Can someone explain intuitively why local maxima and minima in the scale-space domain make for good keypoints?
I understand using LoG or DoG zero-crossing points to identify spacial variations, i.e. corners. And I understand working at different scales to find scale-invariant features. Both of these are pretty intuitive.
But I fail to see what it is about local extrema that makes them good features for various algorithms like SIFT.
LoG and DoG (an approximation of LoG) masks can serve as blob detectors. A blob can exist in an image at a number of locations $(x,y)$-coordinates and scales (some parameter; $t$). In some situation where scale space is divided into 3 discrete 'slices' and there are only 'small,' 'medium' and 'large' sized blobs, a 'medium' sized blob will have some response to both the 'small' and 'large' sized detectors. However, a detector whose scale is matched to the scale of the blob will have the largest response. For this reason, we want to localize the blob by determining the maximum response in both spatial and scale coordinates.
One of the most important characteristics of the key points is its repeatability under different geometric transformations and also lighting. Repeatability ensures that if, for example, you have two images of the same scene, at different sizes and also with a different angle of rotation, the vast majority of key points in both images will coincide and, in this way, you can make a " matching " between both. In the SIFT algorithm these key points are invariant to translation, rotation and scale; for each key point the neighborhood is coded (in a radius proportional to the scale of the DOG operator) using histogram of gradients of 128 components, in this way each key point (x, y) will have associated a vector of 128 components, this vector is known as "descriptor". And what is the usefulness of this?
Object Recognition: given a gallery of images of different objects, a set of descriptors can be stored for each image. When a test image arrives, its descriptors are extracted and compared with the stored descriptors. For this, noma-L2 or classification sparse representation is generally used.
"Partial face recognition: An alignment free approach"
Scene Reconstruction: if you have several images of the different parts of a scene, these can be joined, based on the key points, to reconstruct an image of the whole scene.
SIFT looks for local extrema in the difference-of-Gaussian space. This essentially amounts to band-pass filtering the image and then looking for extrema in order to identify potential keypoints. Strong edges can create extrema in this domain, so you can think of this as an edge detection technique.
