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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 )
, n . 2 This problem
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
)
.
 
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