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On the basis of wavelet theory, we propose an outlier-detection algorithm for satellite gravity gradiometry by applying a wavelet-shrinkage-de-noising method to some simulation data with white noise and outliers. The result shows that this novel algorithm has a 97% success rate in outlier identification and that it can be efficiently used for pre-processing real satellite gravity gradiometry data.

In Medical diagnosis operations such as feature extraction and object recognition will play the key role. These tasks will become difficult if the images are corrupted with noise. So the development of effective algorithms for noise removal became an important research area in present days. Developing Image denoising algorithms is a difficult task since fine details in a medical image embedding diagnostic information should not be destroyed during noise removal. Many of the wavelet based denoising algorithms use DWT (Discrete Wavelet Transform) in the decomposition stage are suffering from shift variance and lack of directionality. To overcome this in this paper we are proposing the denoising method which uses dual tree complex wavelet transform to decompose the image and shrinkage operation to eliminate the noise from the noisy image. In the shrinkage step we used semi-soft and stein thresholding operators along with traditional hard and soft thresholding operators and verified the suitability of dual tree complex wavelet transform for the denoising of medical images. The results proved that the denoised image using DTCWT (Dual Tree Complex Wavelet Transform) have a better balance between smoothness and accuracy than the DWT and less redundant than UDWT (Undecimated Wavelet Transform). We used the SSIM (Structural similarity index measure) along with PSNR (Peak signal to noise ratio) to assess the quality of denoised images.

By substituting an anisotropic diffusion operator for the isotropic Laplace operator in the directed diffusion equation, adding two different coefficients in the two diffusion terms, and choosing the image denoised by soft wavelet shrinkage as the initial approximate image, an improved directed diffusion equation model is proposed. Experiments show that the new model can solve the problem that edges and details will be rapidly blurred during the diffusion process in the directed diffusion equation.

Consider an asymmetric Linex loss. We provide the soft wavelet shrinkage estimation of a Bayesian interpretation under such a loss.

The Gauss–Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss–Markov theorem to include nonparametric mixed-effects models. The extended Gauss–Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss–Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented.

We propose methods of both slow and rapid post-processing of signals for erasure of artifacts that arise in the process of thresholding and quantization. We use wavelets as tools to define constraints and variational functionals as measures of complexity of signals. The methods come from analyses of different possibilities of blending variational calculus and wavelet multiresolution in ways that appear to be natural.

Wavelet shrinkage estimators are obtained by applying a shrinkage rule on the empirical wavelet coefficients. Such simple estimators are now well explored and widely used in wavelet-based nonparametrics. Results of Tao (1996, Appl. Comput. Harmon. Anal.3, 384–387) demonstrated that hard and soft thresholding shrinkage estimators absolutely converge almost everywhere to the original function when the threshold value goes to zero. Such natural and intuitive behavior of threshold estimators is expected, yet this result does not translate to the Fourier expansions. In this paper we show that almost everywhere convergence of shrinkage estimators holds for a range of shrinkage rules, not necessarily thresholding, subject to some mild technical conditions. Comments about the norm convergence are provided as well.

Two-dimensional element distributions generated by scanning secondary ion mass spectrometry (SIMS) are characterised by Poisson statistics of small integer values, specially when the concentration of the measured element is in the sub-ppm range. To achieve high signal-to-noise ratios extremely long measurement time is needed. Because of the signal fluctuations from one measurement point to the next, structures even larger than the resolution of the instrument may not be detectable if the differences in concentration are small. This paper reports the application of a wavelet shrinkage algorithm for de-noising of images following Poisson distribution. In reconstructions of SIMS images resulting from this algorithm the noise is significantly suppressed without great loss of lateral resolution.