# Interpolation vs Interpolation Filter?

Hi guys, I've been reading some papers on - how to remove ghosting artefacts from the Fourier Slice theorem applied to a 3D discrete image volume. The papers mention that in order to remove ghosting artefacts "employ a better(and typically wider) interpolation filter".

Currently, in Matlab I fft a 3D image space volume to Fourier space. I then extract a 2D plane at an arbitrary angle from this 3D frequency domain volume, making sure that my 2D plane passes through the centre of the volume. Since, this 2D plane is at an angle (not all points on the plane correspond to the uniform discrete values in the volume), I the use interpolation on the 3D volume in Fourier space, which I believe uses weighted linear interpolation (8 nearest point average).

What do these papers mean when they say - apply an interpolation filter, is this any different from what I am doing? If it is different, should I be multiplying this interpolation filter with the frequency spectrum and if so what does this filter look like?

• Could you please provide the (links of) papers that you have been referring? – eeerahul Apr 17 '13 at 5:18

Interpolation is the process of adding zeros in between your samples and then low-pass filtering to get rid of the aliases that result. Linear interpolation is just one out of many different possible filters. Linear interpolation filters can always be represented by FIR filters. In one-dimension the interpolation by two linear interpolation filter is $[\frac{1}{2}, 1, \frac{1}{2}]$. There are better low-pass filters.

It is not clear from your quote what they meant by a "wider" filter. I'm guessing they mean "one with more taps", but it could also mean "one with a wider passband".

• Are you saying that the linear interpolation I am using in matlab is actually a type of a filter,( I had thought that it was simply using the slopes to estimate a point that lies in between two given points)? If so how does this filter look, and is it being convolved or multiplied with the frequency spectrum? – user1084113 Apr 16 '13 at 16:32
• Yes, that is what I'm saying. It could use the slopes to calculate the numbers, but it can also do it with a filter. The one-d interp x2 filter is $[\frac{1}{2}, 1, \frac{1}{2}]$, the two-d interp x2 filter is $[\frac{1}{4}, \frac{1}{2},\frac{1}{4};\frac{1}{2}, 1, \frac{1}{2};\frac{1}{4}, \frac{1}{2},\frac{1}{4}]$. It's difficult to represent a three-d filter, but hopefully you get the idea at this point. – Jim Clay Apr 16 '13 at 17:00
• It sounds like the filter is convolved with the frequency spectrum. – Jim Clay Apr 16 '13 at 18:29

Interpolation and interpolation filters are almost synonyms.

• Interpolation is the process of generating additional data points that lie between original sample points.
• An interpolation filter performs the interpolation operation. Adding filter in the name puts an emphasis on the fact that you adopt a Signal Processing point of view and that you take great care of not introducing high frequency artifacts.
• How does an interpolation filter perform the interpolation? The interpolation operation uses linear approximation as such y = y1 + (y1-y0)((x-x0)/(x1-x0)). What is the interpolation filter doing? – user1084113 Apr 18 '13 at 17:10