Statistical measurements of the variability of probability distributions are important in many image analysis applications. For instance, let the appearance of a material in a picture be represented by the distribution of its pixel values. It is necessary to model the variability of these distributions to understand how the appearance of the material is affected by viewpoint, lighting, or scale changes. In medical imaging, an organ's appearance varies not only due to the parameters of the imaging device but also due to changes in the organ, either within a patient day to day or between patients. Classical statistical techniques can be used to study distribution variability, given a distribution representation for which variation forms linear subspaces. For many distributions relevant to image analysis, standard representations are either too constrained or have nonlinear variation, in which case classical linear multivariate statistics are not applicable. This dissertation presents general, non-parametric representations of a variety of distribution types, based on the quantile function, for which a useful class of variability forms linear subspaces. A key consequence is that principal component analysis can be used to efficiently parameterize their variability, i.e., construct a distribution family.

The quantile function framework is applied to two driving problems in this dissertation: (1) the statistical characterization of the texture properties of materials for classification, and (2) the statistical characterization of the appearance of objects in images for deformable model based segmentation. It is shown that in both applications the observed variability forms appropriately linear subspaces, allowing efficient modeling. State of the art results are achieved for both the classification of materials in the Columbia-Utrecht database and the segmentation of the kidney, bladder, and prostate in 3D CT images. While the applications presented in this dissertation use image-based appearance observations in the field of image analysis, the methods and theory should be widely applicable to the variety of observations found in the many scientific fields, and, more specifically, to shape observations in the field of computer vision.