Dimensionality reduction by canonical contextual correlation projections

A linear, discriminative, supervised technique for reducing feature vectors extracted from image data to a lower-dimensional representation is proposed. It is derived from classical Fisher linear discriminant analysis (LDA) and useful, for example, in supervised segmentation tasks in which high-dimensional feature vector describes the local structure of the image. In general, the main idea of the technique is applicable in discriminative and statistical modelling that involves contextual data. LDA is a basic, well-known and useful technique in many applications. Our contribution is that we extend the use of LDA to cases where there is dependency between the output variables, i.e., the class labels, and not only between the input variables. The latter can be dealt with in standard LDA. The principal idea is that where standard LDA merely takes into account a single class label for every feature vector, the new technique incorporates class labels of its neighborhood in its analysis as well. In this way, the spatial class label configuration in the vicinity of every feature vector is accounted for, resulting in a technique suitable for e.g. image data. This spatial LDA is derived from a formulation of standard LDA in terms of canonical correlation analysis. The linearly dimension reduction transformation thus obtained is called the canonical contextual correlation projection. An additional drawback of LDA is that it cannot extract more features than the number of classes minus one. In the two-class case this means that only a reduction to one dimension is possible. Our contextual LDA approach can avoid such extreme deterioration of the classification space and retain more than one dimension. The technique is exemplified on a pixel-based segmentation problem. An illustrative experiment on a medical image segmentation task shows the performance improvements possible employing the canonical contextual correlation projection.