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Oren-Nayar model and state that the upwind scheme can be implemented as a com-
putationally efficient fast marching algorithm for both examined non-Lambertian
reflectance models. Experimentally, the results of the proposed numerical methods
are compared based on synthetically generated images.
The main advantage of the shape from shading approaches outlined in this sec-
tion, compared to the variational method described in Sect. 3.2.2 , is the fact that
they do not require a smoothness or integrability constraint but directly yield a re-
construction of the surface exclusively based on the observed pixel grey values.
They are, however, restricted to surfaces with specific, relatively simple reflectance
properties and in some cases also to specific illumination conditions, e.g. a light
source coincident with the optical centre of the camera, which is a major drawback
in complex real-world scenarios (cf. Chaps. 6 and 8 ).
3.3 Photometric Stereo
The solution of shape from shading based on a single image is ambiguous as long
as the surface albedo is unknown and no assumption about the surface can be made.
Furthermore, for oblique illumination and reflectance maps similar to the Lambert
law ( 3.17 ), the surface gradients perpendicular to the direction of incident light are
much less well defined than those in the direction of incident light. These drawbacks
can be overcome by the analysis of several images, a procedure termed 'photometric
stereo' in analogy to standard triangulation-based stereo image analysis techniques.
3.3.1 Photometric Stereo: Principle and Extensions
A straightforward way to extend the shape from shading method outlined in
Sect. 3.2.2.2 is to acquire a set of L pixel-synchronous images of the surface un-
der known illumination conditions described by the vectors s l . The intensity error
according to Horn ( 1986 ) is extended by Wöhler and Hafezi ( 2005 ) as a sum over
all L images according to
L
I (l)
R I ( s l ,p uv ,q uv ) 2
e i =
uv
(3.43)
u,v
l
=
1
while the smoothness constraint ( 3.20 ) and the integrability constraint ( 3.25 ) remain
unchanged. The corresponding iterative update schemes for the surface gradients are
analogous to ( 3.24 )or( 3.30 ). In this setting we obtain reliable information about
the surface gradients in all directions with L =
2 light sources, but a known uniform
albedo ρ still has to be assumed. In principle, once the value of ρ is determined, it
is possible to obtain without further constraints the surface gradients p uv and q uv
from the L
2 intensities available for each pixel. In many applications, however, it
is advantageous to continue assuming a smooth surface, as this prevents the surface
from being strongly influenced by noise or small-scale specular reflections.
=
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