Abderrazak Chahid^{1}, Hacene Serrai^{2}, Eric Achten^{2}, and Taous-Meriem Laleg-Kirati^{1}

The main of challenge of Magnetic Resonance Imaging (MRI) is
dealing with high levels of noise which may corrupt the image especially since
the noise is almost correlated with the image details. In this regard, we
propose a new MRI enhancement method to overcome this limitation. The proposed
MRI enhancement method relies on square sub-images enhancement depending on the
noise level in each position using spatial adaptation of the Semi-Classical
Signal Analysis (SCSA) method, where an enhancement parameter *h* is
subject to a Gaussian distribution. The results show significant improvement in
noise removal and preserving small details in the image.

Experiments are performed on one healthy male subject, on a 3T scanner (MAGNETOM Tim-Trio, Siemens Healthcare) equipped with a 32-channel head coil for signal reception. Turbo Flash sequence is used with the following parameters: TR/TE = 250/2.46 ms; matrix size, 256x256 resolution and 33 contiguous slices; FOV: 220mm; voxel size: 0.9x0.9x3 mm3; flip angle:10; and receiver bandwidth set to 320 Hz/pixel. To reduce SNR, two sets of data are acquired: one with Nex =2, considered as an original data set (see images below), and the second set with Nex = 1, and Parallel Imaging (PI) acceleration factor set to 2, taken as noisy data set.Two denoising tests are conducted, with fixed and variable *h*. In the latter, the noisy image is first divided into sub-images $$$I_{ij}$$$ (Figure. 1) then the SCSA method is applied on each sub-image with the corresponding $$$h_{ij}$$$ which follows a Gaussian distribution.

The denoised sub-image $$$I_{h_{ij}}(x,y)$$$ is defined as follows:

\begin{equation}I_{h_{ij}}(x,y)=\left(\frac{h_{ij}^{2}}{L^{cl}_{2,\gamma}} \displaystyle\sum_{k=1}^{K_{h_{ij}}} \left(-\mu_{k,h_{ij}} \right)^{\gamma}\psi^{2}_{k,h_{ij}}(x,y)\right)^{\frac{1}{1+\gamma}}~~~,~~\quad i,j=1,\cdots,N_{s} ~~, \end{equation}

where

\begin{equation} L^{cl}_{2,\gamma}=\frac{1}{2^{2}\pi}\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+2)} ~~~~,\end{equation}

$$$I_{h_{ij}}(x,y)$$$ is the pixel $$$(x,y)$$$ in the denoised sub-image $$$I_{h_{ij}}$$$. Γ refers to the standard Gamma function. Moreover, $$$\mu_{k,h_{ij}} $$$ and $$$\psi_{k,h_{ij}}$$$ denote the negative eigenvalues with $$$K_{h_{ij}}$$$ is a finite number of negative eigenvalues, $$$\mu_{1,h_{ij}} < \cdots <\mu_{K_{h_{ij}},h_{ij}}< 0$$$ and associated $$$L^2$$$-normalized eigen functions of the operator $$$\mathcal{H}_{2,h_{ij}}\left(I_{ij}\right)$$$ such that:

\begin{equation}\mathcal{H}_{2,h_{ij}}(I_{ij})\,\psi_{k,h_{ij}} = \mu_{k,h_{ij}}\,\psi_{k,h_{ij}} ~~, ~~\quad ~~ k=1,\cdots,K_{h_{ij}} ~~ ,~~ i,j=1,\cdots,N_{s} . \end{equation}

where, $$$\mathcal{H}_{2,h_{ij}}(I_{ij})= -h_{ij}^{2}\Delta - I_{ij}$$$ , with $$$\Delta=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}} $$$ .

The obtained results demonstrate that the proposed signal enhancement algorithm for MRI data is efficient in noise reduction while preserving the image details. Optimization of size of the sub-images is underway for better noise level estimation for a more accurate spatial distribution of the parameter *h* than the proposed standard Gaussian distribution one.

Research reported in this publication was supported by King Abdullah University of Science and Technology (KAUST).

1. J. Zhang , T.M. Laleg-Kirati, Stephanie Bogaert, Eric Achten , and Hacene Serrai, "Magnetic Resonance Imaging de-noising using the squared eigenfunctions of the Schrödinger operator: Application to brain MRI data", Accepted in ISMRM 24th Annual Meeting, Singapore, 2016.

2. Z. Kaisserli, T.M. Laleg-Kirati and A. Lahmar-Benbernou, "Image reconstruction using squared eigenfunctions of the Schrödinger operator", Digital Signal Processing Journal, Vol. 40, pp.80–87, 2015.

3. G. Blanchet, L. Moisan, proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 1065-1068, 2012.

Figure.1. Gaussian *h* values distribution [up left], Image square grid [up right], *h*_{ij} Gaussian distribution function [bottom left], distributed *h*_{ij} values from * h*_{border}=0.5 to* h*_{center}=2.45 [bottom right].

Figure.2. Comparison of the denoised MR Image performance between SCSA using a fixed *h*=1.36 and a distributed *h* .

Figure.3. Noisy (a.1), and denoised image with fixed *h *(b.1) and variable *h* (c.1), zoom on the region A (displayed on a.1), in the noisy (a.2), and denoised image with fixed *h* (b.2) and variable *h* (c.2).