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Geometric correction of NOAA
AVHRR imagery with few GCPS
Masataka Takagi, Toshiaki Hashimoto, Shunji Murai
Institute of Industrial Science, University of Tokyo
7-22, Roppongi, Minato-ku,Tokyo 106, Japan
Tel : 03-402-6231, ext. 2560
Telex : 02427317 Kosmur J
Fax : 813-479-2762
Masataka Takagi, Toshiaki Hashimoto, Shunji Murai
Institute of Industrial Science, University of Tokyo
7-22, Roppongi, Minato-ku,Tokyo 106, Japan
Tel : 03-402-6231, ext. 2560
Telex : 02427317 Kosmur J
Fax : 813-479-2762
Introduction
There are three types of geometric correction for satellite imagery : (1) system correction; (20 black box correction with GCPs; (3) combination of (1) and (2). The third type is better than the others in accuracy and processing time. The authors have developed the algorithm based on the third type which utilizes phtogrammetric method. In this algorithm, the position and attitude of a satellite are treated as the exterior orientation parameters of photogrammetry and determined by the GCPs.
NOAA is a satellite for weather observation, therefore the cloud informations have to be taken for some applications. However, it is hard to get enough number of GCPs in the cloudy image. Still more, we must carry out geometric correction as fast as possible because NOAA data are received 4 or 6 times every day. In this paper, the determination of orientation parameters with few GCPs had been studies.
Method of geometric correction
The exterior orientation parameters of satellite imagery are determined by space resection in the same way as photogrammetry's one. The center of projection (X0, Y0, Z0), imagery point (x,y) and terrestrial object (X,Y,Z) is on a straight line. The equation of collinearity condition is shown in Eq. 1
| Fx= x - f | m11(X - Xq(L)) + m12(Y - Yq(L)) + m13(Z-Zq(L)) | = 0 |
| m31(X - Xq(L)) + m32(Y - Yq(L)) + m33(Z-Zq (L)) | ||
| Fy=y-f | m21(X - Xq(L))+ m22(Y - Yq(L))+ m23(Z-Zq(L)) | = 0 |
| m31(X - Xq(L)) + m32(Y - Yq(L) + m33(Z-Zq (L)) |
The orientation is carried out on the geocentric coordinates. If many GCPs can be acquired on the imagery, the equation of collinearity condition can be solved and the exterior orientation parameters are determined by using the least square method.
As NOAA AVHRR imagery is scanned in a line. The exterior orientation parameters change line by line. The exterior orientation parameters become the functions of time or line number. The functions are expressed by polynomials. Fourier series, spline functions, etc. The polynomial function is used in this study.
The position (x0, Y0, Z0) of satellite on eh geocentric coordinates is expressed as a function of the distance between geocenter and satellite (r), inclination (i), the latitude of ascending node (V), the angle or between ascending node and satellite on the orbital plane (u).
Eq.2
The eccentricity of satellite is so small, that is, the satellite moves on the circular orbit. Therefore it moves in the uniform velocity. In this case, the parameters r, i, V and u are expressed by the function of time t.
| r = rq
................(const) i = iq ....................(const) W = Wq + WI *t u = uq + uI *t |
..................Eq.3 |
After all , the position of satellite is expressed by the functions of sine and cosine at time t (or line L). since the first ordered approximation of sine function is expressed by a cubic polynomial, the position of satellite can also be expressed by cubic polynomials. The real position of satellite is more complicated, so it should be expressed at least by the 3rd degrees polynomials are adopted in this study.
| Xoq (L) = Xoq + XoI * L + Xo2 *
L2 + Xo3 * L3 Yoq(L) = Yoq + YoI * L + Yo2 * L2 + Yo3 * L3 Xoq(L) = Zoq + ZoI * L + Xo2 * L2 + Zo3 * L3 woq(L) = woq + woI * L + wo2 * L2 + wo3 * L3 foq(L) = foq + foI * L + fo2 * L2 + fo3 * L3 koq(L) = koq + koI * L + ko2 * L2 + ko3 * L3 |
..................Eq.4 |
A cubic polynomial has 4 coefficients. The number of an exterior orientation parameters is six. So 24 parameters are unknown. As there are two equations per one GCP. 12 GCPs are at least necessary for determining the unknown parameters.
Geometric correction with few GCPs
The test imagery was taken by NOAA AVHRR on December8, 1986 (UT). The size of imagery is (2048 Pixel x 4500 line).
The exterior orientation parameters were determined by eq.4 with 2 GCPs. In this case, the RMS error of 6 verification points was less than 1.6 pixels.
These cubic polynomials were shown in Fig.1 from this graph, it is possible to approximate each exterior orientation parameter by quadratic polynomial. Therefore, the orientation was carried out using quadratic polynomials with 9 GCPS. On the same 6 verification points, the error was less than 2.3 pixels.
When the number of GCPs is less than 9 in the imagery, it s necessary to decrease unknown coefficients. On this consideration exterior orientation parameters were determined with GCPs less than 9. However in this test imagery, computation was not converged, perhaps this imagery (x,y) or terrestrial object (X, Y, Z) has error. And least square method has problem with few data.
Conclusion
At this state, we analyzed exterior orientation parameters with few GCP. As a result, number of GCP can be decreased using quadratic equation. But exterior orientation parameters cannot be determined with less than 9 GCPs. The further study has to be carried to reduced with less than 09 GCPs. The further study has to be carried to reduced the number of GCPs to be less than 9 points. The idea are :
- Compute exterior orientation parameters with weight given GCPs
- Computer exterior orientation parameters using imaginary correction except block adjustment.
Fig.1 Orbit of satellite