C. Vincent Tao
e The presence of the additional parameters helps to account for wider field angles of the scanner, and the additional
parameter a,, has proven most effective for SPOT imagery.
e With this model, although a reasonable degree of geometric stability is obtained, triangulation results can be
expected to deteriorate when only a small number of GCPs are available.
3
A GENERIC SENSOR MODEL - RATIONAL FUNCTION MODEL
The RFM uses a ratio of two polynomial functions to compute the x coordinate in the image, and a similar ratio to
compute the y coordinate in the image.
ml m2 m3
1. XY Zt
LEY ZO ^ Ll
A (8)
Dp2(X.Y,Z7) nl n2 nl
NV X bx YZ
i=0 j=0k=0
ml m2. m3
= Yyi7
porn Sài
pA(X,Y,2) Tw n2. n3
NN Yaxr2z
i=0 j=0k=0
Where x, y are normalized pixel coordinates on the image; X, Y, Z are normalized 3D coordinates on the ground (map
and DEM), and a, bis Cr die are polynomial coefficients.
The above equations can be rewritten as follows:
el zZ y YŸ rx) a, a ++ a) (9a)
üz yu — ¥ X) 0,7
A4 zy A reg) (9b)
(7 Y X Y x).
The polynomial coefficients are called rational function coefficients (RFCs). In general, distortions caused by optical
projection can be represented by ratios of first-order terms, while corrections such as earth curvature, atmospheric
refraction, and lens distortion etc., can be well approximated by second-order terms. Some other unknown distortions
with high order components can be modeled using a RFM with third-order terms.
Direct vs. Iterative Lease Square Solution
Given a DEM and an image acquired from the same earth surface using some physical sensor, we assume that many
object-image coordinate pairs of ground points (GPs) and corresponding image points (IPs) can be collected evenly
over the entire ground area. Now, what we should do is to solve RFCs under rational function model using these GP/IP
coordinate pairs, some of which are used as control coordinate pairs (CCPs) to solve RFCs, others are used as check
ones (CKPs).
To solve the RFCs by a linear least squares method, first of all, equations (8) have to be linearized with respect to the
RFCs. From equations (9), error equations can be written as:
I Z Y X y X? rZ rY rY! rY* Jot (10a)
v= Zod 2 I À. m o.u To Je
"dB. B B B h .B B B B B B
uev, Ys Xi ieu ey Qeyt tb ce (10b)
D D-D' D DD D D D D D
or
By el Ze X Y' X*o-.-ri -rY) -rX!BRJ-r (11a)
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000 877
