= = ′ + L l M N l lm y m y x S y x 1 ) , ( ) , ( α ≠ = + ' 0 ' ) , ( m m for m m for m y x S M Ny Parallel MRI Reconstruction Using B-Spline Approximation J. Petr 1 , J. Kybic 1 , S. Müeller 2 , M. Bock 2 , V. Hlavác 1 1 Center for Machine Perception, Dpt. of Cybernetics, Faculty of Elect. Engineering, Czech Technical University in Prague, Prague, Czech Republic, Czech Republic, 2 Interventional Methods, Division of Medical Physics in Radiology, German Cancer Research Center, Heidelberg, Germany, Germany Introduction In parallel MRI, k-space is sampled less dense than in conventional spin warp MRI which allows reducing the total acquisition time by factors of 2 or more. In this work, a parallel MRI reconstruction algorithm using B-splines (PBS algorithm) is presented. The PBS algorithm is designed to minimize the reconstruction error. State of the art The reduced array-coil image is an original image modulated by the coil sensitivity and aliasing. The modulation by the coil sensitivity (Eq. 1) and aliasing (Eq. 2) are both linear transformations, where S l are the array-coil images, S l A are S l with aliasing, S is the ideal reconstruction and M is the reduction factor. Assuming the invertibility of this composite transformation, the reconstruction consists of finding a proper inverse linear transformation = = + L l A l lm M N y x S y x m y x S y 1 ) , ( ) , ( ) , ( α (Eq. 3) with weights lm , where L is the number of array coils. The reconstruction can be performed in either image or k-space domain. The image domain method SENSE [1] estimates the sensitivities C l with the knowledge of S and S l (Eq. 1). An LxM sensitivity matrix is created for each pixel. The value at (m,l) in the sensitivity matrix is set to . Coefficients lm (x,y) are determined from the inversion of the appropriate sensitivity matrix. The k-space methods (e.g. GRAPPA [2]) reconstruct the missing lines as a linear combination of several acquired lines in all coils (Eq. 4), where s is the k-space image. The coefficients jn,lm are estimated in the least square sense using the Eq. 4, where for several k y was the line ) , ( y y x k m k k s ∆ + acquired as an auto-calibration line. PBS Algorithm The PBS method operates in the image domain. The reconstruction coefficients are determined directly without estimating the coil sensitivity maps. Taking into account that value in an aliased image S l A is a combination of M pixel values of the original images (Eq. 2) and the fact that the intensity values in two distinct pixels are independent, we define the ortogonality condition for coefficients lm (Eq. 3) (Eq. 5) The sensitivities and, thus, the reconstruction coefficients are smooth in space. It is not required to estimate the coefficients independently in each pixel as it is usually done. Instead, we look for the solution in a restricted space represented by a B-spline basis. This has a regularization effect and is advantageous from the computational point of view as well. The coefficients lm are represented as - - = j i h x p h y p mijl lm j i g y x x y , ) ( ) ( ) , ( β β α (Eq. 6), where g mijl are B-spline coefficients and p are B-splines functions of order p with knot spacing h y and h x . The B-spline coefficients are estimated in order to minimize the reconstruction error Eq. 3. The reference images used in Eq. 5 can be retrieved in full resolution in a pre-scan or in low resolution during the accelerated scan. Additional lines are acquired near the k-space center to form a low resolution image, while the outer part of the k-space is sampled more sparsely. The method is also capable of estimating the original image S l for each coil separately. This is done by replacing the reference image S by S l in (Eq. 5). The individual array-coil images are combined using the conventional sum-of-squares reconstruction, which provides an optimal SNR, when the coil sensitivities are unknown [5]. Measurements The method was tested on phantom images acquired by a Siemens Magnetom Symphony 1.5T scanner using an 8 channel head coil. A FLASH pulse sequence was used with the following parameters: TR = 30 ms, TE = 8.7 ms, → = 25°, FOV = 150x150 mm 2 , slice thickness = 5 mm and matrix = 512x512. Reconstruction was done with an acceleration factors 2 and 4 with 24 extra reference lines. Images were reconstructed with the PBS algorithm an compared to the manufacturer-provided methods GRAPPA [2] and mSENSE. Results and Discussion In Fig. 1 the PBS algorithm is compared with mSENSE and GRAPPA. The mSENSE image showed the lowest SNR, however without significant artifacts. The SNR was higher in the GRAPPA image, however, there were many visible artifacts. Images reconstructed by the PBS method (Fig.1c) showed both no visible artifacts and a high SNR. The excellent reconstruction properties of the PBS algorithm (cf. Fig. 1) are also seen in the reconstruction error - = y x y x S y x S E , 2 ) , ( ˆ ) , ( , where is the reconstruction and S is the sum of squares reconstruction of a fully sampled dataset. The proposed PBS algorithm uses the B-spline regularization, which significantly reduces the number of variables as compared to SENSE. Furthermore, the reconstruction is not computed independently in each point as in SENSE. The reconstructed images have lower reconstruction error than two commercially used methods mSENSE and GRAPPA. The method imposes no limitations on the coil configuration. It is robust to noise during the estimation. The future development will include a sta- tistical approach considering the noise properties during the estimation and the reconstruction. References [1] K Pruessman, et al. Mag. Res. Med., 42:952-62,1999. [2] M Griswold, et al. Mag. Res. Med., 47:1202-10,2002. [3] M Unser. IEEE Sig. Proc. Mag. 22-38, 1999. [4] J Petr, et al. Research report. Czech Technical University in Prague. TR-09-2004. [5] EG Larsson, et al. J. Mag. Res. 163:121-3,2003. Fig. 1. Reconstructed images with error E for an acceleration factor 2. E4 is error for an acceleration factor 4. a) GRAPPA, E=170, E4=342; b) mSense, E=278, E4=1228;c) PBS, E=101, E4=771. ) , ( ) , ( ) , ( y x C y x S y x S l l = - = + = 1 0 ) , ( ) , ( M m M N l A l y m y x S y x S ) , ( M N l y m y x C + ) , ( y y x j k m k k s ∆ + ∆ + = ∆ + n l y y x l x lm jn y y x j k n k k s k k m k k s , , ) , ( ) ( ) , ( α View publication stats View publication stats