Noise estimation and removal in MR imaging: the variance-stabilization approach
Dr. Alessandro Foi
Department of Signal Processing
Tampere University of Technology
DEI - Conference Room
November 11th, 2010
The magnitude of magneto-resonance (MR) images can be modeled by the Rice distribution. This distribution has two parameters: the unknown noise-free magnitude of the data, and the standard deviation of the additive noise that corrupts the real and imaginary part of the data.
Estimation of the magnitude is a particularly challenging denoising problem because of two main reasons, namely heteroskedasticity and bias: first, the standard-deviation of the noise corrupting the magnitude depends also on the unknown magnitude itself; second, the expectation of the noisy magnitude differs from the unknown noise-free magnitude by a nonlinear function of the noise standard-deviation and of the noise-free magnitude. Special ad-hoc algorithms need to be designed for filtering MR images, in order to address both the heteroskedasticity and bias in the Rician-distributed data.
Our work is motivated by the following pragmatic consideration. Nowadays there exist many highly developed, efficient and extremely powerful methods for estimating and removing the noise from images, provided that the noise is homoskedastic and has zero mean. These methods, which we term "homoskedastic algorithms", cannot be applied to Rician-distributed data.
In order to leverage these powerful methods for MR image filtering, we develop optimal forward and inverse variance-stabilizing transformations for the Rice distribution. The forward transformation makes the data accurately homoskedastic, and thus the noise removal can be accomplished by applying a homoskedastic denoising algorithm; the inverse transformation is designed to be applied on the denoised data and to return a ML estimate of the noise-free magnitude.
A second fundamental contribution of this work consists in a stable and fast iterative procedure for estimating the noise level from a single Rician-distributed image. At each iteration, the procedure exploits variance-stabilization composed with a homoskedastic variance-estimation algorithm. As opposed to existing approaches to noise-level estimation used in MR imaging, our technique requires neither the presence of a dark uniform background, a presegmentation of the data, nor a high signal-to-noise ratio.
An extensive theoretical and experimental study demonstrates the success of our approach to Rician noise estimation and removal through variance stabilization. In particular, we show that utilizing the classical estimator of the noise standard deviation based on the median absolute deviation (MAD) of wavelet detail coefficients and the BM3D denoising algorithm (both are homoskedastic algorithms designed for additive white Gaussian noise) we significantly outperform the best existing techniques specifically designed for the Rician-distributed data.
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