Intensity-Based and Polarisation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 160

3.4 Shape from Polarisation

An important drawback of the single-image shape from shading approach is the
fact that an ill-posed problem has to be solved, since it is desired to determine two
surface gradients (and sometimes also an albedo value) for each pixel based on
a single intensity measurement. This drawback is overcome either by introducing
constraints on the reconstructed surface, such as knowledge about the surface at the
boundaries, smoothness, or uniform albedo, or by regarding several images of the
surface acquired under different illumination conditions.
A further possibility to transform the ill-posed problem into a well-posed one
is offered by extending the measurement of intensities to the additional measure-
ment of polarisation properties of the light reflected from the surface. This approach
yields up to three measurements per pixel, i.e. intensity, degree of polarisation, and
direction of polarisation, which may allow an unambiguous reconstruction of the
surface gradient field under the conditions typically encountered in real-world sce-
narios.

3.4.1 Surface Orientation from Dielectric Polarisation Models

In many shape from shading algorithms a Lambertian reflectance map is explicitly
assumed, as is e.g. the case for the photometric stereo method described in Sect. 3.3 .
The computed surface shape is inaccurate when specular reflections are present in
the image. An early approach to incorporate polarisation information into the shape
from shading framework consists of the detection and subsequent removal of spec-
ular reflections in the image based on the analysis of colour and polarisation data, as
proposed by Nayar et al. ( 1993 ). They use the degree of polarisation to select pixels
characterised by specular reflection and derive a technique to estimate the specular
colour. Furthermore, they show that the colour of the diffuse reflection component
can be constrained to a one-dimensional colour subspace, based on which neigh-
bouring pixels of similar colour are identified. The result is an image of the surface
that displays only the diffuse (Lambertian) component of the reflected light.
The existing methods that aim for a reconstruction of surface shape from polari-
sation measurements deal with dielectric materials. Wolff ( 1987 ) presents a method
to estimate the orientation of a surface by polarisation, where the direction of inci-
dent light, the viewing direction, and the polarisation behaviour of the surface are
assumed to be known. This approach is extended by Wolff ( 1989 ) to the estimation
of surface orientation based on a stereo pair of polarisation images.
In the approach of Rahmann ( 1999 ), a polarisation image is generated by acquir-
ing images of the surface with a camera equipped with a linear polarising filter. The
intensity of a pixel depends on the rotation angle ω of the polarising filter according
to

I v cos 2 (ω Φ) .

I(ω) = I c + − (3.52)