Keypoint position and scale in SIFT 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 ?

Thanks Michals
The $\sigma$ parameter is both. The Gaussian function can generate a scale-space where $\sigma$ is the scale parameter. It doesn't mean the image is scaled, instead it is the scale at which the features are being evaluated.

For example, with higher $\sigma$ the image is more blurred and therefore only larger image features contribute to the gradient histogram. SIFT is made scale-invariant by choosing the greatest feature response from a set of responses from different scales. The Gaussian function is used because with increasing scale one can guarantee no new local extrema (features) are introduced.

I'm not sure about the interpolated extrema, but I guess it is to obtain sub-pixel accuracy.