# Preserving integral through downsampling

I've got a high-resolution image segmentation, which I'm downsampling (and non-linearly deforming) to fit on an ROI image. I typically blur the high-res image before downsampling with a Gaussian-fwhm related to the downsampling factor. The result is high-res and low-res images with the same mean.

I'd like now for my high-res and low-res images to have the same sum. If I scale up the low-res image by the downsampling factor, will local sums be preserved across the image? Is there a better way to downsample and preserving local sums?

• If I understand you correctly, that seems impossible! If my high resolution image is $[100\ 100]$ and my low resolution image is $[100]$ (same mean value) then there is no way the two can also have the same sum. Perhaps I'm missing something! :-? – Peter K. Aug 22 '13 at 14:38
• Oh, I mean make another low-resolution image; one which preserves the mean, and the other which preserves the sum! – Andrew Wood Aug 22 '13 at 14:46
• :-) Thanks. In that case, just scaling by the downsampling factor should get very close to what you need. It probably won't be precisely correct, though. – Peter K. Aug 22 '13 at 15:24