|
The appearance of a particular object depends on both the viewpoint
from which it is observed and the light sources by which it is
illuminated. If the appearance of two objects is never identical for
any pose or lighting conditions, then -- in theory -- the objects can
always be distinguished or recognized. The question arises: What is
the set of images of an object under all lighting conditions and pose?
In this paper, we consider only the set of images of an object under
variable illumination (including multiple, extended light sources and
attached shadows). We prove that the set of n-pixel images of a
convex object with a Lambertian reflectance function, illuminated by
an arbitrary number of point light sources at infinity, forms a convex
polyhedral cone in R^n and that the dimension of this
illumination cone equals the number of distinct surface normals.
Furthermore, we show that the cone for a particular object can be
constructed from three properly chosen images. Finally, we prove that
the set of n-pixel images of an object of any shape and with an
arbitrary reflectance function, seen under all possible illumination
conditions, still forms a convex cone in R^n. These results
immediately suggest certain approaches to object recognition.
Throughout this paper, we offer results demonstrating the empirical
validity of the illumination cone representation.
For more information about Illumination cones, see:
The set of all possible n-pixel images of a scene in fixed pose
under variable illumination including shadows is a convex cone in
RR^n. When the object can be modeled as Lambertian, the cone can
be learned from three or more images. In this case, four images (no
shadowing) are used to construct the cone of a desktop still life
which could just as well have been a scene within our urban
surveillance scenario. The set of images under multiple, extended
light sources without shadowing lies in the intersection of a 3-D
linear subspace and the positive orthant of RR^n [Nayar and Murase
1996, Shashua 1992]. Basis images for this 3-D linear subspace can be
estimated using SVD from three or more images; the direction of light
sources is not needed. The extreme rays of the illumination cone can
then be constructed from the 3-D linear subspace. At the bottom of
the figure are artificially generated images of the scene that lie in
the cone. Note the strong shadow lines and that their locations are
consistent with the expected surface normals and light source
directions. None of the original images had such shadowing, yet they
appear here.
-
The MPEG movie (450kB) shows a
sequence of images of a face constructed by sampling the illumination
cone using a single light source that moved about a great circle.
-
The MPEG movie (1.1MB) shows a
video of a face constructed by sampling the illumination cone using a
two light sources that are moving along two great circles at two
different velocities.
|
