State of Art Shape from Polarization

A thorough description of polarization of electromagnetic waves is given by Wolf and Born (1959). It is start from assume the light is a transverse electromagnetic wave. Wolf described a polarisation state of the electric field.

The use of polarisation in the computer vision literature begin at work by Koshikawa (1979). In his research, koshikawa has used only the linear polarisation state. Since it is sufficient for most applications where unpolarised light is reflected by a surface (Wolff and Boult 1991).

Wolff (1991) assumes that surfaces are lit with unpolarised light, and uses a Fresnel reflectance model which predicts the polarisation state of light reflected from dielectric and metallic surfaces. This model is based on the Fresnel reflection coefficients F⊥, which specifies the attenuation of the light polarised perpendicular to the plane of reflection, and F||, which defines the attenuation of light polarised parallel to the plane of reflection. If these coefficients differ from each other, reflection of unpolarised light results in polarised reflected light. The values of the coefficients depend on the type of material (dielectric or conductive), its index of refraction, the type of reflection (diffuse or specular) and the angle of incidence or emittance (Wolff 1991). For specular reflection of smooth, dielectric surfaces, F⊥ >= F||, resulting in linear polarisation perpendicular to the plane of reflection. The degree of polarisation depends on the incidence angle θi, and reaches 1 at the Brewster angle, arctan(n), where n is the index of refraction, and approaches 0 at θi = 0 and θi = 90◦. Diffuse, or body reflection results is largely unpolarised light (Wolff and Boult 1991), except for large large viewing angles θe. For this case, the polarisation direction is parallel to the emittance plane defined by the surface normal vector ~n and the viewing vector ~v.

The dependencies of the Fresnel coefficients on the material type can be used for material classification by computing the ratio of the Fresnel coefficients of specular reflections. Wolff (1991) proposed further applications of the Fresnel reflectance model including classifications of edges according to their origin (occluding boundary, specularities, albedo and shadow edges), separation of diffuse and specular reflectance components, and the estimation of surface normals using specular reflections on dielectric surfaces.

Since the polarisation state of reflected light is a function of the orientation of the surface, polarisation measurements can be used for estimation of surface orientation. For specular reflection from smooth dielectric surfaces, the polarisation angle ϑ defines a plane in which the surface normal is located. Together with the specular angle of incidence, the surface normal can be determined. Estimation of the specular angle however requires knowledge index of refraction and is subject to a two way ambiguity, except at the Brewster angle, where the degree of polarisation reaches 1 (Wolff 1991).

In other work by Miyazaki et al (2003) a spherical surface normal distribution is assumed and the 3D surface shape and intensity reflectance properties are estimated from a single image lit with multiple light sources.

Miyazaki et al. (2004) propose a related method for reconstruction of transparent objects, where the object is illuminated with light from all directions, producing specular reflection over the whole surface. The two way ambiguity is resolved by using a second image with slightly rotated object.

A Fresnel reflectance model with complex index of refraction has been used by Morel et al. (2005) for the reconstruction of very smooth, mirror like metallic surfaces. In a later publication (Morel et al. 2006), the specular angle ambiguity is resolved by varying the illumination.

Note that all surface estimation algorithms above are limited to reconstruction of smooth, dielectric and metallic surfaces without interreflections. Since the polarisation of the diffuse reflectance component of dielectrics is very low, most approaches assume specular reflections, and thus require a uniform, spherical illumination.