Geometric modeling

Geometric image modeling relies on the interpretation of an image as a function defined on a grid domain. Such models describe and analyze the local spatial relationships, or geometry, between the function values via tools relying on calculus. In this way, such models invariably connect to the fields of differential geometry and differential equations. Such models treat images as functions that can be considered as points in high-dimensional Sobolev spaces. A Sobolev space is a normed space of functions such that all the derivatives upto some order $k$, for some $k \ge 1$, have finite $L_p$ norms, given $p \ge 1$. Modeling image functions in such spaces, however, does not accommodate for the existence of discontinuities, or edges, in images. Edges are formed at the silhouettes of objects and are vital features in image analysis and processing. To accommodate edges in images, two popular models exist. Mumford and Shah [110] invented the object-edge model assuming that the grid image domains can be partitioned into mutually-exclusive and collectively-exhaustive sets such that the resulting functions on each partition belong to Sobolev spaces. Moreover, the partitions have regular boundaries, not fractals, with finite lengths or areas as characterized by the Hausdorff measure. In this way, the partition boundaries can coincide with the edges in the image, segmenting the image into continuous functions that belong to Sobolev spaces. Rudin, Osher, and Fatemi [145] proposed the bounded-variation image model where they assumed images to possess bounded variation. Both these image models, however, impose strong constraints on the data and do not apply well to textured images. To explicitly deal with textured images, researchers have proposed more sophisticated image models that decompose an image into the sum of a piecewise-constant part and an oscillatory texture part. Such models are known as cartoon-texture models [13].