This paper proposes a novel method for texture classification using high-order local pattern descriptor: Local Derivative Pattern (LDP). LDP is used to encode directional pattern features based on local derivative variations. The nth order LDP is proposed to encode the (n-1)th order local derivative direction variations, which can capture more detailed information. The local texture information...

Hermite polynomials arise when dealing with functions of normally distributed variables, and are commonly thought as the analog simple on regular variables. Therefore expansion should be an Taylor expansion. Indeed there is a strong connection between two – general coefficient in weighted integral nth derivative, compared to derivative evaluated at zero case Taylor. This fact can used derive fo...

For any field K, a field K(ζn) where ζn is a root of unity (of order n) is called a cyclotomic extension of K. The term cyclotomic means circle-dividing, and comes from the fact that the nth roots of unity divide a circle into arcs of equal length. We will see that the extensions K(ζn)/K have abelian Galois groups and we will look in particular at cyclotomic extensions of Q and finite fields. T...

We will start with a review of some calculus. We start with one dimension to give intuition for higher dimensions. Given a function from one variable to one variable, for example, f(x) = x, we can approximate it via its derivatives. The most trivial approximation is without derivatives: we can approximate f(x) near c by just the constant f(c). But if f is rapidly changing (has large derivative)...

The nonextensive entropy of Tsallis can be seen as a consequence of postulates on a self-information, i.e., the constant ratio of the first derivative of a self-information per unit probability to the curvature (second variation) of it. This constancy holds if we regard the probability distribution as the gradient of a self-information. Considering the form of the nth derivative of a selfinform...