Intensity-Based and Polarisation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 152

two. In contrast to Simchony et al. ( 1990 ), who compute a least-squares solution
of minimum norm, Agrawal et al. ( 2005 ) propose that at image locations for which
the curl of the measured surface gradients is below a given threshold, the gradients
remain unchanged. A graph-based optimisation scheme is presented for correcting
a maximum possible number of gradients at image locations with large curl values.
In the applications of three-dimensional surface reconstruction techniques to
real-world problems described in this work (cf. Chaps. 6 and 8 ) we utilise a compu-
tationally efficient implementation of the approach by Simchony et al. ( 1990 ) based
on enforcing the 'weak integrability' constraint ( 3.25 ). Any physically reasonable
surface z uv must satisfy this constraint. In our application scenarios, computational
efficiency is relevant, since the employed methods require a reconstruction of the
height map z uv from the gradients p uv and q uv at many different intermediate stages
of the optimisation algorithm.

3.2.4 Surface Reconstruction Based on Partial Differential
Equations

In the mathematical framework discussed in this section, the observed image inten-
sities are assumed to be represented by a continuous function I(x,y) instead of the
discrete pixel grey values I uv . According to Bruss ( 1989 ), for shape from shading
problems involving a reflectance function of the special form R(p,q) p 2 q 2 , = +

( 3.16 ) becomes

p 2 q 2 + = I(x,y) (3.36)

which is referred to as an 'eikonal equation'. Under certain conditions, which are
derived in detail by Bruss ( 1989 ), a unique solution for the reconstructed surface
can be obtained from a single image based on ( 3.36 ).
According to Bruss ( 1989 ), reflectance functions of the form R(p,q) = f(p 2

+
q 2 ) with f as a bijective mapping yield image irradiance equations that can be trans-
formed into a form equivalent to ( 3.36 ). An important practically relevant scenario
in which all conditions derived by Bruss ( 1989 ) under which a unique solution of
the image irradiance equation is obtained are fulfilled is a Lambertian surface illu-
minated by a light source, the position of which coincides with that of the camera
(Bruss, 1989 ; Kimmel and Sethian, 2001 ). Kimmel and S ethian ( 2001 ) point out
that, in this special case, I(x,y) is given by I(x,y)

1 / 1

= + p 2 + q 2 , leading to

the eikonal equation

1
I(x,y) 2 −

∇

z(x,y) = 1 . (3.37)

For orthographic projection, i.e. infinite distance between the surface and the cam-
era, and parallel incident light this situation corresponds to the zero phase angle
( α =

0) case.