Designing an experiment to show blue noise not adding noise below nyquist and not aliasing above

Question

I read this paper from 1983 "Spectral Consequences of Photoreceptor Sampling in the Rhesus Retina".

This paper talks about blue noise sampling patterns in the retina and says that:

The results (Fig. I) indicate that throughout the retina the cones provide a novel form of optimal spatial sampling: optimal in the sense that minimal noise is introduced for spatial frequencies below the nominal Nyquist limits implied by local receptor densities (the limits that would obtain if the cones formed a regular lattice), while spatial frequencies above the local Nyquist limits are not aliased back into conspicuous moire patterns but instead are scattered into broadband noise. Thus, the visual system avoids the aliasing distortion of high frequencies inherent in any regular arrangement of image sampling elements and simultaneously minimizes sampling noise for low frequencies that fall within its potential Nyquist bandwidths. These advantages stem from a quasi-random (that is, random but not Poisson) spatial sampling scheme that apparently has not been used in man-made image-recording devices.

I take that to mean that blue noise:

1. Doesn't add (much) noise when sampling frequencies below nyquist.
2. Doesn't alias for frequencies above nyquist.

This is in contrast to:

1. Uniform sampling which doesn't add noise to any frequency, but aliases frequencies above nyquist.
2. White noise (uniform random) sampling, which adds noise to all frequencies, but doesn't alias for frequencies above nyquist.

That makes sense, but now I'm trying to think of how I could make a simple experiment to show this being true - preferably in the context of 2d sampling and 2d images (image processing / image sampling), but 1d would be ok too.

I get that I could use blue noise to sample sine waves of varying frequencies but am unsure how exactly that sampling would work.

I know how to generate blue noise sample points in any dimension using Mitchel's best candidate algorithm.

I'm guessing I would generate N sample points and then do a reconstruction filter on those sample points, but thinking about that, I can't see how noise would present itself (although, aliasing would just be "missed frequencies"), and frankly I'm not real sure how to do a decent reconstruction filter.

Am I on the right track for making a decent but simple experiment to show these statements being true empirically? Or should I be doing something different?

Answer 1

Interesting! Please do share your results with us!

The reconstruction is indeed the difficult bit. Maybe you can look into NFFT (non-uniform FFT) to obtain the Fourier transform of your irregular sampled image and determine its frequency content. You’d expect a clear peak in the case of a sine wave below Nyquist, or an aliased sine wave. Noise would show up as additional (small) local maxima.