Compressed Sensing for Sparse Magnetic Resonance Spectroscopy

Introduction: Multidimensional magnetic resonance spectroscopy (MRS) can provide additional information at the expense of longer acquisition time than 1D MRS. Assuming 2D MRS is sparse in wavelet domain, Iddo[1] first introduced compressed sensing (CS) [2,3] to reconstruct multidimensional MRS from partial and random free induction decay (FID) data. However, the darkness in 1D MRS derives from the discrete nature of chemical groups [4]. Significant peaks in these MRS takes up partial location of the full MRS while the rest locations own very small or even no peaks. This type of MRS can be considered to be sparse itself, named sparse MRS. In the concept of sparsity and coherence for CS[5], we will demonstrate that wavelet is unnecessary to sparsify sparse MRS, and it makes the reconstructed MRS even worse than using identity matrix. Furthermore, a lp quasi-norm compressed sensing reconstruction is employed to improve the quality of reconstruction. Methods: For a signal x that can be represented by S-term non-zero entries with vector α in basis Ψ , x can be recovered by solving l1 norm optimization when number of measurements M satisfies ( ) 2 , log M C S N μ ≥ ⋅ ⋅ ⋅ Φ Ψ where μ stands for the coherence between sensing matrix Φ and basis matrix Ψ [4]. It implies that the number of required measurements is proportional to the number of nonzero entries in α and the square of coherence μ . Although wavelet is a general transform to sparsify MRS, those MRS from chemical groups carry large peaks in only small portion of locations. These MRS can be viewed as sparse MRS which means spectroscopy is sparse in identity matrix I. In fact, Stern et al proposed to directly do iterative thresholding on the spectroscopy to recover the truncated 1D NMR spectroscopy [6] which implicitly supports this idea. In addition, pioneer works on CS theory pointed out that the coherence between Fourier and wavelet is larger than that of Fourier and time because time-frequency is with the minimal coherence [5]. At this point there is no need to do the wavelet transform on the spectroscopy which is sparse in identity matrix I. In addition, Rick Chartrand [7] proposed to replace l1 norm with lp quasi-norm to reconstruct the signal with fewer measurements than l1 requires. Thus, we propose to employ compressed sensing to reconstruct multidimensional sparse MRS from partial FID data as follows: