Spatially regularized multi-exponential transverse relaxation times estimation from magnitude MRI images under Rician noise

Abstract : Synopsis This work aims at improving the estimation of multi-exponential transverse relaxation times from noisy magnitude MRI images. A spatially regularized Maximum-Likelihood estimator accounting for the Rician distribution of the noise was introduced. This approach is compared to a Rician corrected least-square criterion with the introduction of spatial regularization. To deal with the large-scale optimization problem, a majoration-minimization approach was used, allowing the implementation of both the maximum-likelihood estimator and the spatial regularization. The importance of the regularization alongside the rician noise incorporation is shown both visually and numerically on magnitude MRI images acquired on fruit samples. Purpose Multi-exponential relaxation times and their associated amplitudes in an MRI image provide very useful information for assessing the constituents of the imaged sample. Typical examples are the detection of water compartments of plant tissues and the quanti cation of myelin water fraction for multiple sclerosis disease diagnosis. The estimation of the multi-exponential signal model from magnitude MRI images faces the problem of a relatively low signal to noise ratio (SNR), with a Rician distributed noise and a large-scale optimization problem when dealing with the entire image. Actually, maps are composed of coherent regions with smooth variations between neighboring voxels. This study proposes an e cient reconstruction method of values and amplitudes from magnitude images by incorporating this information in order to reduce the noise e ect. The main feature of the method is to use a regularized maximum likelihood estimator derived from a Rician likelihood and a Majorization-Minimization approach coupled with the Levenberg-Marquardt algorithm to solve the large-scale optimization problem. Tests were conducted on apples and the numerical results are given to illustrate the relevance of this method and to discuss its performances. Methods For each voxel of the MRI image, the measured signal at echo time is represented by a multi-exponential model: with The data are subject to an additive Gaussian noise in the complex domain and therefore magnitude MRI data follows a Rician distribution : is the rst kind modi ed Bessel function of order 0 and is the standard deviation of the noise which is usually estimated from the image background. For an MRI image with voxels, the model parameters are usually estimated by minimizing the least-squares (LS) criterion under the assumption of a Gaussian noise using nonlinear LS solvers such as Levenberg-Marquardt (LM). However, this approach does not yield satisfying results when applied to magnitude data. Several solutions to overcome this issue are proposed by adding a correction term to the LS criterion. In this study, the retained correction uses the expectation value of data model under the hypothesis of Rician distribution since it outperforms the other correction strategies: stands for the sum of squares. We refer to this method as Rician corrected LS (RCLS). A more direct way for solving this estimation problem is to use a maximum likelihood (ML) estimator which comes down to minimize: To solve this optimization problem when dealing with the entire image, a majorization-minimization (MM) technique was adopted. The resulting MM-ML algorithm is summarized in gure 1, the LM algorithm used in this method minimizes a set of LS criteria derived from the quadratic majorization strategy. A spatial regularization term based on a cost function was also added to both criteria (and) to ensure spatial smoothness of the estimated maps. In order to reduce the numerical complexity by maintaining variable separability between each voxel and it's neighboring voxels , the function is majorized by : where stands for the iteration number of the iterative optimization algorithm.
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Christian Hajj, Guylaine Collewet, Maja Musse, Saïd Moussaoui. Spatially regularized multi-exponential transverse relaxation times estimation from magnitude MRI images under Rician noise. ISMRM, Jun 2018, Paris, France. ⟨hal-02318145⟩

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