Intensity-Based and Polarisation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 154

x , and I
are defined in the light source coordinate system. The relation between the original
and the transformed image radiance then corresponds to
I(x,y) = I(s z x + s x z, y).

˜ ˜ ˜

which formally corresponds to the eikonal equation ( 3.37 ). In ( 3.40 ), p , q ,

(3.41)

Inserting ( 3.41 )into( 3.40 ) allows one to determine the depth values z uv of the
surface. As an alternative, Kimmel and Sethian ( 2001 ) propose an extended fast
marching scheme that directly yields the solution for z uv .
The shape from shading approach based on partial differential equations is ex-
tended by Vogel et al. ( 2008 ) to a formulation in terms of the Hamilton-Jacobi
equation which takes into account a finite distance between the object and the cam-
era, i.e. a full projective camera model, and a non-Lambertian Phong reflectance
model. The light source is assumed to be located in the optical centre of the cam-
era, corresponding to a phase angle of α 0 ◦ . According to Vogel et al. ( 2008 ), the

differential equation to be solved for the depth map z(x,y)

=

= z( x ) defined in the

continuous domain corresponds to

2

Rb 2
z Q 2 b 2

x ) 2 +

1
z 2

∇ z

2

+ ( ∇ z · z 2 = (3.42)

with R as the reflectance function, b as the camera constant, and Q = Q( x ) =

b/

b 2 . A discretised solution of the Hamilton-Jacobi equation is obtained
using the 'method of artificial time'. The approach is evaluated by Vogel et al. ( 2008 )
based on synthetic images with and without noise and on simple real-world images.
The method of Vogel et al. ( 2008 ) is applied to more complex real-world images
by Vogel et al. ( 2009b ). The object of interest is segmented by subsequently apply-
ing the approach of Chan and Vese ( 2001 ) for initialisation, which relies on image
grey values rather than edges, and the edge-based geodesic active contour method
of Caselles et al. ( 1997 ) for refinement. A correction for non-uniform surface albedo
is performed using a diffusion-based image inpainting method. For several objects
with non-Lambertian surfaces (a cup, a computer mouse, and a topic with a plas-
tic cover), realistic three-dimensional reconstruction results are obtained. However,
no comparison to ground truth data is provided. A computationally highly efficient
method for solving the Hamilton-Jacobi equation ( 3.42 ) based on a fast marching
scheme is developed by Vogel et al. ( 2009a ).
Beyond an integration of the Phong model, Vogel and Cristiani ( 2011 ) discuss
the approach of Ahmed and Farag ( 2007 ), who employ the reflectance model of
Oren and Nayar ( 1995 ) for shape from shading. The Oren-Nayar model has been
developed for rough surfaces composed of small facets which are individually gov-
erned by a Lambertian reflectance. Vogel and Cristiani ( 2011 ) assume that the light
source is located in the optical centre of the camera. They compare the numerical
approach used by Ahmed and Farag ( 2007 ) with the upwind scheme proposed by
Rouy and Tourin ( 1992 ), concluding that the upwind scheme converges faster and
can be implemented more easily. Furthermore, Vogel and Cristiani ( 2011 ) derive
upper bounds for the step size of the upwind scheme for the Phong model and the

2

+

x