Parallel magnetic resonance imaging (MRI) reconstruction methods are now used in nearly all scanners. Radio frequency coil arrays are commonly used to accelerate image acquisition during these reconstructions. The simultaneous acquisition of spatial harmonics method was the first successful technique of this kind, in which parallel receivers are used to reduce scan times.^{1} This method has since been reformulated and is known as the sensitivity encoding (SENSE) approach.^{2} Other reconstruction techniques have also been proposed, such as automatic simultaneous acquisition of spatial harmonics (AUTO-SMASH) and variable density AUTO-SMASH,^{3} as well as the generalized autocalibrating partially parallel acquisitions (GRAPPA) approach.^{4} The SENSE and GRAPPA methods are both very useful for clinical routines.^{5} Several methods have been developed in recent years to improve the GRAPPA technique, e.g., multicolumn multiline interpolation,^{6} nonlinear GRAPPA,^{7} localized coil calibration and variable density sampling,^{8} iterative reconstruction,^{9} and regularization.^{10} Finding a way to optimize the regularization parameter, however, has remained a problem.

GRAPPA is an autocalibrating parallel imaging method that works in the frequency domain. It is particularly beneficial in areas where accurate coil sensitivity maps are difficult to obtain. In this technique, the missing k-space (the 2D or 3D Fourier transform of the measured MRI) data is reconstructed via a linear combination of the acquired data. Coefficients for this combination are estimated using autocalibration signal lines that are obtained in the central k-space. Errors in the technique originate from noise in the measured data. There are also noise-induced errors that are associated with the estimation of the linear combination coefficients.

Regularization is often used to solve the coefficient inverse problem. Although several regularization methods have previously been described, including the Tikhonov regularization technique,^{11} we have developed a new method for optimizing the regularization parameter in the GRAPPA technique. We use the well-known L-curve criterion as a practical choice for the regularization parameter. This curve is a plot of the norm of a regularized solution versus the norm of the corresponding residual norm. In log–log plots, this curve resembles an ‘L’ and the optimal regularization parameter corresponds to the corner of the curve (see Figure 1). In our work, we have adapted theoretical aspects of L-curve criterion algorithms (in which the L-curve criterion is used to calculate the optimum regularization parameter).^{12, 13} We use the Tikhonov method to calculate the L-curve solution in the analysis of discrete ill-posed problems. We have also conducted a set of comparative MRI reconstruction experiments.^{14}

**Figure 1. **An L-curve plot computed using the Tikhonov method and a random undersampling pattern. The regularization parameter changes until a compromise is achieved between minimizing the residual and solution norms. In the Tikhonov regularization method, the regularized solution (χ_{λ}) is calculated by solving the optimization problem. The vertical part of the curve illustrates where the solution norm is very sensitive to changes in the regularization parameter. This is where the error in the measurements dominates χ_{λ}. The horizontal part of the curve corresponds to solutions where the residual norm is the most sensitive to the regularization parameter (i.e., where χ_{λ}is dominated by the regularization error). The optimum regularization parameter (0.085599) is located at the corner of the curve and represents the most stable approximate solution of χ_{λ}to the problem.

In our experiments, we used the GRAPPA technique and different regularization parameters to reconstruct an MRI data set. We used two different undersampling patterns—uniform and random—for each reconstruction. With the uniform undersampling pattern we achieved an acceleration factor of two, and an acceleration factor of four with the random undersampling pattern. We also used the Tikhonov method to calculate the normalized mean squared errors (NMSE) for each reconstruction. The NMSE of our GRAPPA reconstructions are plotted as a function of kernel size (i.e., a measure of efficiency) for both undersampling techniques in Figure 2. The NMSE for each reconstruction are also plotted as a function of regularization parameter in Figure 3. Our NMSE results indicate that there are larger residual aliasing artifacts for our uniform—see Figure 3(a)— undersampling technique compared with the random—see Figure 3(b)—undersampling. The optimum regularization parameter value we find (i.e., when the relative errors are minimized) is 0.0856. This result indicates that the L-curve we used for the Tikhonov regularization was correct and that our approach provides a robust estimation of the regularization parameter.

**Figure 2. **The normalized mean squared errors (NMSE) plotted as a function of kernel size for the generalized autocalibrating partially parallel acquisitions (GRAPPA) reconstructions made with (a) uniform undersampling and (b) random undersampling. The minimum NMSE value for each plot is indicated.

**Figure 3. **NMSE versus regularization parameter for each GRAPPA reconstruction. Values for reconstructions that used the (a) uniform undersampling and (b) random undersampling patterns are shown. The choice of regularization parameter—derived from the L-curve (see Figure

1)—is close to the optimum choice (0.0856), i.e., where the relative errors are at a minimum.

Results from our reconstruction experiments are shown in Figure 4. With uniform undersampling, we are able to reconstruct good-quality images from the data. The NMSE of our GRAPPA reconstructions are slightly lower from the randomly undersampled reconstructions than from the uniformly undersampled reconstructions. With regularization, however, the difference in our reconstructions becomes more significant. Our GRAPPA reconstructions that involve regularization have—on average—about 60% better NMSE than the reconstructions that were conducted without regularization.

**Figure 4. **(a) GRAPPA reconstructions of a reference magnetic resonance image (shown on the left). Uniform and random undersampling reconstructions are both shown. An iterative self-consistent parallel imaging reconstruction (SPIRiT) is also shown on the right. The NMSE value for each reconstruction is given. (b) The mean residual errors (i.e., between the reference and reconstructed image) are given for the GRAPPA reconstructions.

We have also performed an iterative self-consistent parallel imaging reconstruction (SPIRiT)^{9, 15} on the same data, as shown in Figure 4(a). We chose to also test this method because it is particularly effective. We found that the optimum parameters for this reconstruction were a kernel size of 7×7, 12 iterations, and a regularization parameter of 0.01. The result of this reconstruction is almost the same as the result from the GRAPPA reconstruction with regularization and random undersampling.

We have formulated a new method to determine the optimum regularization parameter for the GRAPPA MRI reconstruction technique. We have demonstrated that the L-curve in the Tikhonov approach can be used to choose an accurate optimum regularization parameter. The corner of the L-curve provides less of an aliasing artifact and a lower noise level in the reconstruction compared with an unregularized reconstruction. In our future work we will compare results from the SENSE, GRAPPA, SPIRiT, and eigenvector-SPIRiT (ESPIRiT) methodologies. We also wish to adapt our L-curve technique for the SPIRiT and ESPIRiT methods. In addition, we are planning to compare our results with those we obtain with the 1D and 2D multifractal analysis technique,^{16} which enables accurate and efficient characterization of image irregularities and defects.

Smain Femmam

University of Haute Alsace (UHA)

Mulhouse, France

Smain Femmam obtained his MS, PhD, and HDR in signal processing and computers from the University of Versailles and UHA in 1997, 1999, and 2013, respectively. He is now director of research. His main research areas are signal processing, safety systems, and communication. He is a senior IEEE member.

Amel Iles, Abdelhafid Bessaid

University of Abou Bekr Belkaïd (UABB)

Laboratory of Biomedical Engineering, Algeria

Amel Iles obtained her MS in electronics from UABB in 2001. She is now an assistant professor. Her main research areas are biomedical engineering and medical imaging.

Abdelhafid Bessaid is the head of the Medical Imaging team. His research focuses on the areas of biomedical engineering and medical imaging.

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doi:10.1117/12.896655