Graphics Reference
In-Depth Information
H
, filling in the elements from left to right in
4. Reshape
h
into a 3
×
3 matrix
each row.
5. Recover the final projective transformation estimate as
H
T
−
1
HT
.
=
While the DLT is easy to implement, the resulting estimate does not minimize a
symmetric, geometrically natural error. Under the assumptions that the measure-
ment errors in each feature location are independently, identically distributed (i.i.d.)
with a zero-mean Gaussian pdf, we can obtain a maximum likelihood estimate for
the projective transformation by minimizing
x
i
y
i
x
i
y
i
2
2
+
2
n
x
i
ˆ
ˆ
ˆ
x
i
ˆ
−
−
(5.6)
y
i
y
i
i
=
1
2
{
(
ˆ
ˆ
)
...
(
ˆ
ˆ
)
}
over the nine elements of
H
and
, which are estimates of the
true feature locations exactly consistent with the projective transformation. Each
(
ˆ
x
1
,
y
1
,
,
x
n
,
y
n
x
i
,
y
i
)
by
H
. This cost func-
tion is nonlinear and can be minimized by the Levenberg-Marquardt algorithm
(Appendix
A.4
).
When the feature matches have errors not well modeled by an i.i.d. Gaussian
distribution — for example in the case of outliers caused by incorrect matches —
the RANSAC algorithm [
142
] should be used to obtain a robust estimate of
H
by
repeatedly sampling sets of four correspondences, computing a candidate
H
, and
selecting the estimate with the largest number of inliers (see Problem
5.18
). Hartley
and Zisserman [
188
] give a detailed description of these and further methods for
estimating projective transformations. We will not focus heavily on parametric cor-
respondence estimation here, since correspondence inmost real-world scenes is not
well modeled by a single, simple transformation.
ˆ
is the transformation of the corresponding
(
ˆ
x
i
,
y
i
ˆ
)
5.2
SCATTERED DATA INTERPOLATION
A different way to think about the problem of obtaining dense correspondence from
sparse featurematches is to view thematches as samples of a continuous deformation
fielddefinedover thewhole imageplane. That is, we seek a continuous function
f
(
x
,
y
)
x
i
,
y
i
)
defined over the first image plane so that
f
(
x
i
,
y
i
)
=
(
,
i
=
1,
...
can be viewed as
scattered data interpolation
since the
are sparsely, unevenly
distributed in the first image plane (as opposed to being regularly distributed in a grid
pattern, in which case we could use a standard method like bilinear interpolation).
The scattered data interpolation problem is sketched in Figure
5.2
.
For all scattered data interpolation methods, it may be necessary for a user to
manually add additional feature matches to further constrain the deformation field
in areas where the estimated correspondence seems unnatural. We discuss this issue
further in Section
5.7
.
(
x
i
,
y
i
)
2
The motion vector
(
u
,
v
)
at a point
(
x
,
y
)
is thus
(
u
,
v
)
=
f
(
x
,
y
)
−
(
x
,
y
)
.