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 summarizes the predominant directions of the gradient in a specified neighborhood of a point, and the degree to which those directions are coherent. The structure tensor is often used in image processing and computer vision.

For a function ${\displaystyle I}$ of two variables p=(x,y), the structure tensor is the 2×2 matrix

where ${\displaystyle I_{x}}$ and ${\displaystyle I_{y}}$ are the partial derivatives of ${\displaystyle I}$ with respect to x and y; the integrals range over the plane ${\displaystyle \mathbb {R} ^{2}}$; and w is some fixed "window function", a distribution on two variables. Note that the matrix ${\displaystyle S_{w}}$ is itself a function of p=(x,y).

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