ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002
parameter space. Similar registration problems can be
formulated as Hough-like approaches, but the computational
complexity will remain an overwhelming challenge.
In this paper we present a new approach to automatic image
registration, which is compatible with the general ideas of
Hough-like approaches, but differs in the way of how the
computational complexity is handled and the application
task. This approach exploits the duality between the
observation space and the parameter space in the sense of
HT. In this approach, as well as the Hough-like ones, the
problem of image registration is characterized, not by the
geometric or radiometric properties, but by the mathematical
transformation that describes the geometrical relationship
between two images. The proposed approach considers
different strategy to reduce the computational complexity,
and is tailored to handle 2-D registration, which is a typical
case in most of remote sensing imagery. The basic idea
underpinning the proposed approach is to pair each data
element belonging to two sets of imagery, with all other data
in the set, through a mathematical transformation that
describes the geometrical relationship between them. The
results of pairing are encoded and exploited in histogram-like
arrays (parameter space) as clusters of votes. Binning in the
specified range of the registration parameters generates these
clusters. The process of using geometrically invariant
features is considered as a strategy to reduce the
computational complexity generated by the high
dimensionality of the mathematical transformation. This
approach does not require feature matching. Matched
features will be recovered as a by-product of this approach.
The developed approach is accommodated with full
uncertainty modeling and analysis using a least squares
solution.
This paper is organized as follows. Section 2 presents the
proposed methodology, section 3 presents the experimental
results, section 4 discusses the obtained results, and finally
section 5 concludes the paper.
2. METHODOLOGY
The basic idea underpinning the proposed approach is to
compare common data elements of two images with all other
data contained in those images through a mathematical
transformation that describes the geometrical relationship
between them. This approach considers two basic
assumptions. First, the characteristics of the object space give
rise to detectable features such as points and lines in both
images, and at least part of these features are common to
both images. Second, the two images can be aligned by a 2-D
transformation. The basic process starts with feature
extraction, followed by geometric invariant features
construction, and then parameter space clustering. For the
interest of developing an intuitive understanding of the basic
process, each step is highlighted briefly, while a through
discussion is deferred to the subsections below. First, in the
presented study point features are dealt with. Second, in
order to construct geometric invariant features, each point in
the first image is related to a collection of other points in the
same image defining a geometric arrangement whose
properties remain invariant under a chosen transformation.
The same process is applied to the second image. By
constructing geometric invariant features, we did not impose
any geometric constraint on the original image features such
as straightness. However, the geometric properties of the
mathematical transformation are considered. Invariant
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features, constructed from the two images to be registered,
characterize these properties. Invariant features gave rise to a
set of mathematical transformations with a reduced
dimensionality. This set of mathematical transformations was
used as voting (clustering) functions in the parameter space.
Third, the basic idea of parameter space clustering is to
compare the data element gathered from two sets according
to a pre-specified observation equation (voting function). The
results of comparison will point to different locations in the
parameter space. The pointing is achieved by incrementing
each admissible location by one during the voting process. A
coexisting location in the parameter space, defined by the
data elements that satisfy the observation equation, will be
incremented several times forming a global maximum in the
parameter space. This maximum will be evaluated as a
consistency measure between the two data sets.
In the sequel of the three subsections below, we present a
detailed derivation of geometric invariant features, the
principle of parameter space clustering and least squares
solution.
2.1 Construction of Geometric Invariant Features
In general, geometric invariants can be defined as properties
(functions) of geometric configurations that do not change
under a certain class of transformations (Mundy and
Zisserman, 1992). For instance, the length of a line does not
change under rigid motion such as translations and rotation.
In this subsection geometric invariance will be developed for
point sets under similarity transformation. Assume that we
have two point sets, P and Q, extracted from two images,
where Pz (Gs. y). |i=1,...,m} and
O= (yy, |j-L.,n]. A registration is to find a
correspondence between a point p, in P and a certain point
q ; in Q; that makes this corresponding pair consistent under
a selected mathematical transformation. The similarity
transformation, f (T uL wu) is used as registration
T
y are the
function between the two sets, where 1, E
translation along the x and y-axes, s is the scale factor, and 0
is the rotation angle between the two images. Let
(Dj ; D) and (dj > dq) be two corresponding pairs in
P and Q respectively. Geometric invariant quantities under
the similarity transformation can be derived as follows:
En. T, ES ame -sinO | xj (1)
y T, sin 0 cos0 Vil
Xj2 i T, +5 2056 —sin6 | x;2 (2)
yj T, sind cosÓ | yi2
By deriving the vector quantities between ( Pip 2) and
|
(dj G 55) we will end up with:
0 > cos —sinO | xj2 — Xj
= S
0 sind cos@ | vin — Vi
R
3)
x2 2
Yj2 3X7
