Mathematical Analysis of “Phase Ramping” for Super-Resolution Magnetic Resonance Imaging

Abstract

Super-resolution image processing algorithms are based on the principle that repeated imaging together with information about the acquisition process may be used to enhance spatial resolution. In the usual implementation, a series of low-resolution images shifted by typically subpixel distances are acquired. The pixels of these low-resolution images are then interleaved and modeled as a blurred image of higher resolution and the same field-of-view. A high-resolution image is then obtained using a standard deconvolution algorithm. Although this approach has been applied in magnetic resonance imaging (MRI), some controversy has surfaced regarding the validity and circumstances under which super-resolution may be applicable. We investigate the factors that limit the applicability of super-resolution MRI.

Measuring information gain for frequency-encoded super-resolution MRI

“…One of the conclusions we reach in this article is that there is indeed new information in each acquisition when the object is shifted in the FE direction prior to imaging, as proposed earlier [15][16][17][18]. The caveat is that the amount of new information present in each acquisition after the first is relatively small and may be difficult to detect in the presence of noise.…”

“…First, we calculate the dot product between L(k m , 0) and R(k m , Àr n )L(k m , +r n ), which we denote by C(r n ) [18]:…”

“…The ramp must be applied prior to the application of the antialiasing filter for the acquisitions to yield independent information [15][16][17][18]. In this case, we cannot undo the effect of the phase ramp because multiplication and integration do not commute in general:…”

“…The order of the processing steps carried out on the measured data, including the spatial shift, is significant. If the object is shifted prior to imaging, new information can, in fact, be present for FE data [15][16][17][18]. This observation suggested that FE SRMRI may still be a viable method for resolution and image quality enhancement.…”