|
In this paper we will be concerned with the recognition of 3D objects from a single 2D view obtained via a generalized weak perspective projection. Our methods will be independent of camera/sensor position and any camera/sensor parameters, as well as independent of the choice of coordinates used to express the feature point locations on the object or in the image. Our focus will be on certain natural metrics on the associated shape spaces (which are called object space and image space, respectively). These metrics provide a distance between two object shapes or between two image shapes and are a generalization of the Procrustes metrics of Statistical Shape Theory. They can be shown to be induced from the L2 metric on the space of all n-tuples of feature points via a modified orbit metric, i.e. as the minimum distance between two orbits under the action of the affine group, modified to account for scale and shear. Finally we will define two notions of “distance” between an object and an image (with distance zero being a match under some weak perspective projection). This makes use of the object-image equations and computes the distance entirely in either the object space or in the image space. A Metric Duality Theory shows these two notions of “distance” are the same. Ultimately, we would like to know if two configurations of a fixed number of points in 2D or 3D are the same if we allow affine transformations. If they are, then we want a distance of zero, and if not, we want a distance that expresses their dissimilarity - always recognizing that we can transform the points. The Procrustes metric, described in the shape theory literature [4], provides such a notion of distance for similarity transformations. However, it does not allow for weak perspective or perspective transformations and is fixed in a particular dimension. By the later we mean that it cannot be regarded as giving us a notion of “distance” between, say, a 3D configuration of points and a 2D configuration of points, where zero distance corresponds to the 2D points being, say, a generalized weak perspective projection of the 3D points. In this paper, we show that generalizations of the Procrustes metric exist in the above cases. Moreover these new metrics are quite natural in the context of the algebro-geometric formulation of the object/image equations discussed in Part I of this paper and reviewed below. These metrics also provide a rigorous foundation for error and statistical analysis in the object recognition problem.
|
