Structure tensor

In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant respect the observing coordinates. The structure tensor is often used in and computer vision. For a function of two variables p = (x, y), the structure tensor is the 2×2 matrix where and are the partial derivatives of with respect to x and y; the integrals range over the plane ; and w is some fixed "window function" (such as a Gaussian blur), a distribution on two variables. Note that the matrix is itself a function of p = (x, y). The formula above can be written also as , where is the matrix-valued function defined by If the gradient of is viewed as a 2×1 (single-column) matrix, where denotes transpose operation, turning a row vector to a column vector, the matrix can be written as the matrix product or tensor or outer product . Note however that the structure tensor cannot be factored in this way in general except if is a Dirac delta function. In image processing and other similar applications, the function is usually given as a discrete array of samples , where p is a pair of integer indices. The 2D structure tensor at a given pixel is usually taken to be the discrete sum Here the summation index r ranges over a finite set of index pairs (the "window", typically for some m), and w[r] is a fixed "window weight" that depends on r, such that the sum of all weights is 1. The values are the partial derivatives sampled at pixel p; which, for instance, may be estimated from by by finite difference formulas. The formula of the structure tensor can be written also as , where is the matrix-valued array such that The importance of the 2D structure tensor stems from the fact eigenvalues (which can be ordered so that ) and the corresponding eigenvectors summarize the distribution of the gradient of within the window defined by centered at .