Complexity of Indexing : E cient and Learnable Large Database

Object recognition starts from a set of image measurements (including locations of points, lines, surfaces, color, and shading), which provides access into a database where representations of objects are stored. We describe a complexity theory of indexing, a meta-analysis which identiies the best set of measurements (up to algebraic transformations) such that: (1) the representation of objects are linear subspaces and thus easy to learn; (2) direct indexing is eecient since the linear subspaces are of minimal rank. Index complexity is determined via a simple process, equivalent to computing the rank of a matrix. We readily rederive the index complexity of the few previously analyzed cases. We then compute the best index for new and more interesting cases: 6 points in one perspective image, 6 directions in one para-perspective image, and 2 perspective images of 7 points. With color we get the following result: 4 color sensors are suucient for color constancy at a point, and the sensor-output index is irreducible; the most eecient representation of a color is a plane in 3D space. For future applications with any vision problem where the relations between shape and image measurements can be written down, we give an automatic process to construct the most eecient database that can be directly obtained by learning from examples.