Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 77

can be assumed to be distributed in a Gaussian manner. In contrast, the resulting
coordinates x , y , and z in scene space show distributions which are not symmetrical.
Correct and incorrect three-dimensional points are thus more favourably separated
according to their disparity values, since the distance between the camera and the
plane is inversely proportional to the disparity.
If it is assumed that a part of the scene can be modelled by a plane
ε s : z(x,y) = a s x + b s y + c s

(1.131)

in the scene space spanned by x , y , and z , the corresponding plane in disparity space
spanned by the pixel coordinates u and v and the disparity d follows from the basic
equations of the pinhole model according to
x
z =

u
f ,

y
z =

v
f ,

lf
d

z = (1.132)

with f as the camera constant in pixel units and l as the baseline distance of the
stereo camera system in metric units. Inserting the expressions for x , y , and z de-
rived from ( 1.132 )into( 1.131 ) yields a transformed plane in disparity space accord-
ing to

ε d : d(u,v) = a d u + b d v + c d (1.133)

with

l
c s a s ,

l
c s b s ,

lf
c s .

a d =− b d =− c d = (1.134)

The offset parameter c s of the plane z(x,y) in scene space, whose normal vector
is given by the previously determined parameters a s and b s , is obtained by trans-
forming it into disparity space according to ( 1.133 ) and ( 1.134 ) and then comput-
ing the mean distance between all three-dimensional points with coordinates (u, v)
covered by the plane ε d in disparity space and the disparities describing that plane
(cf. ( 1.133 )). The distances between the three-dimensional points and the plane are
weighted according to the M-estimator method (Rey, 1983 ), resulting in the error
function

M d i − 2

a s l
c s

b s l
c s

lf
c s

E ME (c s ) = − u i − v i + (1.135)

(u i ,v i ) ∈ ε d

) − 1 . The brackets with M(x) denote the average. The func-

tion E ME (c s ) is minimised with respect to the distance parameter c s of the plane ε s
in scene space using the nested intervals method. The parameter k ME is user defined.
The behaviour of E ME (c s ) isshowninFig. 1.25 b for scene 1 regarded in Sect. 1.6.4 .
The plane model adaptation method described in this section does not require
initial pose parameters. The image region characterised by repetitive structures and
its metric extension in scene space are extracted without a priori knowledge about
the scene. The normal vector of the plane is determined based on the assumption that
the repetitive structures are characterised by a uniform distance parameter in scene
space. Only the offset parameter of the model plane is estimated based on the three-
dimensional point cloud generated by model-free stereo analysis; the experimental

= 1 − ( 1 +| x/k ME | ...