Consider a set of deformable objects undergoing geometric and radiometric transformations. As a result of the action of these transformations, the set of different realizations of each object is generally a manifold in the space of observations. Assuming the geometric deformations an object undergoes, belong to some finite dimensional family, it has been shown that the universal manifold embedding (UME) provides a set of nonlinear operators that universally maps each of the different manifolds, where each manifold is generated by the set all of possible appearances of a single object, into a distinct linear subspace of an Euclidean space. In this paper, we generalize this framework to the case where the observed object undergoes both an affine geometric transformation, and a monotonic radiometric transformation, and present a novel framework for the detection and recognition of the deformable objects. Applying to each of the observations an operator that makes it invariant to monotonic amplitude transformations, but is geometry-covariant with the affine transformation, the set of all possible observations on that object is mapped by the UME into a single linear subspace-invariant with respect to both the geometric and radiometric transformations. The embedding of the space of observations is independent of the specific observed object; hence it is universal. The invariant representation of the object is the basis of a matched manifold detection and tracking framework of objects that undergo complex geometric and radiometric deformations: the observed surface is tessellated into a set of tiles such that the deformation of each one is well approximated by an affine geometric transformation and a monotonic transformation of the measured intensities. Since each tile is mapped by the radiometry invariant UME to a distinct linear subspace, the detection and tracking problems are solved by evaluating distances between linear subspaces. Classification in this context becomes a problem of determining which labeled subspace in a Grassmannian is closest to a subspace in the same Grassmannian, where the latter has been generated by radiometry invariant UME from an unlabeled observation.

We present a novel framework for detection, tracking and recognition of deformable objects undergoing geometric and radiometric transformations. Assuming the geometric deformations an object undergoes, belong to some finite dimensional family, it has been shown that the universal manifold embedding (UME) provides a set of nonlinear operators that universally maps each of the different manifolds, where each manifold is generated by the set all of possible appearances of a single object, into a distinct linear subspace. In this paper we generalize this framework to the case where the observed object undergoes both an affine geometric transformation, and a monotonic radiometric transformation. Applying to each of the observations an operator that makes it invariant to monotonic amplitude transformations, but is geometry-covariant with the affine transformation, the set of all possible observations on that object is mapped by the UME into a distinct linear subspace - invariant with respect to both the geometric and radiometric transformations. This invariant representation of the object is the basis of a matched manifold detection and tracking framework of objects that undergo complex geometric and radiometric deformations: The observed surface is tessellated into a set of tiles such that the deformation of each one is well approximated by an affine geometric transformation and a monotonic transformation of the measured intensities. Since each tile is mapped by the radiometry invariant UME to a distinct linear subspace, the detection and tracking problems are solved by evaluating distances between linear subspaces.

We present a novel framework for detection and recognition of deformable objects undergoing geometric deformations. Assuming the geometric deformations belong to some finite dimensional family, it is shown that there exists a set of nonlinear operators that universally maps each of the different manifolds, where each manifold is generated by the set all of possible appearances of a single object, into a unique linear subspace. In this paper we concentrate on the case where the deformations are affine. Thus, all affine deformations of some object are mapped by the above universal manifold embedding into the same linear subspace, while any affine deformation of some other object is mapped by the above universal manifold embedding into a different subspace. It is therefore shown that the highly nonlinear problems of detection and recognition of deformable objects can be formulated in terms of evaluating distances between linear subspaces. The performance of the proposed detection and recognition solutions is evaluated in various settings.