This paper introduces an image denoising procedure based on a 2D scale-mixing complex-valued wavelet transform. Both the minimal (unitary) and redundant (maximum overlap) versions of the transform are used. The covariance structure of white noise in wavelet domain is established. Estimation is performed via empirical Bayesian techniques, including versions that preserve the phase of the complex-valued wavelet coefficients and those that do not. The new procedure exhibits excellent quantitative and visual performance, which is demonstrated by simulation on standard test images.

We apply a non-parametric regression technique based on second generation wavelets to irregularly spaced network data. Conventional wavelet based non-parametric regression can be modelled as, fi= gi+ isini, where fi= f(ti), gi= g(ti) and i = 1,..., n. Key requirements for this model are that n = 2Jfor some J isin Nopf, data are observed on a regular grid ti= i/n and error term isini~ N(0, sigma2). Communication networks have irregularities that standard wavelet regression cannot handle. We adopt the technique developed by Jansen et al [7], called "lifting one coefficient at a time", for irregularly spaced network data. We demonstrate the linear shrinkage and non-linear single coefficient shrinkage induces large bias in the estimation. We propose a new 'across-scale' block shrinkage method for coefficient processing that produces much better estimates.

MOTIVATION: Many standard statistical techniques are effective on data that are normally distributed with constant variance. Microarray data typically violate these assumptions since they come from non-Gaussian distributions with a non-trivial mean-variance relationship. Several methods have been proposed that transform microarray data to stabilize variance and draw its distribution towards the Gaussian. Some methods, such as log or generalized log, rely on an underlying model for the data. Others, such as the spread-versus-level plot, do not. We propose an alternative data-driven multiscale approach, called the Data-Driven Haar-Fisz for microarrays (DDHFm) with replicates. DDHFm has the advantage of being 'distribution-free' in the sense that no parametric model for the underlying microarray data is required to be specified or estimated; hence, DDHFm can be applied very generally, not just to microarray data. RESULTS: DDHFm achieves very good variance stabilization of microarray data with replicates and produces transformed intensities that are approximately normally distributed. Simulation studies show that it performs better than other existing methods. Application of DDHFm to real one-color cDNA data validates these results. AVAILABILITY: The R package of the Data-Driven Haar-Fisz transform (DDHFm) for microarrays is available in Bioconductor and CRAN.