Many of the variational models of image processing and computer vision involve optimizing an energy over interfaces. An important example is image segmentation, where the goal is to partition the image domain into regions containing distinct objects. Typically, the models include a geometric penalty term, such as perimeter or Euler's elastica energy. We will discuss some of the popular algorithms for computing these models, including level set, diffuse interface (phase field), and diffusion generated motion-based approaches. However, the emphasis will be on more recent algorithms that convert some of these notoriously hard, non-convex optimizations to equivalent convex optimization problems.