Spatiotemporal Blur and Frame Differences

Blurring a cube

Steering the blur by velocity

The frame difference

Derivatives obey the constancy relation

A video is convolved with a spatiotemporal Gaussian having small spatial width $\sigma = 1$ and large temporal width $\sigma_t = 4$. What does the output look like?

A video contains a pedestrian walking right at 1 pixel per frame and a cyclist crossing left at 4 pixels per frame. You filter with the steered Gaussian $g(x - v_x t, y, t)$ using $v_x = 1$. What happens?

A pattern satisfies brightness constancy with $v_x = 2$ pixels per frame and no vertical motion. At some pixel the spatial derivative $\partial \ell / \partial x$ equals $3$. Using $\partial \ell/\partial t = -v_x\, \partial \ell/\partial x$, compute the temporal derivative there.