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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
rather than point features, conventional photogrammetric rules 
may not be appropriate (Mikhail and Kanok, 1994); c) most of 
these models are valid for the projective geometry imagery of a 
photograph which is not exactly the case for linear array sensor 
imagery; d) the models become quite complicated when 
modified for the geometry and time dependency characteristics 
of linear array scanners; e) numerical problems could be 
encountered because of the initial approximation; and finally, f) 
constraints improve accuracy of the adjustment and increase the 
redundancy in estimation but each constraint adds an additional 
parameter to the adjustment and multiple constraints may lead 
to over parameterization (Habib et al., 2003). 
To date, there has been a substantial body of work dealing with 
non-rigorous mathematical models (such as rational functions, 
affine, polynomial, and DLT models) to circumvent the absence 
of satellite information and to rectify HRSI (see for example 
Fraser et al. (2002), Fraser and Hanley (2003), Shi and Shaker 
(2003), and Grodecki and Dial (2003)). These models are point 
based and have focused on two main aspects concerning 
accuracy: the accuracy attainable in image rectification, and the 
accuracy of DTM extraction by stereo spatial intersection. All 
reports demonstrate that the models described in them produce 
acceptable results. 
It is obvious that linear features can be used with rigorous 
mathematical models and points can be applied to non-rigorous 
mathematical models. That leads to the question of “Can linear 
features be used with non-rigorous mathematical models in 
order to circumvent the absence of satellite information and 
maintain satisfactory results?” This research answers the 
question with the development of a new model named the Line 
Based Transformation Model (LBTM). 
With the LBTM, most of the problems of using linear features 
with the present gencration of rigorous models have been 
overcome. The model can either solely use linear features or 
use linear features plus a number of control points to define the 
image transformation parameters. It is a very simple model 
which is time independent, can be applied to images from any 
linear array sensor, does not require any information about 
sensor calibration or satellite orbit, and does not require any 
initial approximation values. The underlying principle of the 
model is that the relationship between line segments of straight 
lines in the image space and the object space can be expressed 
by affine or conformal relationships. The model adopts the 
same structure for 3D transformation as the eight-parameter 
affine model, and the same structure for 2D transformation as 
the six-parameter affine and four-parameter conformal models. 
Adopting these structures further allowed direct comparison 
between the developed LBTM and the existing models. 
Synthetic as well as real data have been used to check the 
validity and fidelity of the model and the results show that the 
LBTM can be used to efficiently and accurately rectify HRSI. 
2. THE MATHEMATICAL MODEL 
Successful exploitation of linear features in image rectification 
and terrain modeling requires consideration of the following 
two major aspects: the mathematical description of linear 
features in image and object space and the mathematical 
representation of the relationship between the two spaces. There 
are different options for representing linear features in both 
image and object space. Straight lines, circles, ellipses and free 
form lines are examples of such representation. In this work, 
straight lines as well as natural lines (free form lines) converted 
851 
to straight lines by mathematical functions are used. Circles and 
ellipses are discarded due to their impracticality and because 
they are not transformation invariant. 
Various forms of equations can represent straight lines in two 
and three-dimensional spaces with each of them exhibiting 
some weakness in certain applications. For each of these forms, 
describing a line in 2D or 3D space requires two or four 
independent parameters, respectively. Straight lines can be 
represented in either image or object space in different ways 
such as the intersection between two planes, line descriptors, 
unit vectors and normal to line descriptors. The Line Based 
Transformation Model (LBTM) is based on the relationship 
between the unit vector components of a line in image space 
and the unit vector components of the conjugate line in object 
space. Unit vector representation, which can be obtained from 
any two points along straight line segments, was chosen 
because they can be easily defined from images, existing geo- 
databases or terrestrial mobile mapping systems in both image 
and object space. However, the unit vector is not a unique 
representation of a straight line as it can represent the line in 
question and an infinite number of parallel lines. This problem 
is addressed in section 2.1.1. 
The LBTM applies to high-resolution satellite imagery 
produced from CCD linear array sensors, which are widely used 
in remote sensing applications. Regardless of the capturing 
technique (cross or along track), images from linear array 
sensors consist of independent scanned lines. Each line on the 
image is the result of a nearly parallel projection in the flight 
direction and a perspective projection in the CCD line direction. 
Therefore, rigorous mathematical models based on the 
collinearity equations and including a time dependent function 
could be applied for geo-referencing the images. To circumvent 
the complexity of the time dependent model, in other words, to 
simplify the relationship between image and object coordinate 
systems, several assumptions were adopted: (a) the satellite 
sensor moves linearly in space with stable attitude; (b) the 
sensor orientation angles are constant; and (c) the satellite flight 
path is almost straight. 
Under these assumptions, the scanned lines from the sensor can 
be considered to form a continuous (single) image. These 
characteristics allow the ordinary collinearity equations 
between the satellite imagery and the ground to be replaced by 
simple affine and conformal transformation models similar to 
those introduced in Hanly and Fraser (2001) and Fraser et al., 
(2002). The underlying principle of the developed model is that 
the unit vector components in either image or object space 
could replace the point coordinates in the previous models. 
Both affine and conformal implementations of the LBTM were 
developed. The derivation of the 3D affine LBTM is given in 
the following section and the 2D affine and the 2D conformal 
LBTM form can be obtained by simplification. Here we will 
refer to eight-parameter affine model, six-parameter affine 
model, and four-parameter conformal model as 3D affine 
model, 2D affine model and 2D conformal model respectively. 
2.1 The 3D affine LBTM 
Vectors » and V; are unit vectors for conjugate lines in image 
and object space respectively (see Figure 1). The two unit 
vectors can be defined by any two points along the line segment 
in image and object space. Suppose that point p;= (x, y;) and