Parametric Manifold of an Object under Different Viewing Directions

Abstract

The appearance of a 3D object depends on both the viewing directions and illumination conditions. It is proven that all n-pixel images of a convex object with Lambertian surface under variable lighting from infinity form a convex polyhedral cone (called illumination cone) in n-dimensional space. This paper tries to answer the other half of the question: What is the set of images of an object under all viewing directions? A novel image representation is proposed, which transforms any n-pixel image of a 3D object to a vector in a 2n-dimensional pose space. In such a pose space, we prove that the transformed images of a 3D object under all viewing directions form a parametric manifold in a 6-dimensional linear subspace. With in-depth rotations along a single axis in particular, this manifold is an ellipse. Furthermore, we show that this parametric pose manifold of a convex object can be estimated from a few images in different poses and used to predict object's appearances under unseen viewing directions. These results immediately suggest a number of approaches to object recognition, scene detection, and 3D modelling. Experiments on both synthetic data and real images were reported, which demonstrates the validity of the proposed representation.