# 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? Commented Apr 17, 2013 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? Commented Apr 16, 2013 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. Commented Apr 16, 2013 at 17:00
• It sounds like the filter is convolved with the frequency spectrum. Commented Apr 16, 2013 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? Commented Apr 18, 2013 at 17:10

Maybe, for interpolation using finite-order polynomials, one step up from linear interpolation is 3rd-order Hermite polynomials interpolation.

That would be a 4×4×4 matrix in 3D space.

Linear interpolation(midpoint) of a pure sinusoidal wave is similar to describing a circle by the inscribed diamond.

Maximum error is sqrt(2)/2 -.5 0.207...

Your use of 8 terms would optimally expand the above analogy an inscribed octagon(see below)

This expansion roughly approximates the Taylor series of Cos up to 8 terms, probably less depending on the fitting algorithm.

In any case, where the number of an linear factors are not infinite, artifacts will appear.

By alternately filling the source stream with zeros 0, and filtering, no additional information is added, leading to a cleaner output at a higher computational cost and associated precision loss.

Artifacts are still produced by the non-infinate precision, but they are minimal.

For phase/frequency modulated signals, the error factor scales with the modulation index.

For completeness, linear interpolation becomes more attractive when super-sampling. At Fs=F, instead of Fs=F/2, the octagon becomes the max error. Super-sampling also allows for broad phase correction during decimation.

As stated by Jim Clay in his answer the FIR filter equivalent contains 3 terms that equates to a cos window of size 3 rolling mean applied to the data. The term widen applies to the window size.

A third order FIR filter is inadequate, most FIR filters have 50-500+ terms, and will cause ringing at such a small size.