Integrated Frameworks for Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 210

direction s and the viewing direction v , such that the modelled photopolarimetric
properties of a pixel correspond to the measured values:

I uv =

R I (p uv ,q uv , s , v ) (5.14)

R Φ (p uv ,q uv , s , v ) (5.15)
D uv = R D (p uv ,q uv , s , v ). (5.16)
The reflectance functions ( 5.14 )-( 5.16 ) may depend on further, e.g. material-
specific, parameters which possibly in turn depend on the pixel coordinates (u, v) ,
such as the surface albedo ρ uv which influences the intensity reflectance R I .
As long as a single light source is used, it is possible without loss of general-
ity to define the surface normal in a coordinate system with positive x and zero y
components of the illumination vector s , corresponding to p s < 0 and q s =

Φ uv =

0 with

q, 1 ) T . Furthermore, for simplicity we always choose
the z axis such that the viewing direction corresponds to v

a surface normal n = ( − p, −

= ( 0 , 0 , 1 ) T . The surface

q, 1 ) T in the world coordinate system, in which the azimuth an-
gle of the light source is denoted by ψ , is related to n by a rotation around the z
axis, leading to

normal n ˜ = ( −˜ p, −˜

q cos ψ. (5.17)
It is generally favourable to define the reflectance functions R I , R Φ , and R D in the
coordinate system in which q s =

p ˜ = p cos ψ − q sin ψ and q ˜ = p sin ψ +

0. If several light sources with different azimuth
angles are used, one must then remember to take into account the transformation
between the two coordinate systems according to ( 5.17 ).
In the following paragraphs we describe a global and a local approach to solve
the problem of shape from photopolarimetric reflectance (SfPR), i.e. to adapt the
surface gradients p uv and q uv to the observed photopolarimetric properties I uv , Φ uv ,
and D uv by solving the (generally nonlinear) system of equations ( 5.14 )-( 5.16 ). The
three-dimensional surface profile z uv is then obtained by integration of the surface
gradients according to the method proposed by Simchony et al. ( 1990 ) as described
in Sect. 3.2.3 .

5.3.1.1 Global Optimisation Scheme

The first solving technique introduced by d'Angelo and Wöhler ( 2005a ) is based on
the optimisation of a global error function simultaneously involving all image pix-
els, an approach described in detail by Horn ( 1986 , 1989 ), Horn and Brooks ( 1989 ),
and Jiang and Bunke ( 1997 ) (cf. Sect. 3.2 ). One part of this error function is the
intensity error term ( 3.19 ). As the pixel grey value information alone is not neces-
sarily sufficient to provide an unambiguous solution for the surface gradients p uv
and q uv , a regularisation constraint e s is introduced which requires smoothness of
the surface, e.g. small absolute values of the directional derivatives of the surface
gradients. We therefore make use of the additional smoothness error term ( 3.20 )
(Horn, 1986 , 1989 ; Horn and Brooks, 1989 ; Jiang and Bunke, 1997 ). In the sce-
narios regarded in this work, the assumption of a smooth surface is realistic. For