# Why resizing an image smoothed by Gaussian by factor of 2 also increase sigma by factor of 2

In the paper about SIFT algorithm (Distinctive Image Features from Scale-Invariant Keypoints) it says at the end of section 3.3:

We double the size of the input image using linear interpolation prior to building the first level of the pyramid. [...]. We assume that the original image has a blur of at least σ = 0.5 (the minimum needed to prevent significant aliasing), and that therefore the doubled image has σ = 1.0 relative to its new pixel spacing.

The paper says that when doubling the size of an image smoothed by a Gaussian with σ = 0.5, the new image doubles its spread, so σ = 1.0 in the new image.

I can understand intuitively why it can be true, and I also checked it in Matlab to see the effect. However, I can't prove it to myself mathematically.

I'll be happy to see proof or a direction to reference that can help me understand it mathematically.

A first discrete is pixel wise: imagine an image with only one active pixel. If you upsample it by two in both directions, you get a $$2\times 2$$ pixel block. Below, I did not extend the size of the right-hand side image, as it is all black. Now, think of a natural image of a linear combination of many single-pixel images.
A second more continuous vision is to consider the image as a function $$f$$. It can be treated separately in each $$x$$ and $$y$$ directions. A measure of spread $$\varsigma$$ for $$f(x)$$ would be:
$$\varsigma = \sqrt{\int \mathrm{d}x f^2(x)}\,.$$
Then, the double-sized image would be represented by $$f(x/2)$$. A change of variable will give you a $$2\varsigma$$ spread. The same reasoning would work in two dimensions, if you integrate $$f(x,y)$$ and $$f(x/2,y/2)$$.