Why Does the Rudin Osher Fatemi (ROF) Method Use Variational Methods for Image Denoising When Denoising Problems Are Not Boundary Value Problems?
This turned out to be easier than I thought. Image processing is a boundary value problem, and the boundary is the set of pixels along the edge of the image. The common notation for this boundary in the Vese Guyader book is $\partial(\Omega)$, so that is not the most obvious notation for an image. Of course $\partial$ is often the boundary operator in differential topology too. I think for the Rudin-Osher-Fatemi method, there is a Neumann boundary condition, meaning that the first derivative at the boundary pixels is not changing.
So the answer was very easy after all.