Graphics Reference
In-Depth Information
5.11
HOMEWORK PROBLEMS
5.1 Suppose we want to estimate an affine transformation
T
from a set
of
feature matches
{
(
x
1
,
y
1
)
,
...
,
(
x
n
,
y
n
)
}
in the first image plane and
x
1
,
y
1
)
x
n
,
y
n
)
}
{
(
in the second image plane. Determine the linear least-
squares problem and closed-form solution for the affine parameters that
correspond to minimizing
,
...
,
(
n
x
i
,
y
i
)
−
2
1
((
T
(
x
i
,
y
i
))
(5.65)
i
=
5.2 Determine the 3
3 homogeneous matrix
T
corresponding to the similarity
transform that normalizes the input points
×
{
(
x
1
,
y
1
)
,
...
,
(
x
n
,
y
n
)
}
for the DLT.
5.3
Show that the
h
obtained in Step 3 of the DLT corresponds to minimizing
Ah
2
subject to the constraint that
h
=
1. Hint: use Lagrange multipliers.
5.4
Suppose a projective transformation
H
relates the coordinates
(
x
,
y
)
of
I
1
of
I
2
. Determine the projective transformation
H
that relates the
coordinates of the two images after theyhavebeen transformedby similarity
transformations represented by 3
x
,
y
)
and
(
3 matrices
T
and
T
, respectively.
5.5 Show that if only three arbitrary feature matches are supplied to the thin-
plate spline interpolation algorithm, solving Equation (
5.9
) results in an
affine transformation (with no nonlinear part).
5.6 A thin-plate spline interpolant
f
×
(
x
,
y
)
is estimated from a set of feature
matches
{
(
x
1
,
y
1
)
x
n
,
y
n
)
}
x
1
,
y
1
)
...
(
)
}
{
(
...
(
in the
second image plane using Equation (
5.9
). We then reverse the roles of
the two image planes and estimate a second thin-plate spline interpolant
g
,
,
x
n
,
y
n
in the first image plane and
,
,
x
,
y
)
. Provide a simple sketch to explain why
f
and
g
are not generally
inverses of each other. Under what circumstances are
f
and
g
inverses of
each other?
5.7 Each approach in Section
5.2
produces a forward mapping (i.e., flow field)
from the coordinates
(
x
,
y
)
of
I
2
. We often
want to create a warped image
I
1
that corresponds to rendering the pixel
colors of
I
1
in the coordinate system of
I
2
based on this flow field. Propose
a method for coloring the pixels of
I
1
, keeping in mind that the forward
mapping is not usually invertible in closed form.
5.8 Determine an analogous constraint to the brightness constancy assumption
when we want to compute optical flow on color images. How should the
Horn-Schunck cost function in Equation (
5.21
) change when dealing with
color images?
(
x
,
y
)
of
I
1
to the coordinates
(
5.9
Show that the Taylor expansion of the brightness constancy assumption in
Equation (
5.16
) leads to Equation (
5.17
).
5.10
Show that Equation (
5.17
) implies that the component of the flow vector in
the direction of the image gradient is given by Equation (
5.19
).
