Automated target recognition (ATR) is a problem of great importance in a wide variety of applications: from military target recognition to recognizing flow-patterns in fluid- dynamics to anatomical shape-studies. The basic goal is to utilize observations (images, signals) from remote sensors (such as videos, radars, MRI or PET) to identify the objects being observed. In a statistical framework, probability distributions on parameters representing the object unknowns are derived an analyzed to compute inferences (please refer to  for a detailed introduction). An important challenge in ATR is to determine efficient mathematical models for the tremendous variability of object appearance which lend themselves to reasonable inferences. This variation may be due to differences in object shapes, sensor-mechanisms or scene- backgrounds. To build models for object variabilities, we employ deformable templates. In brief, the object occurrences are described through their typical representatives (called templates) and transformations/deformations which particularize the templates to the observed objects. Within this pattern-theoretic framework, ATR becomes a problem of selecting appropriate templates and estimating deformations. For an object (alpha) (epsilon) A, let I(alpha ) denote a template (for example triangulated CAD-surface) and let s (epsilon) S be a particular transformation, then denote the transformed template by sI(alpha ). Figure 1 shows instances of the template for a T62 tank at several different orientations. For the purpose of object classification, the unknown transformation s is considered a nuisance parameter, leading to a classical formulation of Bayesian hypothesis- testing in presence of unknown, random nuisance parameters. S may not be a vector-space, but it often has a group structure. For rigid objects, the variation in translation and rotation can be modeled through the action of special Euclidean group SE(n). For flexible objects, such as anatomical shapes, higher-dimensional groups such as a diffeomorphisms are utilized.