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Wavelet Shrinkage using DWT has been widely used in de-noising although DWT has a translation variance problem. In this study, we solve this problem by using the translation invariant DWT. For this purpose, we propose a new complex wavelet, the Real-Imaginary Spline Wavelet (RI-Spline wavelet). We also propose the Coherent Dual-Tree algorithm for the RI-Spline wavelet and extend it to the 2-Dimensional. Then we apply this translation invariant RI-Spline wavelet for translation invariant de-noising. Experimental results show that our method, when applied to ECG data, the medical image and the textile surface inspection can obtain better de-noising results than that of conventional Wavelet Shrinkage.

Medical images generally have low contrast and they get complex type of noise due to the use of various devices and applications of various algorithms. However, most of the denoising methods consider only additive noise or some special noise model dependent on their systems and conditions only. Such methods when applied to real medical images yield poor results.

The present work proposes a method for denoising of medical images using soft-thresholding in wavelet domain on multiple levels. We have developed a method to compute the threshold values for denoising of medical images, which depend on the median as well as the contrast ratio of the wavelet coefficients and also on the level number. We have performed experiments by adding various proportions of Gaussian, Salt-and-Pepper and Speckle noise, and found that the proposed method performs better for these cases. The method is efficient because the threshold values can be calculated directly and it is adaptive as these values depend on mean, median and standard deviation of wavelet coefficients of the particular image. The proposed method also gives a criterion for level-dependent thresholding. Application of the proposed method to Ultrasound, X-ray and MRI images is demonstrated in experiments. In the present work, we have also done studies on how to select the mother wavelet for a particular problem.

Nonlinear diffusion filtering and wavelet shrinkage are two successfully applied methods for discontinuity preserving denoising of signals and images. Recently, relations between both methods have been established taking into account wavelet shrinkage at one scale. In this paper, we propose a new explicit scheme for nonlinear diffusion which directly incorporates ideas from multiscale Haar wavelet shrinkage. We prove that our scheme permits larger time steps while preserving convergence to the mean signal value. Numerical examples demonstrate the behavior of our scheme for two and three scales.

Since Donoho et al. proposed the wavelet thresholding method for signal denoising, many different denoising approaches have been suggested. In this paper, we present three different wavelet shrinkage methods, namely NeighShrink, NeighSure and NeighLevel. NeighShrink thresholds the wavelet coefficients based on Donoho's universal threshold and the sum of the squares of all the wavelet coefficients within a neighborhood window. NeighSure adopts Stein's unbiased risk estimator (SURE) instead of the universal threshold of NeighShrink so as to obtain the optimal threshold with minimum risk for each subband. NeighLevel uses parent coefficients in a coarser level as well as neighbors in the same subband. We also apply a multiplying factor for the optimal universal threshold in order to get better denoising results. We found that the value of the constant is about the same for different kinds and sizes of images. Experimental results show that our methods give comparatively higher peak signal to noise ratio (PSNR), are much more efficient and have less visual artifacts compared to other methods.

In this study, in order to verify the effectiveness of the variable filter band discrete wavelet transform (VFB-DWT) and construction method of the variable-band filter (VBF), a fetal ECG extraction has been carried out and the main results obtained are as follows. The approach to configuration VBF by selecting the frequency band only where the fetal ECG component is present was effective to configure the optimal base sensible signal. The extraction of the fetal ECG was successful by applying the wavelet shrinkage to VFB-DWT, which used the constructed VBF. The information entropy was selected as an evaluation index, and two kinds of ECG signals are used to evaluate the wavelet transform basis between the wavelet packet transform (WPT) and the VFB-DWT. One is a synthesized signal composed of white noise, the maternal ECG and the fetal ECG. The other signal is the real target signal separated by independent component analysis (ICA) and has the mother's body noise, the maternal ECG and the fetal ECG. The result shows that the basis by VBF of the VFB-DWT is better than the basis of the WPT that was chosen by the best basis algorithm (BBA).

Nonparametric regression techniques provide a very effective and simple way of finding structure in data sets without the imposition of a parametric regression model. Wavelet theory has the potential to provide statisticians with powerful new techniques for nonparametric inference. In this paper, we consider the wavelet shrinkage kernel estimator of regression function with a common one-dimensional probability density function. We investigate a new nonparametric curve estimator and convergence ratio of given estimator by using cross-validation method to choice of wavelet threshold when the observations are taken on the regular grid. At the end we used simulation study to examine our proposed estimator. We survey the theoretical outcomes with numerical computation by using $R$ software to compare purpose estimator with another estimators.

Wavelet shrinkage and nonlinear diffusion denoising are two frequently used techniques for denoising images, while preserving their discontinuities. In this paper, we study the relations and differences between these two methods. Relations between 2D discrete Haar wavelet transform and nonlinear diffusion are educed. The results show that discrete 2D scheme for nonlinear diffusion can be explained in wavelet shrinkage schemes, and that nonlinear diffusion is superior to wavelet shrinkage method in image denoising, under certain circumstances.

A novel technique for finding pathogenicity islands in genome data with multiscale neural network (MNN) is present. Firstly, decomposes different scales of genomic sequences signal into different scales of wavelets (local frequencies), and shrinking the coefficients of each scale of wavelets by means of a separate multilayer perception neural network, reconstruct the result sequence signal from the smoothed coefficients by applying the Inverse Wavelet Transform. Secondly,detect G+C patterns in genomes by comparing the result sequence with original sequences. The results on G+C patterns analysis of Dradiodurans chromosome I and N.serogroup A strain Z2491 are presented. The findings show that MNN is a powerful tool to detect pathogenicity islands in genome data.