Optical flow

Figure 8.13: The optical flow of features in an image due to motion in the world. These were computed automatically using image processing algorithms. (Image by Andreas Geiger, from Max Planck Institute for Intelligent Systems in Tübingen.)

Figure 8.14: Example vector fields: (a) A constant vector field, for which every vector is $(-1,0)$, regardless of the location. (b) In this vector field, $(x,y) \mapsto (x+y,x+y)$, the vectors point away from the diagonal line from $(-1,1)$ to $(1,-1)$, and their length is proportional to the distance from it.

Recall from Section 6.2, that the human visual system has neural structures dedicated to detecting the motion of visual features in the field of view; see Figure 8.13. It is actually the images of these features that move across the retina. It is therefore useful to have a mathematical concept that describes the velocities of moving points over a surface. We therefore define a vector field, which assigns a velocity vector at every point along a surface. If the surface is the $xy$ plane, then a velocity vector

$$(v_x,v_y) = \left( \frac{dx}{dt}, \frac{dy}{dt} \right)$$ (8.32)

is assigned at every point $(x,y)$. For example,

$$(x,y) \mapsto (-1,0)$$ (8.33)

is a constant vector field, which assigns $v_x = -1$ and $v_y = 0$ everywhere; see Figure 8.14(a). The vector field

$$(x,y) \mapsto (x+y,x+y)$$ (8.34)

is non-constant, and assigns $v_x = v_y = x + y$ at each point $(x,y)$; see Figure 8.14(b). For this vector field, the velocity direction is always diagonal, but the length of the vector (speed) depends on $x + y$.

To most accurately describe the motion of features along the retina, the vector field should be defined over a spherical surface that corresponds to the locations of the photoreceptors. Instead, we will describe vector fields over a square region, with the understanding that it should be transformed onto a sphere for greater accuracy.