Fast Interscale Wavelet Denoising of Poisson-Corrupted Images

Thierry Blu, Florian Luisier, Michaël Unser, Cédric René Jean Vonesch
Elsevier, 2010
Journal paper

Abstract

We present a fast algorithm for image restoration in the presence of Poisson noise. Our approach is based on (1) the minimization of an unbiased estimate of the MSE for Poisson noise, (2) a linear parametrization of the denoising process and (3) the preservation of Poisson statistics across scales within the Haar DWT. The minimization of the MSE estimate is performed independently in each wavelet subband, but this is equivalent to a global image-domain MSE minimization, thanks to the orthogonality of Haar wavelets. This is an important difference with standard Poisson noise-removal methods, in particular those that rely on a non-linear preprocessing of the data to stabilize the variance. Our non-redundant interscale wavelet thresholding outperforms standard variance-stabilizing schemes, even when the latter are applied in a translation-invariant setting (cycle-spinning). It also achieves a quality similar to a state-of-the-art multiscale method that was specially developed for Poisson data. Considering that the computational complexity of our method is orders of magnitude lower, it is a very competitive alternative. The proposed approach is particularly promising in the context of low signal intensities and/or large data sets. This is illustrated experimentally with the denoising of low-count fluorescence micrographs of a biological sample.

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Fast and automated wavelet-regularized image restoration in fluorescence microscopy

Cédric René Jean Vonesch

The subject of this thesis is image restoration, that is, deconvolution and denoising. Our work is motivated by applications in the rapidly expanding field of biological imaging and in particular fluorescence microscopy. Inverse problems in this area typically involve regularly structured but high-dimensional data sets, thus requiring a specific tradeoff between restoration quality, computational complexity and user interaction. Our interest lies in novel algorithmic solutions for addressing these challenges. The common point of our developments is that they rely centrally on wavelets and multilevel concepts. The thesis is organized in three main parts. In the first part, we provide a general overview of fluorescence microscopy in the context of biological research. We then review the current state of the art in deconvolution microscopy, within a unifying variational framework. The second part is concerned with deconvolution using a non-quadratic (typically ℓ1) wavelet-domain regularization. In the context of deconvolution microscopy, this approach is novel. It amounts to imposing a sparsity constraint on the object to be recovered, a concept that has recently attracted considerable interest in signal and image processing. This type of regularization was shown to belong to the most effective restoration methods, but it involves a computationally challenging optimization problem. We propose a multilevel version of the standard "thresholded Landweber" algorithm that yields a substantial acceleration. This allows us to apply wavelet regularization to large-scale deconvolution problems encountered in deconvolution microscopy. In the third part, we present some applications of risk estimation techniques for automating the process of image restoration. We first propose a denoising method that is specifically designed for very low light intensities. The algorithm is designed from the beginning for complete self-adjustment of its parameters and for low computational complexity. It also performs competitively with most existing approaches for Poisson intensity estimation. Our second contribution is a general method for estimating the signal-to-noise ratio during the execution of iterative image-restoration procedures; it is applicable to most algorithms emanating from variational formulations, in particular those involving wavelet regularization.

EPFL2009

Denoising is an essential step prior to any higher-level image-processing tasks such as segmentation or object tracking, because the undesirable corruption by noise is inherent to any physical acquisition device. When the measurements are performed by photosensors, one usually distinguish between two main regimes: in the first scenario, the measured intensities are sufficiently high and the noise is assumed to be signal-independent. In the second scenario, only few photons are detected, which leads to a strong signal-dependent degradation. When the noise is considered as signal-independent, it is often modeled as an additive independent (typically Gaussian) random variable, whereas, otherwise, the measurements are commonly assumed to follow independent Poisson laws, whose underlying intensities are the unknown noise-free measures. We first consider the reduction of additive white Gaussian noise (AWGN). Contrary to most existing denoising algorithms, our approach does not require an explicit prior statistical modeling of the unknown data. Our driving principle is the minimization of a purely data-adaptive unbiased estimate of the mean-squared error (MSE) between the processed and the noise-free data. In the AWGN case, such a MSE estimate was first proposed by Stein, and is known as "Stein's unbiased risk estimate" (SURE). We further develop the original SURE theory and propose a general methodology for fast and efficient multidimensional image denoising, which we call the SURE-LET approach. While SURE allows the quantitative monitoring of the denoising quality, the flexibility and the low computational complexity of our approach are ensured by a linear parameterization of the denoising process, expressed as a linear expansion of thresholds (LET).We propose several pointwise, multivariate, and multichannel thresholding functions applied to arbitrary (in particular, redundant) linear transformations of the input data, with a special focus on multiscale signal representations. We then transpose the SURE-LET approach to the estimation of Poisson intensities degraded by AWGN. The signal-dependent specificity of the Poisson statistics leads to the derivation of a new unbiased MSE estimate that we call "Poisson's unbiased risk estimate" (PURE) and requires more adaptive transform-domain thresholding rules. In a general PURE-LET framework, we first devise a fast interscale thresholding method restricted to the use of the (unnormalized) Haar wavelet transform. We then lift this restriction and show how the PURE-LET strategy can be used to design and optimize a wide class of nonlinear processing applied in an arbitrary (in particular, redundant) transform domain. We finally apply some of the proposed denoising algorithms to real multidimensional fluorescence microscopy images. Such in vivo imaging modality often operates under low-illumination conditions and short exposure time; consequently, the random fluctuations of the measured fluorophore radiations are well described by a Poisson process degraded (or not) by AWGN. We validate experimentally this statistical measurement model, and we assess the performance of the PURE-LET algorithms in comparison with some state-of-the-art denoising methods. Our solution turns out to be very competitive both qualitatively and computationally, allowing for a fast and efficient denoising of the huge volumes of data that are nowadays routinely produced in biomedical imaging.

EPFL2010

Fast Haar-Wavelet Denoising Of Multidimensional Fluorescence Microscopy Data

Thierry Blu, Florian Luisier, Michaël Unser, Cédric René Jean Vonesch

We propose a novel denoising algorithm to reduce the Poisson noise that is typically dominant in fluorescence microscopy data. To process large datasets at a low computational cost, we use the unnormalized Haar wavelet transform. Thanks to some of its appealing properties, independent unbiased MSE estimates can be derived for each subband. Based on these Poisson unbiased MSE estimates, we then optimize linearly parametrized interscale thresholding. Correlations between adjacent images of the multidimensional data are accounted for through a sliding window approach. Experiments on simulated and real data show that the proposed solution is qualitatively similar to a state-of-the-art multiscale method, while being orders of magnitude faster.

Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa2009

Fast and automated wavelet-regularized image restoration in fluorescence microscopy

Cédric René Jean Vonesch

EPFL2009

EPFL2010