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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004
x, vy coordinate of the conjugate point in the input image
(a.b) affine transformation parameters.
/ Transformation Function
DI
i
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Figure 1. Similarity measure using straight-line segments
2.3 Similarity measure
The next step in the registration paradigm is the selection of the
similarity measure, which mathematically describes the
necessary constraints for ensuring the correspondence of
conjugate primitives. The similarity measure formulation
depends on the selected registration primitives and their
respective attributes. As mentioned before, the registration
primitives, straight-lines, will be represented by their end
points, which need not be conjugate.
Assuming that a line segment (1-2) in the reference image
corresponds to the line segment (3-4) in the input image, Figure
1, the similarity measure should mathematically describe the
fact that the line segment (1-2) will coincide with the
corresponding line segment (3-4) after applying the
transformation function relating the reference and input images.
Such a measure can be derived by forcing the normal distances
between the end points of a transformed line segment in the
reference image, and the corresponding line segment in the
input image to be zero (i.e., n; = n; = 0 , Figure 1). Equation 2
mathematically describes such a constraint for one of the end
points of the line segment in the reference image.
x; cosü * yj sinQ- p -0 (2)
where
(0,0)
Polar coordinates representing the line segment (3-4)
in the input image
(X, x) Transformed coordinates of point 1 in the reference
image after applying the registration transformation
function.
Another constraint in the form of Equation 2 can be written for
point 2 along the line-segment in the reference image.
2.4 Matching strategy
To automate the solution of the registration problem, a
controlling framework that utilizes the primitives, similarity
measure, and transformation function must be established. This
framework is usually referred to as the matching strategy. In
447
this research, the MIHT is used as the matching strategy. Such a
methodology is attractive since it allows for simultaneous
matching and parameter estimation. Moreover, it does not
require complete correspondence between the primitives in the
reference and input images. MIHT has been successfully
implemented in several photogrammetric operations such as
automatic single photo resection and relative orientation (Habib
etal, 2001a, 2001b).
MIHT assumes the availability of two datasets where the
attributes of conjugate primitives are related to each other
through a mathematical function (similarity measure
incorporating the appropriate transformation function). The
approach starts by making all possible matching hypotheses
between the primitives in the datasets under consideration. For
each hypothesis, the similarity measure constraints are
formulated and solved for one of the parameters in the
registration transformation function. The parameter solutions
from all possible matching hypotheses are stored in an
accumulator array, which is a discrete tessellation of the
expected range of the parameter under consideration. Within the
considered matches, correct matching hypotheses would
produce the same parameter solution, which will manifest itself
as a distinct peak in the accumulator array. Moreover, matching
hypotheses that contributed to the peak can be tracked to
establish the correspondence between conjugate primitives in
the involved datasets. Detailed explanation of the MIHT can be
found in Habib et al, 2001b.
The basic steps for implementing the MIHT for solving the
registration problem are as follows:
= Approximations are assumed for the parameters which are
yet to be determined. The cell size of the accumulator array
depends on the quality of the initial approximations; poor
approximations will require larger cell sizes.
= All possible matches between individual registration
primitives within the reference and input images are
evaluated, incrementing the accumulator array at the location
of the resulting solution, pertaining to the sought-after
parameter, from each matching hypothesis.
= After all possible matches have been considered, the peak in
the accumulator array will indicate the most probable
solution of the parameter in question. Only one peak is
expected for a given accumulator array.
* After each parameter is determined (in a sequential manner),
the approximations are updated. For the next iteration, the
accumulator array cell size is decreased to reflect the
improvement in the quality of the parameters. Then, the
above two steps are repeated until convergence is achieved
(for example, the estimated parameters do not significantly
change from one iteration to the next).
* By tracking the hypothesized matches that contribute
towards the peak in the last iteration, one can determine the
correspondence between conjugate primitives. These
matches are then used in a simultaneous least squares
adjustment to derive a stochastic estimate of the involved
parameters in the registration transformation function.
Once the registration primitives, transformation function,
similarity measure, and the matching strategy have been
selected, they are integrated in an automatic registration
procedure. As mentioned earlier the accuracy of the registration
process is the key factor that controls the validity and the
reliability of the change detection outcome. Section 5 will show
that a few pixels accuracy has been achieved.
