived under the assumption that the noise standard deviation is known. In practice, this value is often unknown. As suggested in [3], the Maximum Absolute Deviation (MAD) estimator, calculated over the detail coefficients of the first level of the wavelet decomposition, is a good estimate of . when there is no significant information about the signal in the detail wavelet coefficients. Some theoretical results recently established in [4] make it possible to estimate the noise standard deviation when significant information about the signal are present in the noisy detail wavelet coefficients. These results relate to sparsity in the sense that they hold true for signals with unknown distriubutions that are less present than absent and whose minimum ammplitude is above some positive real value. In fact, these results can apply to transforms other than the wavelet transform. We present an application of these results to speech processing as well as an extension to coloured noise. The conclusion of this talk will present extensions of the results presented above. In particular, a new family of shrinkages will be briefly described for non-parametric estimation of a signal in ndependent and additive white Gaussian noise. Some examples in image denoising will be commented so as to emphasize the new features of these shrinkages. This is a joint work with Abdourrahmane M. Atto.