Superresolution in MRI?
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Recently, Peled and Yeshurun (1) investigated the possibility of calculating superresolution images from a set of spatially shifted, low-resolution images. Eight low-resolution images were acquired with an in-plane shift in phase encoding (PE) direction of 0, 1/4, 1/2, 3/4 pixel, and the same PE-shifts with an additional shift of 1/2 pixel in the read direction. All images were acquired with the same field-of-view (FOV) and resolution. This means that the sampled positions in k-space are identical for all images, since the sampling positions in k-space are uniquely defined by the given FOV and resolution, and vice versa. Let us now consider an in-plane shift along the PE direction. As given by the Fourier-shift theorem a spatial shift in image space is equivalent to a linear phase modulation in k-space along the shift direction (2,3). The sampled positions in k-space do not depend on any in-plane shift along PE direction. As a result, the switched gradient patterns of all three axes as well as the applied RF-pulses are completely identical for any PE-shift. From this it follows immediately that the acquired k-space data is also totally identical, except for the noise and does not depend on the in-plane shift. The actual shift of the image is always done in a postprocessing step, either by multiplying the k-space data K(km,kn) with
[1] J Hale,et al. Applications of voxel shifting in magnetic resonance imaging. , 1990, Investigative radiology.
[2] S Peled,et al. Superresolution in MRI: Application to human white matter fiber tract visualization by diffusion tensor imaging , 2001, Magnetic resonance in medicine.