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Fig. 1.23 ( a ) Spatio-temporal intensity profile of a moving object boundary, measured over a time
interval of 3 time steps. The size of the spatio-temporal matching window is 21 × 7 × 3 pixels. For
visualisation, the (v, t) axis is divided such that each time step comprises an interval of 7 pixels.
( b ) Modelling result according to ( 1.119 ), with p 2 (v, t) of first order and p 3 (v, t) of second order
in v and t
The parametric model according to ( 1.118 ) in its general form requires that a non-
linear least-mean-squares optimisation procedure be applied to each interest pixel,
which may lead to a prohibitively high computational cost of the method. It is possi-
ble, however, to transform the nonlinear optimisation problem into a linear problem
by making the following simplifying assumptions:
I of the spatio-
1. The offset p 4 (v, t) is proportional to the average pixel grey value
w I ,
2. The amplitude p 1 (v, t) of the sigmoid is proportional to the standard deviation σ I
of the pixel grey values in the spatio-temporal matching window with p 1 (v, t)
=
temporal matching window, i.e. p 4 (v, t)
=
I .
These simplifications yield the model equation
artanh I(u,v,t)
w I
I(u,v,t),
p 2 (v, t)u
+
p 3 (v, t)
=
(1.119)
I
where the model parameters, i.e. the coefficients of the polynomials p 2 (v, t) and
p 3 (v, t) , can be determined by a linear fit to the transformed image data I(u,v,t) .
Real-time processing speed is achieved by implementing the artanh function as a
look-up table.
Pixels with
|[ I(u,v,t) w I ] / [ I ]| are excluded from the fit, where θ is
a user-defined threshold with θ< 1, since arguments of the artanh function close
to 1 would lead to a strong amplification of noise in the original pixel grey values.
The factors k and w are further user-defined parameters of the algorithm. A typical
spatio-temporally local modelling result for a moving object boundary is shown in
Fig. 1.23 .
Equation ( 1.118 ) allows for a direct computation of the location u e of the epipolar
intersection, i.e. the position of the intensity change at subpixel accuracy in the
u direction. This value is essential for a precise determination of disparity. The value
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