Introduction to invariant and equivariant problems

Summary

The curve completion problem

Consider the ‘curve completion problem’, which is a subproblem of the much more complex ‘inpainting problem’. Suppose we are given a partially obscured curve in the plane, as in Figure 0.1, and we wish to fill in the parts of the curve that are missing. If the missing bit is small, then a straight line edge can be a cost effective solution, but this does not always give an aesthetically convincing look. Considering possible solutions to the curve completion problem (Figure 0.2), we arrive at three requirements on the resulting curve:

• it should be sufficiently smooth to fool the human eye,

• if we rotate and translate the obscured curve and then fill it in, the result should be the same as filling it in and then rotating and translating,

• it should be the ‘simplest possible’ in some sense.

The first requirement means that we have boundary conditions to satisfy as well as a function space in which we are working. The second means the formulation of the problem needs to be ‘equivariant’ with respect to the standard action of the Euclidean group in the plane, as in Figure 0.3. This condition arises naturally: for example, if the image being repaired is a dirty photocopy, the result should not depend on the angle at which the original is fed into the photocopier.