I was studying the Gunnar Farneback method for dense optical flow and encountered the following paragraph:
By stacking the frames of an image sequence onto each other we obtain a spatiotemporal image volume with two spatial dimensions and a third temporal dimension. It is easy to see that movement in the image sequence induces structures with certain orientations in the volume. A trans- lating point, e.g., is transformed into a line whose direction in the volume directly corresponds to its velocity. A powerful representation of local orientation is the ori- entation tensor . In 3D this tensor takes the form of a $3 \times 3$ symmetric positive semidefinite matrix $T$ and the quadratic form $\hat{u}^T T \hat{u}$ can be interpreted as a measure of how much the signal locally varies in the direction given by $\hat{u}$.
My understanding of the 3D orientation tensor is that it is a rotation matrix in 3D world. Thus, it is a $3 \times 3$ matrix. But I don't understand how $\hat{u}^T T \hat{u}$ is derived and why $\hat{u}^T T \hat{u}$ can represent how much the signal locally varies in the direction given by $\hat{u}$. Thank you.