Gibbs ringing artifact, spatial correlation, and spatial resolution in MRI

Gibbs ringing is an inevitable artifact in MR Fourier Transform (FT) imaging caused by truncating k-space data via a rectangular window. A common practice is to multiply k-space data with a filter function prior to FT reconstruction for softening the amount of ringing and accepting the concomitant blur. Although by properly choosing the filter functions, the apodization approach is effective in removing the overshoot and ringing, it is at the price of both spatial correlation and spatial resolution of pixel intensities in the reconstructed image. Our study reported in this paper shows that the truncation and apodization introduce/change the spatial correlation and the spatial resolution of pixel intensities in FT imaging. We develop a 4-step procedure progressively prove the spatially asymptotic independence of pixel intensities and introduce a delta-train method to derive the spatial resolution in the image, for both the basic and the filtered FT imaging. The methods are intuitive without any assumption and approximation. Different filter functions have been investigated. We also discuss the relationship between the spatial resolution and pixel size and correct some misconceptions on these issues.