'alue of the struc-
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erture problem is
lue of coh. For all
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all certainty.
edge | rank
0 0
1 1
0 2
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measure for dif-
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je. It constitutes
g the absence of
oints where both
al flow f can be
id type measures
linear positioning
'S the entire grid
htness. These ar-
ing points of the
>, respectively.
ype measures for
in Figure 4. The
ittern shows ran-
:gions with addi-
ure these regions
e about the pres-
o reconstruct the
ization technique
1997).
least squares ap-
rron et al. (1994)
uantify the singu-
zing the eigenval-
oposes to thresh-
et al. (1994) ar-
oved to be more
s. The matrix À
two-dimensional
borhood and rep-
nensional images
son, 1998). The
hree-dimensional
of the two eigen-
the eigenvalues
hresholding parts
1antifies the con-
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Figure 4: Illustration of the importance of confidence
and type measures for a ring pattern moving with
(1,1) pixels/frame towards the upper left corner (with ad-
ditive noise on — 1). Upper row: One frame of the mov-
ing ring pattern (left), optical flow computed for any image
point without confidence and type measures (right). Lower
row: Optical flow masked by the confidence measure (left),
local regularization incorporating confidence and knowl-
edge about the presence of an aperture problem (right).
fidence of local orientation and therefore the pres-
ence of an aperture problem. It also approaches zero
for homogeneous brightness. The three-dimensional
coherency constitutes a generalization of the two-
dimensional case. It also includes information on the
coherency of motion and identifies isotropic noise pat-
terns.
5. FILTER OPTIMIZATION
A crucial factor in optical flow computation is the dis-
cretization of the partial derivative operators. Figure
5 illustrates this basic fact with a simple numerical
study. With the standard symmetric difference filter
1/2 [1 0 -1], large deviations from the correct displace-
ments of more than 0.1 pixels/frame occur. With an
0270.20240.60.8 131.2.1.4
Figure 5: Systematic error in the velocity estimate as a
function of the interframe displacement of a moving ran-
dom pattern. Derivatives computed with the symmetric
difference filter 0.5 [1 0 — 1] (left) and an optimized Sobel
filter (right) given by (Scharr et al., 1997).
709
Figure 6: Demonstration of the influence of spatial sensi-
tivity variations of the CCD sensor on motion estimation:
Top row: One image of the elephant sequence (left), con-
trast enhanced relative responsivity (right). Middle row:
Velocity component in z direction (left), smallest eigen-
value of the structure tensor (right). Bottom row: Velocity
component in z direction for the corrected sequence (left),
smallest eigenvalue of the structure tensor for a sequence
corrected for the spatial responsivity changes (right).
optimized 3x3 Sobel-type filter (Scharr et al., 1997),
the error is well below 0.005 pixels/frame.
Analyzing the impact of noise on differential least-
squares techniques, Bainbridge-Smith and Lane
(1997) report the same results for the errors as in the
left image in Figure 5. This error can be identified
as discretization error of the differential operators. In
order to reduce these errors they use a combination of
smoothing and derivative kernels for local regulariza-
tion in one dimension.
We used a new class of regularized separable first-
order derivative filters with a transfer function
D, (kt) — (ik) B([kt]) (15)
for a filter in the direction p. A similar approach has
been used by (Simoncelli, 1993). While Simoncelli
(1993) applied a linear error functional, à nonlinear
one has been used by (Scharr et al., 1997) minimizing
the angle error of the gradient vector.
6. SENSOR UNIFORMITY CORRECTION
Errors of motion analysis with real sensor data are sig-
nificantly higher than those obtained with computer
generated sequences. The higher errors are related to
imperfections of the CCD sensor/camera system. A
radiometric calibration study showed that standard
