Encoding with frames in MRI and analysis of the signal-to-noise ratio

Recently, many new encoding techniques have been suggested for magnetic resonance imaging and an important image reconstruction problem has been raised in order to fully exploit the advantages promised by these new encoding techniques. In terms of frames in an L2 (R) space, we treat these encoding techniques in a unified perspective and propose a solution for this image reconstruction problem. We first develop a matrix form of frame theory and apply it to numerically construct the duals of the encoding functions, which are the transverse excitation profiles generated by radio-frequency pulses along a linear magnetic field gradient. An analysis of the signal-to-noise ratio (SNR) is also presented. Simulations have been carried out and they show that our image reconstruction algorithm minimizes the mean square error between the original and reconstructed images. The SNRs evaluated from simulations agree with our theoretical predictions.