If I understood it right, a keypoint is a tuple $$(x,y,\sigma, r),$$ where $x,y$ define the position of the keypoint and $r$ an orientation (given by the most domiant gradiant vector around the keypoint). I'm still not sure about the intention of $\sigma$.
What is done in the first part of SIFT is to consider $$L(x,y,\sigma):=G(x,y,\sigma) \ast I(x,y),$$ where $I(x,y)$ denote the value of the image at position $(x,y)$, $\ast$ the convolution and $G(x,y,\sigma)$ the gaussian filter with standard-deviation $\sigma$, evaluated at position $(x,y)$.
Now DoGs are considered, for different sizes of the image and for different blurring-values $\sigma$ and interesting points are collected.
What confuses me most about this procedure is that Lowe denotes $\sigma$ as the scale of the image. However, $\sigma$ is just a blurring-parameter.
Hence, I define a blured image of a scaled image by $$L(x,y,\sigma,s):=G(x,y,\sigma) \ast I_s(x,y),$$ where $I_s$ is the Image $I$ scaled by factor $s$ (So multiplying both dimensions of the image by s, and rescaling it). Thus, an interesting point in a DoG is a point $(x,y)$ in the image $L(x,y,k\sigma,s)-L(x,y,\sigma,s)$.
In Lowe's paper, he never considers the parameter 's' but just $\sigma$. Why is that possible ? How do I relate $(x,y)$ of $L(x,y,\sigma,s)$ to a position in the image $I$ and why do I save the blurring-parameter $\sigma$?
In what way is the keypoint now scale (resize)-invariant ?
Should there be in the resized image at some DoG a keypoint, which corresponds to the former keypoint by having more or less the same orientation and magnitude of its surrounding gradiants ? Is that the invariance Lowe is talking about ?
To find a keypoints, a local extrema of $L(x,y,\sigma)$ around a point $(x,y,\sigma)$ is considered. What is idea behind using interpolated extrema via Tayler expansion ?