A general variational framework for image approximation and segmentation is introduced. By using a continuous "line-process" to represent edge boundaries, it is possible to formulate a variational theory of image segmentation and approximation in which the boundary function has a simple explicit form in terms of the approximation function. At the same time, this variational framework is general enough to include the most commonly used objective functions. Application is made to Mumford-Shah type functionals as well as those considered by Geman and others. Employing arbitrary Lp norms to measure smoothness and approximation allows the user to alternate between a least squares approach and one based on total variation, depending on the needs of a particular image. Since the optimal boundary function that minimizes the associated objective functional for a given approximation function can be found explicitly, the objective functional can be expressed in a reduced form that depends only on the approximating function. From this a partial differential equation (PDE) descent method, aimed at minimizing the objective functional, is derived. The method is fast and produces excellent results as illustrated by a number of real and synthetic image problems.