3
$\begingroup$

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.

$\endgroup$

1 Answer 1

4
$\begingroup$

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. Double size image 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 $f(x/2,y/2)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.