# Analytical expression for salient Blob detection in scale space

I am working on exact mathematical expression which can be obtained in image processing tasks.(I do not know much about image processing but my work is more mathmatical) In Gaussian scale space, we obtain the salient blob like structures in image by finding the extremum points in scale-normalized Laplacian of the image in the famous SIFT method. Can anyone give me reference to

1) how can we represent this condition mathematically, for e.g. this will be the set of points given as:

$$S=\{s= (x,y,t): \text {s is the point of extremum of }\sigma^2 \nabla^2 L(x,y,t) \}$$ should I write it as: $$S=\{s= (x,y,t): \frac{d[ \sigma^2 \nabla^2 L(s)]}{ds}=0 \}$$

2) How have we reached to the fact that extremum in scale-normalized laplacian will give us the salient blob like structures? I have read some papers e.g. Blostein and Ahuja and T lindeberg "scale space theory in computer vision", where it is mentioned that you can use laplaican operator to obtain such features, but even in that paper I cannot find any mathmatical expression leading to such conclusion.

If you see paper like "scale and differential structure of images" by florack et. al., they have mathematically proven how we obtain Gaussian kernel in scale space. Why is there no such mathematical treatment of image features like salient blob detection in SIFT and edge detection in non-linear scale space by KAZE(where you use similar extrema for determinant of Hessian.). If there is, can someone give reference of the paper or book?

I will be very thankful if someone can answer this.