The variational representations used in image processing are much more complicated than you suggest. In many cases, they are developed with semantic models that

The variational representations used in image processing are much more complicated than you suggest. In many cases, they are developed with semantic models that inform grammars of what the image can be composed of, and directly model things like occlusion, cusps, edges, blur, etc., when postulating a joint functional space that can describe the “energy” of an image, which then minimizing gives not only pixel values but also parametric curves for boundaries and sometimes also labels. It’s essentially a way to encode semantics as prior distributions over possible images, then variational methods “optimize” for the posterior parameters given some data. I’m trying to say that that whole exercise listed above, conceptually, is basically a failed idea compared with solving the same problems using deep learning to automatically discover what that manifold looks like in parameter space and transform it directly to the output space of practical interest (segmentation boundaries, etc.).