Brisk (binary robust invariant scalable keypoints )

Question

I am trying to do BRISK my own code in matlab.

Where ı am stack, ı don't understand what this expression means.

let us consider one of the N*(N −1)/2 sampling-point pairs (pi, pj).

A = {(pi, pj) ∈ R2 × R2 | i < N ∧ j < i ∧ i, j ∈ N }

The other my question , what is the difference between local gradient and global gradient?

Answer 1

The expression means that you are looking at a pair of pixels (pi, pj), such that both pixels belong to the region R2 x R2, and the two pixels cannot be the same.

Gradient is a vector (Ix, Iy), where Ix is the first derivative in the x direction, and Iy is the first derivative in the y direction. This vector is defined at a point, so gradient is local by definition. I don't know what global gradient means. More context may help here.

Answer 2

Given that we have set of points of size N. N*(N −1)/2 is N choose 2 which equals number of subsets of size 2 that can be taken from a set of size N ( a concept in probability called Combinations). Because you are working with pair of points you need subset size to be 2.

R refers to the set of all real numbers ( a single value). When it is squared it refers to the Cartesian plane, so pi is a pair of real numbers(x,y), a point in the Cartesian plane.

The character '^' is AND operation. So all the following conditions has to be satisfied:

1. the index i of the first point, pi, should be less than N
2. the index j of the second point must be less than the index of the first point.
3. Like i, j must also be less than N

Local gradient is computed locally on the pair of pixels pi and pj. while, global gradient is estimated for the region surrounding the keypoint by accumulating local gradients.