Graphics Reference
In-Depth Information
5.11
Suppose that we assume the optical flow field
is constant within a
window centered at a pixel. Show that summing the Horn-Schunck Euler-
Lagrange equations (
5.22
) over this window is equivalent to solving the
Lucas-Kanade equations (
5.25
) using a box filter corresponding to the
window.
(
u
,
v
)
5.12
Show that replacing the anisotropic diffusion tensor
D
in Equation (
5.29
)
with the identity matrix gives the original Horn-Schunck smoothness term.
5.13
Interpret the anisotropic diffusion tensor
D
in Equation (
5.30
) when (a)
the magnitude of
g
is 0 and (b) the magnitude of
g
is much larger than
(
g
)
β
.
5.14
Show that the Lorentzian robust cost function in Table
5.1
is redescending.
Is the generalized Charbonnier cost function redescending for any positive
value of
?
5.15 Explain why there are seven degrees of freedom in the nine entries of the
fundamental matrix.
β
5.16
Showwhy the fundamental matrix constraint in Equation (
5.34
) leads to the
row of the linear system given by Equation (
5.40
).
5.17
Suppose the fundamentalmatrix
F
relates anypair of correspondences
(
x
,
y
)
in
I
2
. Determine the fundamental matrix
F
that relates the
correspondences in the two images after they have been transformed by
similarity transformations
T
and
T
, respectively.
5.18 The basis of the RANSAC method [
142
] for estimating a projective transfor-
mation or fundamental matrix in the presence of outliers is to repeatedly
sample sets of points that are minimally sufficient to estimate the parame-
ters, until we are tolerably certain that at least one sample set contains no
outliers.
a)
x
,
y
)
in
I
1
and
(
Let the (independent) probability of any point being an outlier be
, and
suppose we want to determine the number of trials
N
such that, with
probability greater than
P
, at least one random sampling of
k
points
contains no outliers. Show that
ε
log
(
1
−
P
)
N
=
(5.66)
k
log
(
1
−
(
1
−
ε)
)
b) Compute
N
for the projective transformation estimation problem
where
k
=
4,
P
=
0.99, and
ε
=
0.1.
5.19
Showwhy the algorithm in Section
5.4.3
results in a pair of rectified images.
That is, show that after applying the given projective transformations, the
fundamental matrix is in the form of Equation (
5.41
).
5.20
Show why any values of
a
,
b
, and
c
in Equation (
5.44
) preserve the property
that
is a pair of rectifying projective transformations.
5.21 Determine the disparity map and binary occlusion map with respect to the
left-hand image in Figure
5.32
(assuming the background is static).
(
H
1
,
H
2
)
