Why is Scale Space Needed to Detect Scale Invariant Features?

Question

I have been doing research on the SIFT (scale invariant feature transform) algorithm, and while I have repeatedly read that in order to detect image features that do not vary with respect to scales of visual perspective, a scale space must be constructed from the source image using repeated Gaussian smoothing and taking differences of those Gaussians. Then local extrema are found in the four dimensional scale space which are designated as scale invariant key points. How this process is performed seems straightforward, but my question is why does this process work at all? What is special about the Gaussain blur that allows us to do this? In the publications that I am reading (listed below), the reason this works is not clear to me.

Lindeberg, Tony. “Detecting Salient Blob-like Image Structures and Their Scales.” Scale-Space Theory in Computer Vision (1994): 249-70. Web. http://www.nada.kth.se/~tony/abstracts/Lin92-IJCV.html

Lindeberg, Tony. “Scale-space.” Encyclopedia of Computer Science and Engineering. Vol. 4. N.p.: n.p., n.d. 2495-504. Web. 16 Oct. 2016. ftp://ftp.nada.kth.se/CVAP/reports/Scale-Space-EncCompSci.pdf

Lowe, David G. “Distinctive Image Features from Scale-Invariant Keypoints.” International Journal of Computer Vision 60.2 (2004): 91-110. Web.

EDIT

I have read many high level explanations of this concept, which generally say that as the sigma of the Gaussian increases, fine grained features should be suppressed and that coarse grained features should be generalizations of fine grained features. But I am looking for a more rigorous and possibly mathematical explanation of why this is the case.

Answer 1

This is several years later (thanks, bot, for bumping the question).

What is special about the Gaussian blur that allows us to do this?

In theory, you could use any bandpass filter at different scales to implement a SIFT-like algorithm, but difference-of-Gaussians has a couple of features which makes it especially convenient.

First, the convolution of two Gaussians is a Gaussian. So applying a Gaussian blur to an image that has already been Gaussian-blurred is mathematically equivalent to applying a Gaussian (with larger sigma) blur to the original image. This saves a lot of work in practice.

Second, the difference-of-Gaussians is a reasonably good approximation to the Laplacian-of-Gaussian. The Laplacian operator measures the difference between the value of a function at a point, and the average of the function in a region around the point. (This is why solutions to Laplace's equation are often the same thing as "equilibrium" in physical systems.) But it's also an excellent way of capturing what it means to be a "feature" in image processing: a "feature" is a value that's different from the average in its neighbourhood.

Answer 2

If a maximum is found at a given octave X in the original image, then, if we have the scale-invariant property, this maxima will be found on the zoomed down (downscaled) image at octave X-1.