Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 71

of u e is defined by the maximum gradient of the intensity profile in the u direction,
corresponding to the root of the hyperbolic tangent. This condition yields

u e (v, t) =− p 3 (v, t)/p 2 (v, t), (1.120)

where the value u e (v c ,t c ) with v c and t c denoting the centre of the matching window
is used for the determination of disparity.
The direction δ of the intensity gradient at the location of the interest pixel, rep-
resenting a feature that will be used for correspondence analysis later on, is given
by

v c ,t c

∂u e
∂v

δ = . (1.121)

The velocity μ of the intensity gradient along the epipolar line corresponds to the
temporal derivative

v c ,t c

∂u e
∂t

μ = (1.122)

of the location of the epipolar transection. Such explicit motion information is not
revealed by the purely correlation-based spacetime approach described by Davis et
al. ( 2005 ). The approach described by Franke et al. ( 2005 ) yields motion informa-
tion for each three-dimensional point but requires a separate processing stage for
individually tracking the corresponding positions and velocities.
For the purpose of correspondence analysis, a similarity measure between two
interest pixels located on the same epipolar line v is determined based on the func-
tions h( P left ,u,v,t) and h( P right ,u,v,t) fitted in the left and the right image to the
spatio-temporal matching windows of the interest pixels, respectively, where the
obtained function parameters are denoted by the vectors P left and P right .
Similarity measures analogous to those well known from classical correlation-
based stereo vision approaches (Franke and Joos, 2000 ), such as the sum of squared
differences (SSD), the sum of absolute differences (SAD), or the cross-correlation
coefficient, can be adapted to this spacetime stereo algorithm. This is achieved by
comparing the fitted functions h( P left ,u,v,t) and h( P right ,u,v,t) rather than the
pixel grey values themselves Davis et al. ( 2005 ). As an example, the SSD similarity
measure then reads

h P l ,u

(v c ,t c ), v, t − h P r ,u (v c ,t c ), v, t 2 dudvdt,

(1.123)

u right u left S SSD =

− −

e

e

where u , v , and t traverse the spatio-temporal matching windows of the left and the
right interest pixel, respectively. Analogous expressions are obtained for the SAD
or cross-correlation similarity measure.
Once a correspondence between two interest pixels on the same epipolar line
has been established by searching for the best similarity measure e.g. according
to ( 1.123 ), the disparity d corresponds to the difference between the epipolar tran-