# downsampling to reduce noise - how much?

Started Nov 26, 2008 | Discussions thread
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 Amusing footnote In reply to ejmartin, Nov 29, 2008

Staring at Phil's "demonstration" about grain size one last time,

I was amused to observe that the middle image was obtained by upsampling the left one by a factor 1.5, and the right image was obtained by upsampling it by 2x (just look at the "exclamation point" artifact halfway down on the right side of each image). Indeed, if one upsamples the "fine grain" image by 2x it matches (left image) the "coarse grain" sample (right image)

except for the fact that Phil increased the contrast after upsampling to increase the noise std dev to match that of the "fine grain" image.

This explains why, in the noise power spectra

the "medium grain" sample's noise power (blue curve) peaks at about 2/3 of Nyquist (Nyquist is 128 on the horizontal axis) for 3:2 upsampling, and the "coarse grain" sample's noise power (red curve) peaks at about 1/2 of Nyquist for 2:1 upsampling. The leakage of noise power above these peaks is due to the smoothing effect of the bicubic resampling used, otherwise there would be little or no power at all beyond the peak.

And so now we see why downsampling eg the 2 pixel grain image by a factor of two didn't do much -- it was generated from the original "fine grain" image by upsampling the 1 pixel grain image by 2x. If the upsample/downsample had been more accurate, the std dev wouldn't have changed at all, since one wouldn't have done anything to the original image!

Phil can be such a practical joker at times

Even so, the example illustrates the connection between the noise power spectrum and what one sees when one views the image. The degree of absence of noise power at the high end of the spectrum is directly related to the "coarseness" of the noise grain. The fact that the coarse grain was achieved by upsampling explains why there is little noise power at the finest scales -- there were no such scales to begin with in the lower resolution image the coarse grain images were derived from! And it explains why there will be little reduction in the noise std dev upon downsampling coarse grain -- since that is just getting one back to where one started, there will be rather little effect on the noise std dev.

It should be emphasized that the std dev is just one measure of noise; it does not convey the character of the noise. The noise power spectrum quantifies that noise character, as one can see by comparing the three grain samples above with their corresponding noise power spectra. And the effect of downsampling (or upsampling) can be predicted at least qualitatively from a cursory examination of that spectrum; with some knowledge of the downsampling algorithm, the prediction can be made quite precise.

It also cannot be emphasized enough that all this has zero to do with comparing noise from cameras with different pixel counts, as I have tried to explain elsewhere using these same methods using power spectra: