Optimal approximations by piecewise smooth functions and associated variational problems

This reprint introduces and studies the most basic properties of three new variational problems suggested by applications to computer vision. The motivating task is to appropriately decompose the domain of a function g(x,y) that represents an image—the strength of the light signal striking a plane domain R when a three-dimensional world is observed from a point through a lens. The paper sets up the physical model of how images are formed in order to frame these problems.

The key observation is that light reflected off the surfaces of distinct solid objects strikes different open subsets of the image domain, so when one object appears partly in front of another, their regions share a common boundary, and the image is generally discontinuous along this edge. The three variational problems formalize how to optimally approximate such an image by piecewise smooth functions over these regions. By rigorously treating this optimal approximation of images separated by edges, the work provided a mathematical foundation for image segmentation and edge detection.