Reduction of acquisition time has been a major target in Magnetic Resonance Imaging (MRI). It allows measuring accurately dynamic and functional changes in diagnostic studies. Several techniques are currently used to reduce acquisition time (e.g: parallel imaging (PI)). However, high speed MRI suffers from low signal to noise ratio (SNR), which makes difficult clinical data analysis. Several denoising methods, attempting to filter out noise from the acquired MRI images has been proposed to improve MR image quality^1,2. In this study, we propose a new denoising technique, which we refer to Semi-Classical Signal Analysis (SCSA)³. This method has the advantage to employ adaptive vectors derived from the Schrödinger operator to decompose the MRI image into a sum of squared eigenfunctions and uses a soft threshold to separate between useful image coefficients and those belonging to noise. The SCSA method has been successfully applied in MRS data denoising for accurate data quantification⁴.

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, 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 PI acceleration factor set to 2, taken as noisy data set. The proposed method uses the discrete spectrum of the Schrodinger operator, consisting of the computed negative eigenvalues, to reconstruct the image $$$V_2$$$ as follows, \begin{equation} V_2(x,y)= \left(\frac{h^{2}}{L^{cl}_{2,\gamma}} \displaystyle \sum_{k=1}^{K_{h}} \left(-\mu_{k,h} \right)^{\gamma}\psi^{2}_{k,h}(x,y)\right)^{\frac{1}{1+\gamma}}, \end{equation} where $$$\gamma>0$$$ (in this study $$$\gamma=4$$$) and $$$L^{cl}_{2,\gamma}$$$ is given by: \begin{equation} L^{cl}_{2,\gamma}=\frac{1}{2^{2}\pi}\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+2)}, \end{equation} $$$\Gamma$$$ refers to the standard Gamma function. Moreover, $$$\mu_{k,h}$$$ and $$$\psi_{k,h}$$$ denote the negative eigenvalues with $$$\mu_{1,h} < \cdots <\mu_{K_{h},h}< 0, K_{h}$$$ is a finite number of negative eigenvalues, and associated L²-normalized eigen functions of the operator $$$\mathcal{H}_{2,h}\left(V_{2}\right)$$$ such that: \begin{equation} \mathcal{H}_{2,h}(V_2)\,\psi_{k,h} = \mu_{k,h}\,\psi_{k,h}, \quad k=1,\cdots,K_h \end{equation} where, \begin{equation} \mathcal{H}_{2,h}(V_{2})= -h^{2}\Delta-V_{2}, \end{equation} with $$$\Delta=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}} $$$. An efficient algorithm to reconstruct the MRI image using the SCSA has been proposed in reference 3 where the 1D SCSA is run on the rows and the columns of the image and then the squared L²-normalized eigenfunctions of these 1D problems are combined using a tensor product approach. The parameter h plays an important role in the SCSA. When h decreases, the approximation improves³. However in the de-noising process, it is recommended to retain the eigenfunctions belonging to the signal and discard those representing noise. The SNR is computed as in reference 2.

Figure 1 shows a brain image selected from the original (SNR =43.41) and noisy (SNR = 34.2) sets above, and the reconstructed SCSA image, where a significant increase in SNR has been obtained (SNR = 47.6). In figure 2, the middle of the images in Figure 1 is zoomed, showing that SCSA preserves the contrast of the image while reducing noise contribution. As mentioned above, SCSA relies on the choice of h. By increasing this parameter, we observe an SNR increase, but with image blurring effect, whereas when h decreases, less SNR increase is obtained, but the image details are preserved. An optimal choice of the value of h, which is strongly linked to the SNR, has to be determined. One approach is to locally vary its value across the processed image given the SNR value of the same image location. As shown, the SCSA method is powerful in significantly reducing noise while preserving image information. This allows for reduction of acquisition time to achieve high performance dynamic and functional MR studies. Better performance of the method is achieved by spatially varying the value of h as function of the SNR across the image. Preliminary results are encouraging.