Given a downscaled image, produce Gaussian blur of the original image

Question

Let's say I a perform convolution of an image with a Gaussian kernel of some size.

Is it possible to produce the exact same result from a downscaled version of the image? If so, what's its minimal size?

I'm not looking for an exact answer, some directions where I should start looking are enough. (I have some basic DSP and image processing knowledge but not much beyond that.)

Edit: A closely related question is - does Gaussian blur lose information, i.e. is it impossible to reconstruct the original from the blurred image?

Answer 1

I don't think you can do that. You can probably get a similar result, in that you won't be able to tell by eye that they are different.

The reason I don't think you can do what you have in mind is this: Imagine that you downscale (downsample, basically throw away pixels) an image, apply a blur, upscale the image. Suppose now that you apply a blur to the original image. If those two are the same, then you could deconvolve both images and from both of them recover the original image.

This would be the best compression format ever. Downscale by 16X, blur. Small file, send by email. Upscale by 16X, deblur, original image. Hell, why stop at 16X? You could compress all of your images to a single pixel each and be able to recover them.

Unfortunately, things don't work that way. Downscaling will (must) throw away data, and you can't recover it.

So, the downscaled and blurred image will be different from the blurred original.

Answer 2

For the purposes of the question, downscaling can be considered as a lowpass filter, which makes it easier to think about, also because then the order of blurring and downscaling has no effect on the result.

A Gaussian blur convolution kernel has a 2-d Fourier transform that is also a 2-d Gaussian function. A Gaussian function has ever-decaying tails that never go all the way to zero value. In the original resolution, high spatial frequencies that would be permanently erased by (properly done) downscaling will therefore be attenuated but not wholly removed by Gaussian blur. That erasing of what is left after Gaussian blur is the source of quantifiable error due to downscaling, and you would need to formulate a metric to describe that error and to decide on an acceptable level of it.

Answer 3

To the first question, the answer is no. Take a Dirac-like image, i.e. zero everywhere except for one pixel. Its filtering by a Gaussian will be the same Gaussian. But any integer subsampling, starting with the appropriate shift, will be able to destroy the Dirac, and yield a zero image. Any subsequent filtering will not be capable of producing the above Gaussian.

To the second question: does Gaussian blur lose information? No in theory, since the Gaussian kernel does not vanish in the frequency domain. Theoretically, you could divide the blurred image (in the frequency domain) by the Fourier transform of your known Gaussian kernel and invert the Fourier transform. That is called the "Division Bell " (pun on you avatar picture) as the Gaussian is sometimes called the Bell curve.

But yes in practice. You would be very lucky if you succeed: the image is of finite size, the kernel in generally approximately known, and you have noise. So due to the low amplitude spectrum of the Gaussian in high frequencies, noise is likely to blow up.

Answer 4

When you Gaussian blur an image, its spectrum sees its high frequencies decrease, following the Gaussian spectrum, so that their amplitude quickly falls under the quantization limit and can be considered zeroed.

If you subsampled over the Nyquist frequency, then perfect reconstruction of the blurred image is possible.

Intuitively, the allowed subsampling rate should be on the order of $\sigma/\sqrt{b}$, where $b$ is the number of significant bits.

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And yes, Gaussian blur does lose information because of truncation (quantization) errors.