Image reconstruction using shift-variant resampling kernel for magnetic resonance imaging Ahmed S. Fahmy, Bassel S. Tawfik, Yasser M. Kadah * Biomedical Engineering Department, Cairo University, Giza, Egypt ABSTRACT Nonrectilinear k-space trajectories are often used in MRI applications due to their inherent fast acquisition and immunity to motion and flow artifacts. In this work, we develop a more general formulation for the problem of resampling under the same assumptions as previous techniques. The new formulation allows the new technique to overcome the present problems with these techniques while maintaining a reasonable computational complexity. The image space is decomposed into a complete set of orthogonal basis functions. Each function is sampled twice, once with a rectilinear trajectory and the other with a nonrectilinear trajectory resulting in two vectors of samples. The mapping matrix that relates the two sets of vectors is obtained by solving the set of linear equations obtained using the training basis set. In order to reduce the computational burden at the reconstruction time, only a few nonrectilinear samples in the neighborhood of the point of interest are used. The proposed technique is applied to simulated data and the results show a superior performance of the proposed technique in both accuracy and noise resistance and demonstrate the usefulness of the new technique in the clinical practice. Keywords: Resampling, image reconstruction, magnetic resonance imaging, least-squares problems. 1. INTRODUCTION In Magnetic Resonance Imaging (MRI), data is collected in the k-space which represents the Fourier transform of the imaged slice. The sampling trajectory along the k-space is determined by the shape of the waveform of the applied magnetic gradients. On-Off gradient waveforms result in evenly spaced k-space samples which can be easily transformed into the image domain using the fast Fourier transform (FFT). Unfortunately, generation of fast switching gradients is not easily accomplished. Thereby smoothly varying gradient waveforms are usually implemented in fast MRI acquisition techniques. However, this results in non-rectilinear sampling trajectories with a multitude of loop-like patterns such as spiral 1,2 , radial sampling 2 , Rossettes 3 , or Lissajous 4 . Such data points must be resampled onto a rectilinear grid in order to take advantage of the speed of the FFT. Ideal reconstruction is theoretically guaranteed by the sampling theory as long as the Nyquist criterion is satisfied 5,6 . That is, if f(k) is the continuos k-space representation of the imaged slice at a spatial frequency coordinate k, and S(k- k r ) is a sampling function consisting of 2-dimensional evenly spaced impulse functions located at the rectilinear k-space points, k r , the sampling theory guarantees that f(k) can be completely recovered from its sampled version f s (k)= f(k).S(k- k r ) as follows 5 , f(k) = f s (k)*C(k) , (1) where C(k) is an infinite Sinc function, and * is the convolution operator. Since working with an infinite sinc function is not feasible in practice, truncating the Sinc function is necessary. Substituting k=k nr , where k nr is the non-rectilinear k- space grid coordinates and assuming (without loss of generality) that both the rectilinear and the non-rectilinear grids carry the same number of points (=N), then equation (1) can be written in vector form as follows: f nr Nx1 = C NxN .f r Nx1 , (2) where f r , and f nr are vectors containing the rectilinear and the non-rectilinear samples, respectively, while C is a matrix whose entries are Sinc(k r -k nr ). In the Uniform Re-Sample algorithm (URS) 4 , f r is directly obtained from f nr by inverting * E-mail: ymk@ieee.org Medical Imaging 2002: Image Processing, Milan Sonka, J. Michael Fitzpatrick, Editors, Proceedings of SPIE Vol. 4684 (2002) © 2002 SPIE · 1605-7422/02/$15.00 825