Image reconstruction in k‐space from MR data encoded with ambiguous gradient fields

PURPOSE: In this work, the limits of image reconstruction in k-space are explored when non-bijective gradient fields are used for spatial encoding. THEORY: The image space analogy between parallel imaging and imaging with non-bijective encoding fields is partially broken in k-space. As a consequence, it is hypothesized and proven that ambiguities can only be resolved partially in k-space, and not completely as is the case in image space. METHODS: Image-space and k-space based reconstruction algorithms for multi-channel radiofrequency data acquisitions are programmed and tested using numerical simulations as well as in vivo measurement data. RESULTS: The hypothesis is verified based on an analysis of reconstructed images. It is found that non-bijective gradient fields have the effect that densely sampled autocalibration data, used for k-space reconstruction, provide less information than a separate scan of the receiver coil sensitivity maps, used for image space reconstruction. Consequently, in k-space only the undersampling artifact can be unfolded, whereas in image space, it is also possible to resolve aliasing that is caused by the non-bijectivity of the gradient fields. CONCLUSION: For standard imaging, reconstruction in image space and in k-space is nearly equivalent, whereas there is a fundamental difference with practical consequences for the selection of image reconstruction algorithms when non-bijective encoding fields are involved.