Santiago Aja-Fernández

Noise and Signal Estimation in Magnitude MRI and Rician Distributed Images: A LMMSE Approach

A new method for noise filtering in images that follow a Rician model-with particular attention to magnetic resonance imaging-is proposed. To that end, we have derived a (novel) closed-form solution of the linear minimum mean square error (LMMSE) estimator for this distribution. Additionally, a set of methods that automatically estimate the noise power are developed. These methods use information of the sample distribution of local statistics of the image, such as the local variance, the local mean, and the local mean square value. Accordingly, the dynamic estimation of noise leads to a recursive version of the LMMSE, which shows a good performance in both noise cleaning and feature preservation. This paper also includes the derivation of the probability density function of several local sample statistics for the Rayleigh and Rician model, upon which the estimators are built.

On the estimation of the coefficient of variation for anisotropic diffusion speckle filtering

In this paper, we focus on the problem of speckle removal by means of anisotropic diffusion and, specifically, on the importance of the correct estimation of the statistics involved. First, we derive an anisotropic diffusion filter that does not depend on a linear approximation of the speckle model assumed, which is the case of a previously reported filter, namely, SRAD. Then, we focus on the problem of estimation of the coefficient of variation of both signal and noise and of noise itself. Our experiments indicate that neighborhoods used for parameter estimation do not need to coincide with those used in the diffusion equations. Then, we show that, as long as the estimates are good enough, the filter proposed here and the SRAD perform fairly closely, a fact that emphasizes the importance of the correct estimation of the coefficients of variation

Noise-Driven Anisotropic Diffusion Filtering of MRI

A new filtering method to remove Rician noise from magnetic resonance images is presented. This filter relies on a robust estimation of the standard deviation of the noise and combines local linear minimum mean square error filters and partial differential equations for MRI, as the speckle reducing anisotropic diffusion did for ultrasound images. The parameters of the filter are automatically chosen from the estimated noise. This property improves the convergence rate of the diffusion while preserving contours, leading to more robust and intuitive filtering. The partial derivative equation of the filter is extended to a new matrix diffusion filter which allows a coherent diffusion based on the local structure of the image and on the corresponding oriented local standard deviations. This new filter combines volumetric, planar, and linear components of the local image structure. The numerical scheme is explained and visual and quantitative results on simulated and real data sets are presented. In the experiments, the new filter leads to the best results.

Restoration of DWI Data Using a Rician LMMSE Estimator

This paper introduces and analyzes a linear minimum mean square error (LMMSE) estimator using a Rician noise model and its recursive version (RLMMSE) for the restoration of diffusion weighted images. A method to estimate the noise level based on local estimations of mean or variance is used to automatically parametrize the estimator. The restoration performance is evaluated using quality indexes and compared to alternative estimation schemes. The overall scheme is simple, robust, fast, and improves estimations. Filtering diffusion weighted magnetic resonance imaging (DW-MRI) with the proposed methodology leads to more accurate tensor estimations. Real and synthetic datasets are analyzed.

Noise estimation in single- and multiple-coil magnetic resonance data based on statistical models

Noise estimation is a challenging task in magnetic resonance imaging (MRI), with applications in quality assessment, filtering or diffusion tensor estimation. Main noise estimators based on the Rician model are revisited and classified in this article, and new useful methods are proposed. Additionally, all the surveyed estimators are extended to the noncentral chi model, which applies to multiple-coil MRI and some important parallel imaging algorithms for accelerated acquisitions. The proposed new noise estimation procedures, based on the distribution of local moments, show better performance in terms of smaller variance and unbiased estimation over a wide range of experiments, with the additional advantage of not needing to explicitly segment the background of the image.

Estimation of fiber Orientation Probability Density Functions in High Angular Resolution Diffusion Imaging

An estimator of the Orientation Probability Density Function (OPDF) of fiber tracts in the white matter of the brain from High Angular Resolution Diffusion data is presented. Unlike Q-Balls, which use the Funk-Radon transform to estimate the radial projection of the 3D Probability Density Function, the Jacobian of the spherical coordinates is included in the Funk-Radon approximation to the radial integral. Thus, true angular marginalizations are computed, which allows a strict probabilistic interpretation. Extensive experiments with both synthetic and real data show the better capability of our method to characterize complex micro-architectures compared to other related approaches (Q-Balls and Diffusion Orientation Transform), especially for low values of the diffusion weighting parameter.

Image quality assessment based on local variance.

A new and complementary method to assess image quality is presented. It is based on the comparison of the local variance distribution of two images. This new quality index is better suited to assess the non-stationarity of images, therefore it explicitly focuses on the image structure. We show that this new index outperforms other methods for the assessment of image quality in medical images

DWI filtering using joint information for DTI and HARDI.

The filtering of the Diffusion Weighted Images (DWI) prior to the estimation of the diffusion tensor or other fiber Orientation Distribution Functions (ODF) has been proved to be of paramount importance in the recent literature. More precisely, it has been evidenced that the estimation of the diffusion tensor without a previous filtering stage induces errors which cannot be recovered by further regularization of the tensor field. A number of approaches have been intended to overcome this problem, most of them based on the restoration of each DWI gradient image separately. In this paper we propose a methodology to take advantage of the joint information in the DWI volumes, i.e., the sum of the information given by all DWI channels plus the correlations between them. This way, all the gradient images are filtered together exploiting the first and second order information they share. We adapt this methodology to two filters, namely the Linear Minimum Mean Squared Error (LMMSE) and the Unbiased Non-Local Means (UNLM). These new filters are tested over a wide variety of synthetic and real data showing the convenience of the new approach, especially for High Angular Resolution Diffusion Imaging (HARDI). Among the techniques presented, the joint LMMSE is proved a very attractive approach, since it shows an accuracy similar to UNLM (or even better in some situations) with a much lighter computational load.

Statistical noise analysis in GRAPPA using a parametrized noncentral Chi approximation model

The characterization of the distribution of noise in the magnitude MR image is a very important problem within image processing algorithms. The Rician noise assumed in single‐coil acquisitions has been the keystone for signal‐to‐noise ratio estimation, image filtering, or diffusion tensor estimation for years. With the advent of parallel protocols such as sensitivity encoding or Generalized Autocalibrated Partially Parallel Acquisitions that allow accelerated acquisitions, this noise model no longer holds. Since Generalized Autocalibrated Partially Parallel Acquisitions reconstructions yield the combination of the squared signals recovered at each receiving coil, noncentral Chi statistics have been previously proposed to model the distribution of noise. However, we prove in this article that this is a weak model due to several artifacts in the acquisition scheme, mainly the correlation existing between the signals obtained at each coil. Alternatively, we propose to model such correlations with a reduction in the number of degrees of freedom of the signal, which translates in an equivalent nonaccelerated system with a minor number of independent receiving coils and, consequently, a lower signal‐to‐noise ratio. With this model, a noncentral Chi distribution can be assumed for all pixels in the image, whose effective number of coils and effective variance of noise can be explicitly computed in a closed form from the Generalized Autocalibrated Partially Parallel Acquisitions interpolation coefficients. Extensive experiments over both synthetic and in vivo data sets have been performed to show the goodness of fit of out model. Magn Reson Med, 2010.

Automatic noise estimation in images using local statistics. Additive and multiplicative cases

In this paper, we focus on the problem of automatic noise parameter estimation for additive and multiplicative models and propose a simple and novel method to this end. Specifically we show that if the image to work with has a sufficiently great amount of low-variability areas (which turns out to be a typical feature in most images), the variance of noise (if additive) can be estimated as the mode of the distribution of local variances in the image and the coefficient of variation of noise (if multiplicative) can be estimated as the mode of the distribution of local estimates of the coefficient of variation. Additionally, a model for the sample variance distribution for an image plus noise is proposed and studied. Experiments show the goodness of the proposed method, specially in recursive or iterative filtering methods.