On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs

Soft wavelet shrinkage, total variation (TV) diffusion, total variation regularization, and a dynamical system called SIDEs are four useful techniques for discontinuity preserving denoising of signals and images. In this paper we investigate under which circumstances these methods are equivalent in the 1-D case. First we prove that Haar wavelet shrinkage on a single scale is equivalent to a single step of space-discrete TV diffusion or regularization of two-pixel pairs. In the translationally invariant case we show that applying cycle spinning to Haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of TV diffusion. We prove that space-discrete TV difusion and TV regularization are identical, and that they are also equivalent to the SIDEs system when a specific force function is chosen. Afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the Laplacian pyramid of the signal. We analyse possibilities to avoid Gibbs-like artifacts for multiscale Haar wavelet shrinkage by scaling the thesholds. Finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and PDE / variational approaches. These methods are based on iterated shift-invariant wavelet shrinkage at multiple scales with scaled thresholds

Improving the performance of translation wavelet transform using BMICA

Research has shown Wavelet Transform to be one of the best methods for denoising biosignals. Translation-Invariant form of this method has been found to be the best performance. In this paper however we utilize this method and merger with our newly created Independent Component Analysis method – BMICA. Different EEG signals are used to verify the method within the MATLAB environment. Results are then compared with those of the actual Translation-Invariant algorithm and evaluated using the performance measures Mean Square Error (MSE), Peak Signal to Noise Ratio (PSNR), Signal to Distortion Ratio (SDR), and Signal to Interference Ratio (SIR). Experiments revealed that the BMICA Translation-Invariant Wavelet Transform out performed in all four measures. This indicates that it performed superior to the basic Translation- Invariant Wavelet Transform algorithm producing cleaner EEG signals which can influence diagnosis as well as clinical studies of the brain

Wavelet methods in statistics: Some recent developments and their applications

The development of wavelet theory has in recent years spawned applications in signal processing, in fast algorithms for integral transforms, and in image and function representation methods. This last application has stimulated interest in wavelet applications to statistics and to the analysis of experimental data, with many successes in the efficient analysis, processing, and compression of noisy signals and images. This is a selective review article that attempts to synthesize some recent work on ``nonlinear'' wavelet methods in nonparametric curve estimation and their role on a variety of applications. After a short introduction to wavelet theory, we discuss in detail several wavelet shrinkage and wavelet thresholding estimators, scattered in the literature and developed, under more or less standard settings, for density estimation from i.i.d. observations or to denoise data modeled as observations of a signal with additive noise. Most of these methods are fitted into the general concept of regularization with appropriately chosen penalty functions. A narrow range of applications in major areas of statistics is also discussed such as partial linear regression models and functional index models. The usefulness of all these methods are illustrated by means of simulations and practical examples.Comment: Published in at http://dx.doi.org/10.1214/07-SS014 the Statistics Surveys (http://www.i-journals.org/ss/) by the Institute of Mathematical Statistics (http://www.imstat.org

Integrodifferential equations for multiscale wavelet shrinkage : the discrete case

We investigate the relations between wavelet shrinkage and integrodifferential equations for image simplification and denoising in the discrete case. Previous investigations in the continuous one-dimensional setting are transferred to the discrete multidimentional case. The key observation is that a wavelet transform can be understood as derivative operator in connection with convolution with a smoothing kernel. In this paper, we extend these ideas to the practically relevant discrete formulation with both orthogonal and biorthogonal wavelets. In the discrete setting, the behaviour of the smoothing kernels for different scales is more complicated than in the continuous setting and of special interest for the understanding of the filters. With the help of tensor product wavelets and special shrinkage rules, the approach is extended to more than one spatial dimension. The results of wavelet shrinkage and related integrodifferential equations are compared in terms of quality by numerical experiments