Analysis of multiple views of the same scene is an area of active research in computer vision. The study of the structure of points and lines in two views received much attention in the eighties and early nineties [2,4,8]. Studies on the constraints existent in three and more views have followed since then [3,10,11,12]. These multiview studies have concentrated on how geometric primitives like points, lines and planes are related across views. Specifically, the algebraic constraints satisfied by the projections of such primitives in different views have been the focus of intense studies.
An important issue in multi-view analysis is recognition. The problem of recognition in the context of multiple views is as follows: Given the image of an object in one or more views, can we recognise the object in novel views, specifically, when the viewing parameters of the camera are unknown.
An object can be recognised either based on the object-boundary or the textural and structural content inside the boundary. In this paper, we limit the recognition problem to that of recognising planar objects from their boundaries. A notable object recognition approach is due to Ullman and Basri [13] who formulated the recognition problem using linear combination of models, for orthographic views. This recognition differs from the conventional shape recognition approaches. This provides algebraic constraints between different views of the same object in contrast to a set of features invariant to similarity transformations. This algorithm was later generalised by Shashua [11] for perspective views. These results demonstrate that the various views of an object lie in a lower dimensional linear subspace and there can exist some algebraic constraints for recognition of objects in multiple views.
A number of approaches have been proposed for planar shape recognition. Algorithms for planar shape recognition include recognition by alignment [5], polygonal approximation [9], based on geometrically invariant features [6],etc. . Boundaries are also recognised by modelling the boundary in a transform domain like the Fourier one [14]. All these algorithms limited their attention to similarity transformation between views. In most practical situations, the image to image homography is more general than the simple similarity transformation. When a planar object is imaged from multiple viewpoints, the image to image transformation is general projective and the conventional algorithms based on Euclidean and similarity frameworks will not work for them. Klaus Arbter et al. [1] formulated techniques for affine invariant recognition. Their emphasis was on choosing a suitable set of affine invariant features and then perform matching in an affine invariant space.
In this paper, we try to analyse the properties of a collection of points, such as a planar object's contour, in multiple views instead of analysing them as independent points. Collections of points such as a boundary have more information than isolated points. The sequencing inherent in such a collection makes a transform domain approach, such as the Fourier one, a good tool to study their properties. The linear image-to-image relationships combined with the properties of the contour in the Fourier domain enable rich constraints that essentially characterize the contour independent of the viewpoint. We come up with a number of view-independent characterizations of the planar shape boundary using measures computed in the Fourier domain. Recognition of a shape in different views is a natural consequence if the description is invariant to the types of view transformations. We present the problem of recognition in multiple views in the form of a rank constraint on a matrix computed from the contours. Some preliminary results were presented in [7].
We formulate the basic problem of view-independent characterization and give our notation in Section 2. Section 3 presents the main results in terms of a number of rank constraints on the measurement matrices computed from the Fourier domain representation of contours. Results of experiments on synthetic and real data are given in Section 4. Some concluding remarks are in Section 5.