Image Analysis in Poincaré-Peters Perceptual Representative Spaces

The problem considered in this paper is how to detect similarities in the content of digital images, useful in image retrieval and in the solution of the image correspondence problem,

i.e.

, to what extent does the content of one digital image correspond to content of other digital images. The solution to this problem stems from a recent extension of J.H. Poincaré’s representative spaces from 1895 introduced by J.F. Peters in 2010 and near sets introduced by J.F. Peters in 2007. Elements of a perceptual representative space are sets of perceptions arising from n-dimensional image patch feature vector comparisons. An image patch is a set of subimages. In comparing digital images, partitions of images determined by a particular form of indiscernibility relation ~ℬ is used. The

L

1

(taxicab distance) norm in measuring the distance between feature vectors for objects in either a perceptual indiscernibility or a perceptual tolerance. These relations combined with finite, non-empty sets of perceptual objects constitute various representative spaces that provide frameworks for image analysis and image retrieval. An application of representative spaces and near sets is given in this chapter in terms of a new form of content-based image retrieval (CBIR). This chapter investigates the efficacy of perceptual CBIR using Hausdorff, Mahalanobis as well as tolerance relation-based distance measures to determine the degree of correspondence between pairs of digital image. The contribution of this chapter is the introduction of a form of image analysis defined within the context of Poincare-Peters perceptual representative spaces and near sets.