Consider a filter to be anything that multiplies the MRI signal.
Point spread function is the Fourier transform of a filter. Why:
Assume
. Using the convolution theorem in this
case implies that will be the Fourier transform of the filter.
Point spread function answers the question, how much blurring would occur if you are
trying to image a point.
Data always contains the effect of truncation and sampling. What
does this imply? Windowing and sampling function can be described as
(13.1)
Remember multiplication in one domain is convolution in the other,
(13.2)
so to find effect in -space on a point function convolve this function with
function, which just gives back Fourier transform of
(13.3)
Doing this sum out gives
(13.4)
This is a sinc like function, as we might expect, knowing
that the Fourier transform of a rect is a sinc.
Also note that the wider the sampling window becomes, the
narrower the sinc becomes, positive thing that as we decrease
the point-spread gets narrower.