Symmetric measures via moments

Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection, namely, one between probabilistic models invariant under a finite group of (non-singular) linear transformations and polynomials invariant under the same group. Two specific aspects of the connection are discussed: generalization of the (uniqueness part of the multivariate) problem of moments and log-linear, or toric, modeling by expansion of invariant terms. A distribution of minuscule subimages extracted from a large database of natural images is analyzed to illustrate the above concepts.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6144 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Ordinal Coding of Image Microstructure

Abstract-Applications of rank-order-based methods to image and signal analyses have primarily focused on filtering. Classical median, min, and max filters have long been part of standard image processing toolboxes. More recent work has focused on more elaborate versions of such filters and associated computational issues. However, the application of these nonlinear methods to problems such as image interpretation has been scarce. We attempt to show that simple rank-order-based methods for coding image patches provide informative and computationally efficient local image descriptors