Based on "Multiscale vessel enhancement filtering" by A.F. Frangi, 1998. Link to paper.
"A common approach to analyze the local behavior of an image, L, is to consider its Taylor expansion in the neighborhood of a point Xo" - A.F. Frangi
L(Xo+deltaXo, s) ~= L(Xo, s) + (tranpose(deltaXo))(gradient vector of Xo at scale s) +tranpose(deltaXo)(Hessian matrix at Xo and scale s)*(deltaXo)
Looking at an image as a matrix of values, examining the Taylor expansion/Taylor series in the neighborhood of a point will give us information with which we can analyze or modify. A Taylor expansion is a sum of terms composed of the derivatives of a function around a single point. We know this works because a Taylor expansion approximates the function that models the "surface" of intensity.
Per Frangi, we're looking to the second-derivative expansion (the Hessian) as shown in the equation above. To be fair, this isn't the theoretical Taylor expansion because that one has a function as input and expand it, here we are approximating the terms that makes up the expansion, working backwards in a way.
A Hessian matrix provides valuable information in the form of its eigenvalues.