Maarten Jansen

Multiple wavelet threshold estimation by generalized cross validation for images with correlated noise

Denoising algorithms based on wavelet thresholding replace small wavelet coefficients by zero and keep or shrink the coefficients with absolute value above the threshold. The optimal threshold minimizes the error of the result as compared to the unknown, exact data. To estimate this optimal threshold, we use generalized cross validation. This procedure does not require an estimation for the noise energy. Originally, this method assumes uncorrelated noise. In this paper, we describe how we can extend it to images with correlated noise.

Obg and Membrane Depolarization Are Part of a Microbial Bet-Hedging Strategy that Leads to Antibiotic Tolerance

Within bacterial populations, a small fraction of persister cells is transiently capable of surviving exposure to lethal doses of antibiotics. As a bet-hedging strategy, persistence levels are determined both by stochastic induction and by environmental stimuli called responsive diversification. Little is known about the mechanisms that link the low frequency of persisters to environmental signals. Our results support a central role for the conserved GTPase Obg in determining persistence in Escherichia coli in response to nutrient starvation. Obg-mediated persistence requires the stringent response alarmone (p)ppGpp and proceeds through transcriptional control of the hokB-sokB type I toxin-antitoxin module. In individual cells, increased Obg levels induce HokB expression, which in turn results in a collapse of the membrane potential, leading to dormancy. Obg also controls persistence in Pseudomonas aeruginosa and thus constitutes a conserved regulator of antibiotic tolerance. Combined, our findings signify an important step toward unraveling shared genetic mechanisms underlying persistence.

Empirical Bayes approach to improve wavelet thresholding for image noise reduction

Wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients. This is basically a local procedure, because wavelet coefficients characterize the local regularity of a function. Although a wavelet transform has decorrelating properties, structures in images, like edges, are never decorrelated completely, and these structures appear in the wavelet coefficients: a classification based on a local criterion-like coefficient magnitude is not the perfect method to distinguish important, uncorrupted coefficients from coefficients dominated by noise. We therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a Bayesian framework. The local characterization is incorporated into the conditional model, whereas the prior model describes only configurations, not coefficient values. More precisely, local characterization favors configurations with clusters of important coefficients. In this way, we can compute, for each coefficient, the posterior probability of being “sufficiently clean.” This article proposes and motivates the particular and original choice of the conditional model. Instead of introducing this Bayesian framework, we could also apply heuristic image processing techniques to find clustered configurations of large coefficients. This article also explains the benefits of the Bayesian approach compared to these simple techniques. The parameter of the prior model is estimated on an empirical basis using a pseudolikelihood criterion.

Stabilised wavelet transforms for non-equispaced data smoothing

This paper discusses wavelet thresholding in smoothing from non-equispaced, noisy data in one dimension. To deal with the irregularity of the grid we use the so-called second generation wavelets, based on the lifting scheme. The lifting scheme itself leads to a grid-adaptive wavelet transform. We explain that a good numerical condition is an absolute requisite for successful thresholding. If this condition is not satisfied the output signal can show an arbitrary bias. We examine the nature and origin of stability problems in second generation wavelet transforms. The investigation concentrates on lifting with interpolating prediction, but the conclusions are extendible. The stability problem is a cumulated effect of the three successive steps in a lifting scheme: split, predict and update. The paper proposes three ways to stabilise the second generation wavelet transform. The first is a change in update and reduces the influence of the previous steps. The second is a change in prediction and operates on the interval boundaries. The third is a change in splitting procedure and concentrates on the irregularity of the data points. Illustrations show that reconstruction from thresholded coefficients with this stabilised second generation wavelet transform leads to smooth and close fits.

Scattered data smoothing by empirical Bayesian shrinkage of second-generation wavelet coefficients

We propose a novel approach for scattered data smoothing based on second generation wavelets. This wavelet transform automatically adapts to the irregularity of the grid. Our implementation also pays attention to numerical stability, a crucial property in estimation procedures. The wavelet coefficients are shrunk either with simple soft-thresholding or with an empirical Bayesian estimation.

Asymptotic behavior of the minimum mean squared error threshold for noisy wavelet coefficients of piecewise smooth signals

This paper investigates the asymptotic behavior of the minimum risk threshold for wavelet coefficients with additive, homoscedastic, Gaussian noise and for a soft-thresholding scheme. We start from N samples from a signal on a continuous time axis. For piecewise smooth signals and for N/spl rarr//spl infin/, this threshold behaves as C/spl radic/(2logN)/spl sigma/, where /spl sigma/ is the noise standard-deviation. The paper contains an original proof for this asymptotic behavior as well as an intuitive explanation. The paper also discusses the importance of this asymptotic behavior for practical cases when we estimate the minimum risk threshold.

Second-generation wavelet denoising methods for irregularly spaced data in two dimensions

This paper discusses bivariate scattered data denoising. The proposed method uses second-generation wavelets constructed with the lifting scheme. Starting from a simple initial transform, we propose predictor operators based on a stabilized bivariate generalization of the Lagrange interpolating polynomial. These predictors are meant to provide a smooth reconstruction. Next, we include an update step which helps to reduce the correlation amongst the detail coefficients, and hence stabilizes the final estimator. We use a Bayesian thresholding algorithm to denoise the empirical coefficients, and we show the performance of the resulting estimator through a simulation study.

Image de‐noising by integer wavelet transforms and generalized cross validation

De-noising algorithms based on wavelet thresholding replace small wavelet coefficients by zero and keep or shrink the coefficients with absolute value above the threshold. The optimal threshold minimizes the error of the result as compared to the unknown, exact data. To estimate this optimal threshold, we use Generalized Cross Validation. This procedure has linear complexity and is fully automatic, i.e., it does not require an estimate for the noise energy. This paper uses the method for wavelet transforms that map integer gray-scale pixel values to integer wavelet coefficients. An image with artificial noise is used to illustrate the optimality properties of the estimator. Not all theoretical requirements for a successful application of the method are strictly fulfilled in the integer transform case. However, this has little influence on practical results.

Minimum risk thresholds for data with heavy noise

In the estimation of data with many zeros (sparse data), such as wavelet coefficients, thresholding is a common technique. This letter investigates the behavior of the minimum risk threshold for large values of the noise standard deviation. It finds that the threshold depends quadratically on the noise standard deviation. The relevance of this result is situated in the context of both Bayesian and universal thresholding.

A Wavelet Approach for Estimating Chlorophyll-A From Inland Waters With Reflectance Spectroscopy

This letter presents an application of continuous wavelet analysis, providing a new semi-empirical approach to estimate Chlorophyll-a (Chl-a) in optically complex inland waters. Traditionally spectral narrow band ratios have been used to quantify key diagnostic features in the remote sensing signal to estimate concentrations of optically active water quality constituents. However, they cannot cope easily with shifts in reflectance features caused by multiple interactions between variable absorption and backscattering effects that typically occur in optically complex waters. We use continuous wavelet analysis to detect Chl-a features at various wavelengths and frequency scales. Using the wavelet decomposition, we build a 2-D correlation scalogram between in situ pond reflectance spectra and in situ Chl-a concentration. By isolating the most informative wavelet regions via thresholding, we could relate all five regions to known inherent optical properties. We select the optimal feature per region and compare them to three well-known narrow band ratio models. For this experimental application, the wavelet features outperform the NIR-red models, while fluorescence line height (FLH) yield comparable results. Because wavelets analyze the signal at different scales and synthesize information across bands, we hypothesize that the wavelet features are less sensitive to confounding factors, such as instrument noise, colored dissolved organic matter, and suspended matter.