22-02-2013, 02:42 PM
Robust Albedo Estimation from a Facial Image With Cast Shadow Under General Unknown Lighting
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Abstract
Albedo estimation from a facial image is crucial
for various computer vision tasks, such as 3-D morphablemodel
fitting, shape recovery, and illumination-invariant face
recognition, but the currently available methods do not give good
estimation results. Most methods ignore the influence of cast
shadows and require a statistical model to obtain facial albedo.
This paper describes a method for albedo estimation that makes
combined use of image intensity and facial depth information
for an image with cast shadows and general unknown light. In
order to estimate the albedo map of a face, we formulate the
albedo estimation problem as a linear programming problem that
minimizes intensity error under the assumption that the surface
of the face has constant albedo. Since the solution thus obtained
has significant errors in certain parts of the facial image, the
albedo estimate needs to be compensated. We minimize the mean
square error of albedo under the assumption that the surface
normals, which are calculated from the facial depth information,
are corrupted with noise. The proposed method is simple and the
experimental results show that this method gives better estimates
than other methods.
INTRODUCTION AND PRELIMINARIES
Introduction
ALBEDO is the ratio of reflected light at a surface point
when it is illuminated, which is determined by the
property of material. Albedo estimation from a facial image is
necessary in many applications of computer vision and graphics
such as illumination-invariant face recognition [1]– [3],
3D morphable-model fitting [4], [5], image relighting [1],
[3], [6], [7], and so on. Since albedo does not change due
to illumination variation, it is useful for face recognition
whose performance can be greatly affected by illumination
variation.
Lambert’s Cosine Law for Albedo Estimation
Albedo can be obtained by using Lambert’s cosine law [14],
[18]. The reflectance surface is either specular, Lambertian,
or both. An ideal specular surface behaves like an ideal
mirror, and reflected light can leave only along the specular
direction which reflects the direction of incoming light with
respect to the surface normal. Consequently, luminance is
observed differently depending on the direction of view-point.
A Lambertian reflection, in which incoming light is reflected
in various directions, is different from specular reflection, and
the luminance of a Lambertian surface is determined by the
incident light regardless of view-point. The surface of the face
is approximated as a combination of a Lambertian component
and a specular component. We assume that the surface of a
face is Lambertian because only a small part of a facial image
has a specular component.
Generating Images With Cast Shadow
If a dense set of illumination maps with cast shadow is
obtained, the illumination map of an image under general
light conditions can be estimated by finding a proper linear
combination of these illumination maps. In order to generate
a dense set of illumination maps with cast shadow, we need
to disperse the light sources evenly so that light can travel in
every direction. When the light sources are not balanced, the
illumination maps generated by these light sources are biased.
Then, it can be a problem finding a proper linear combination
of these illumination maps for an image. Since the light is not
usually balanced when we distribute light sources randomly,
it is important to make the distribution balanced.
Estimate Error Compensation by Minimizing Mean Square
Error
The albedo estimate in the previous subsection is good in the
sense that it is not affected by cast shadow and is mostly constant
for skin area (Fig. 10). Also, the albedos for outlier areas,
such as eyes, eyebrows, hair, lips, and so on, are estimated
fairly well. However, the errors are significant in particular
areas, e.g., around the eyes and nose. Thus, it is necessary to
compensate for the albedo estimates. The previous work [17]
compensated the albedo estimates by using a median filter,
which removed a large part of the estimate errors. However,
there were still areas of significant error and the compensated
images looked somewhat unnatural (Fig. 10(b)). In this
subsection, we propose another albedo compensation method
which can give better results than the previous work [17].
CONCLUSION
In this paper, an albedo estimation method has been
proposed for a facial image when the intensity and depth
information of a facial image is given. The lighting condition
of an image may be general and unknown. In order to remove
the influence of cast shadow, we generated images for each of
the light sources, which are evenly distributed. The dimension
of the generated images with cast shadows was reduced by
PCA and the intensity of an image was represented by a
linear combination of the generated images. We formulated
the illuminance estimation as an optimization problem using
the l1-norm, and using the illuminance estimate, we obtained
the albedo estimate. We further compensated for the albedo
estimation error by minimizing the mean square error under
the assumption that the albedo estimates were corrupted with
noise.