computational scenarios with the physical sensor model available.
The establishment of imaging geometric model is the basis of photogrammetric stereotaxic processing, which reflects the
relationship between 2D image coordinates and their corresponding 3D ground coordinates. Rigorous model
describes the imaging process in the case of known sensor information, thus it is closely related to the physical and
geometric characteristics of the sensors. Rigorous geometric modeling consists of exterior orientation and interior orientation
and coordinate calculation based on collinearity equation. In this section, a general description of pushbroom optical
rigorous physic model is given below.
2.1.1 Interior Orientation
Interior orientation refers to the transformation from image coordinates lines and samples to their focal plane coordinates
centered at the principal point of the image. Given the focal length f of the camera and the pointing angle
X
,
Y
of a pixel r, c, the focal plane coordinates of this pixel can be calculated
by the interior orientation formula as follows:
tan tan
X Y
x f
y f
1 Image space coordinate system is defined using the perspective
center S as its origin. X- and Y-axis are coincident with the X- and Y- axis of focal plane coordinate system, Z axis is
coincident with principal optic axis, forming a right-handed coordinate system. Thus, the pointing vector u of the pixel in
image space coordinate system can be presented as:
tan tan
X Y
f u
f f
2
2.1.2 Exterior Orientation
Exterior orientation refers to the coordinate transformation from image space coordinate system to an object space coordinate
system. Exterior orientation process can be expressed as:
IG OI
BO TB
tan tan
S X
S Y
S
X X
f Y
Y f
u f
Z Z
R R R R
R
3 where X, Y, Z and Xs, Ys, Zs are the ground point and the
perspective center position in planetary body-fixed coordinate system PBCS, respectively; R
ib
is the rotation matrix from image space coordinate system ISCS to the spacecraft body
coordinate system BCS; R
bo
is the rotation matrix from BCS to orbit coordinate system OCS; R
ol
is the rotation matrix from the OCS to to PBCS;
is a scale factor; R represents the overall rotation matrix from ISCS to PBCS. Usually, the
transformation relationship between coordinate systems is determined by three
Euler angles φ, ω, , the attitude angles φ, ω, and the perspective center position Xs, Ys, Zs are
called exterior orientation parameters EOPs. For push-broom CCD sensors, EOPs are changing all the time;
in other words, each imaging moment has a set of EOPs. Considering the stability of the motion platform, change of
EOPs over short trajectories can be modeled using polynomials. A third-order polynomial model with imaging time as the
independent variable is usually used for fitting of EOPs.
2 3
1 2
3 2
3 1
2 3
2 3
1 2
3 2
3 1
2 3
2 3
1 2
3 2
3 1
2 3
s s
s
X t a
a t a t
a t Y t
b b t
b t b t
Z t c
c t c t
c t t
d d t
d t d t
t e
e t e t
e t t
f f t
f t f t
4
where a0, a1, … f3 are the polynomial coefficients of the six EOPs
Xs, Ys, Zs, φ, ω, .
2.1.3 Coordinate calculation
Collinearity equation is the basic and important formula in photogrammetry, which is widely used in space intersection,
back projection and so on. It is established based on the principle that the perspective center, an image point and the
corresponding ground point lie in a straight line. The matrix form is expressed in Eq.3. With the collinearity equation, we
can calculate the 3D ground coordinates from image coordinates with elevation or calculate the image coordinates
from 3D ground coordinates. Planets can be considered as an ellipsoid which is formed by
rotation around its rotational axis. We define the equatorial radius of a planet is a and the polar radius is b; assuming that a
ray from the perspective center SXs, Ys, Zs intersects the planet at a ground point MXM,YM,ZM which has an
elevation of h See Fig.1,
Figure 1. Ground point calculation by space It can be deduced according to Eq.3:
1 2
3 M
S M
S M
S
X X
u Y
Y u
u Z
Z
5
where
T 1
2 3
u u
u is the normalized form of vector Ru in
Eq.3. Eq.6 can be further written as Eq.7.
1 2
3 M
S M
S M
S
X u
X Y
u Y
Z u
Z
6
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XLI-B4-441-2016
442
Considering that the ray actually intersects an ellipsoid with three semi-principal axes of length a+h, a+h, b+h, thus, it can be
also deduced as follows:
2 2
2 2
2 2
1
M M
M
X Y
Z a
h a
h b
h
7
Substituting Eq.7 into Eq.8, we can derive a quadratic equation about as Eq.9.
2 1
2 3
p p
p
8 Where
2 2
2 2
2 1
1 2
3 2
2 2
1 2
3 2
2 2
2 2
2 2
3
[ ]
2[ ]
[
S S
S S
S S
p b
h u
u a
h u p
b h
X u Y u
a h Z u
p b
h X
Y a
h Z a
h b
h
9 Hence, the two roots of the equation are:
2 2
2 1
3 1
4 2
p p
p p p
10 By submitting the value of to Eq.6, the ground coordinates
X
M
, Y
M
, Z
M
can be solved. One of the solutions is the actual ground coordinates of M, and the other is a fake one which is
intersected at the reverse side by the ray throughout the ellipsoid. By judging whether X
M
, Y
M
, Z
M
is in a reasonable range or which has the relative short distance from perspective
center S to the two intersection points, the fake solution can be eliminated and the correct solution is kept.
Back projection refers to the transformation from a ground point to its corresponding image point. For push-broom CCD
sensor, back projection can be performed using binary searching method. Given the ground coordinates, the image coordinates
are calculated with EOPs of the middle scan line using Eq.5; by comparing the calculated x value to the theoretical x value
row direction, focal plane coordinates continue to be calculated using the upper or lower half of EOPs until the
difference between the calculated value and theoretical value is less than a pre-defined threshold. The EOPs which can get the
closest x value are finally chosen to calculate the image coordinates of this point see Fig.2.
Ground coordinates Exterior orientation
parameters
Is the difference between x and theoretical value less
than the threshold Continue searching
with the upper or lower half of EOPs
No Yes
Taking EOPs of the middle scan line Calculating focal plane coordinatesx, y
Calculating x, y with EOPs in a window centered at this scan line
Taking the EOPs which get the closest x value as the true EOPs of this point
Figure 2. Back projection using binary searching
2.2 Construction of rational function model
