Space-variant computer vision: a graph-theoretic approach

• Authors: Leo John Grady;Eric L. Schwartz
• Venue: Space-variant computer vision: a graph-theoretic approach
• Year: 2004

Space-variant sampling of visual input is ubiquitous in the higher vertebrate brain, because a large input space may be processed with high peak precision without requiring an unacceptably large brain mass. Space-variant sampling has been studied in computer vision for decades. A major obstacle to exploiting this architecture in machines, and understanding its role in biology, is the lack of algorithms that generalize beyond regular samplings. Most image processing algorithms implicitly assume a Cartesian grid underlying the sensor. This thesis generalizes image processing to a sensor architecture described by an arbitrary graph. This data structure separates the sensor topology, expressed by the graph edge structure, from its geometry, represented by coordinates of the vertex set. The combinatorial Laplacian of the sensor graph is a key operator underlying a series of novel image processing algorithms. First, a new graph partitioning algorithm for segmentation is presented that heuristically minimizes the ratio of the perimeter of the partition border and the area of the partitions, under a suitable definition of graph-theoretic area. This approach produces high quality image segmentations. Interpolation of missing data on graphs is developed, using a combinatorial version of the Dirichlet Problem, i.e., minimizing the average gradients of the interpolated values while maintaining fixed boundary conditions. This leads to the solution of the Laplace Equation, which represents the steady-state of the diffusion process for stated boundary conditions. Results compare favorably to both isotropic and anisotropic diffusion for filling-in of missing data. A pyramid graph is defined by connecting vertical and horizontal levels of the Laplacian pyramid data structure. The isoperimetric algorithm, run on the graph pyramid, yields an improved segmentation at little extra computational cost. Finally, a small-world graph topology is employed by randomly introducing a few new edges to the image graph. This results in a large speed-up in computation time, with identical final results. The algorithms developed in this thesis do not require that the data associated with the graph are embedded in two-dimensions or even have a metric structure. Therefore, this approach to generalized image processing may find wider application in other areas of discrete data processing.