SVD eigenimage based SENSE

Introduction: In this study, an eigenimage theory based on singular value decomposition (SVD) is proposed for SENSE [1]. Using SVD, a set of eigenimages can be generated from the reduced FOV images and the SENSE image can be represented as the linear combination of these eigenimages. As a result, the matrix inversion in SENSE is treated as the calculation of a set of linear coefficients for weighting the eigenimages. Based on this theory, a group of techniques for SENSE optimization has been developed. As an example, this abstract presents a data-driven SENSE regularization technique, where the regularization parameters are generated from the SNR of acquired data. This technique has advantages over several previously proposed regularized SENSE techniques, where the regularization parameters have to be determined either from experience [2] or from a priori information [3]. Moreover, our experiments demonstrated that this technique can suppress not only the noise amplification, but also the artifacts introduced by sensitivity map calibration. Theory: Consider a set of reduced FOV data acquired from an N-channel RF coil array with a reduction factor of R. The standard SENSE unfolding matrix (SΨS)SΨ [1] can be written as pinv(ΦS), where "pinv" represents the pseudo-inverse, the matrix Φ is generated from the noise correlation matrix Ψ by Ψ=ΦΦ and S is the sensitivity matrix. By applying SVD, ΦS can be represented by Eq. 1, where di's are eigenvalues, and ui's and vi's are the corresponding eigenvectors for the matrices ΦSS H Φ and SΦΦS respectively. From SVD, it is easy to obtain Eq. 2, which gives a different approach to calculate the SENSE unfolding matrix. In addition, the new concept of eigenimages is defined in Eq. 3, where Ei is the ith eigenimage ordered by magnitude of eigenvalues, and a is the reduced FOV image (or aliased image). As a result, the final SENSE image is a weighted summation of eigenimages (Eq. 4). Using this new eigenimage concept, it can be seen that the accuracy of SENSE reconstruction depends on the estimate of weighting coefficients ki's and the SNR of eigenimages. It is therefore possible to optimize the total SNR in SENSE by modifying the coefficients for weighting the eigenimages according to their individual SNRs. This leads to Eq. 5, which describes the key idea in the data-driven regularized SENSE reconstruction. It is important to mention that in the definition according to Eq.5, λ is a normalization parameter and has the same value for all pixels and for all eigenimages. For the special case when λ=0 in Eq. 5, the Eq. 4 will be reduced to the original SENSE formulation by Pruessmann. In our current implementation we chose λ to be defined according to Eq. 6. This choice appears to be a good compromise between maximizing SNR and minimizing residual artifact.