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depends on the reflectance function, which is not necessarily exactly known, and
may be affected by imperfections of the CCD sensor used for image acquisition,
e.g. due to saturation effects, or by scattering or specular reflection of incoming
light at the surface. Furthermore, even if the slopes p uv and q uv are reasonably well
known, small but correlated errors of p uv and q uv will cumulate and finally result
in large errors of z uv over large scales. Therefore, depth differences estimated based
on shape from shading are usually significantly more reliable on small scales than
on large scales.
The algorithm presented in this section copes with any reflectance function
R(p,q) that is varying in a smooth and monotonic manner with cos θ i .Forsim-
plicity we will exemplify the reconstruction method using a Lambertian reflectance
function, keeping in mind that the proposed framework is suitable for arbitrary re-
flectance functions with physically reasonable properties (cf. Chap. 8 ), in order to
illustrate how to incorporate a shadow-related error term into the shape from shading
formalism. For the application scenario of three-dimensional reconstruction of small
sections of the lunar surface as described in Chap. 8 , the Lambertian reflectance val-
ues may be inaccurate by a few percent, but this error is usually smaller than the er-
rors caused by nonlinearities and other imperfections of the CCD sensor. Anyway,
the shadow-related error term permits significant deviations of the reconstruction
result from the solution obtained from reflectance only.
For reflectance functions accurately modelling the photometric surface properties
of planetary bodies, one should refer to the detailed descriptions by Hapke ( 1981 ,
1984 , 1986 , 2002 )orbyMcEwen( 1991 ) (cf. Sect. 8.1.2 ). For the metallic surfaces
of the industrial parts examined in Sect. 6.3 , we performed measurements which
indicate that, for moderate surface gradients, viewing directions roughly perpen-
dicular to the surface, and oblique illumination, the assumption of a Lambertian
reflectance function is a good approximation. For shallow slopes ( p,q
1) and
oblique illumination, many realistic reflectance functions can be approximated over
a wide range in terms of a linear expansion in p and q at a good accuracy. So-
phisticated shape from shading algorithms have been developed for this important
class of reflectance functions by Horn ( 1989 ), which should all be applicable within
the presented framework for surfaces with shallow slopes, perpendicular view, and
oblique illumination.
With a Lambertian reflectance function, the result of the iterative shape from
shading scheme according to ( 3.24 ) depends on the initial values p ( 0 )
uv and q ( 0 )
uv of
the surface gradients and on the value chosen for the surface albedo ρ . A differ-
ent initialisation usually yields a different result of the algorithm, because although
the regularisation (e.g. smoothness or integrability) constraint strongly reduces the
range of possible solutions, it still allows for an infinite number of them. Without
loss of generality, it is assumed that the scene is illuminated exactly from the right-
hand or the left-hand side.
To obtain a solution for z uv which both minimises the error term ( 3.21 ) and is
at the same time consistent with the average depth difference derived from shadow
analysis, we propose an iterative procedure.
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