Intensity-Based and Polarisation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 156

The classical approach to photometric stereo for surfaces of non-uniform albedo
is the method introduced by Woodham ( 1980 ), which relies on three pixel-
synchronous images of the surface acquired under different illumination conditions
(cf. also Horn 1986 , Jiang and Bunke 1997 , and Klette et al. 1996 ), defined by the
illumination vectors s l with 1. The surface is assumed to display a Lamber-

tian reflectance behaviour according to ( 3.17 ) and a non-uniform albedo ρ uv .The
light sources are situated at infinite distance, and their irradiances are identical. For
each pixel (u, v) , three intensity values are measured and expressed as the three
components of the vector I uv . If the surface normal n uv is defined as a unit vector,
the pixel grey values are given by
I uv =

s l =

·

ρ uv S n uv , (3.44)

where the rows of the 3 3matrix S contain the illumination vectors s l . Taking the

norm on both sides of ( 3.44 ) immediately yields
ρ uv =

×

S − 1 I uv ,

(3.45)

and the surface normal amounts to

1
ρ uv

S − 1 I uv . n uv = (3.46)

This algorithm has the considerable advantage that it copes with surfaces of arbitrary
non-uniform surface albedo. However, a drawback is that the inverse matrix S − 1
exists only if the three illumination vectors s l are not coplanar, and that it is restricted
to surfaces of purely Lambertian reflectance.
The 'multi-image shape from shading' method proposed by Lohse and Heipke
( 2003 , 2004 ) and Lohse et al. ( 2006 ) relies on the principle of photometric stereo
but assumes a uniform surface albedo. This technique is based on the direct recon-
struction of the surface in an object-centred coordinate system. It is used to obtain
digital elevation models of the lunar polar regions based on images acquired by the
Clementine spacecraft (cf. Chap. 8 ).
A detailed overview of classical and recent photometric stereo methods is given
in the survey by Herbort and Wöhler ( 2011 ), some of which are described in the
following text. Several recent approaches specifically address the problem of re-
constructing surfaces with non-Lambertian reflectance functions and moving ob-
jects.
A photometric stereo approach is used by Alldrin et al. ( 2008 ) to obtain a three-
dimensional surface reconstruction along with the locally non-uniform reflectance
function. The reflectance function of the surface material is represented by a linear
combination of the reflectance functions of several known 'basis materials', where
smoothness and monotonicity constraints are imposed on the reflectance function.
The experimental evaluation by Alldrin et al. ( 2008 ) involves more than 100 images
per object acquired under different illumination conditions. Furthermore, it is shown
that photorealistic images can be rendered for new viewing directions based on the
estimated three-dimensional shapes and reflectance functions.
Goldman et al. ( 2010 ) propose a photometric stereo approach which performs
a three-dimensional surface reconstruction along with an estimation of the locally