Graphics Reference
In-Depth Information
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)
=
kσ
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)
kσ
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
]
/
[
kσ
I
]|
>θ
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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