Geometric correction of satellite stereo images by DEM matching without ground control points and map projection step: tested on Cartosat-1 images

Abstract

Ground Control Points (GCPs) are needed in most geometric processing of satellite images. Generally used geometric model is based on the use of rational polynomials. The Coefficients of Rational Polynomials (RPCs) are provided by image vendors which are contaminated by some biases. In this paper, a no-GCP bias correction method for RPCs of satellite stereo images is introduced. The method uses global DEMs as ground control information: First, a point cloud is generated from stereo pair, then using the DEM matching strategy it is aligned to the global DEM for estimation of 3D rigid transformation parameters. This transformation is performed with our originally developed method which separates three planimetric parameters from three leveling parameters. They are then employed for bias correction in object space. For DEM matching and also for bias correction, we developed new formulae given directly in geodetic longitude and latitude format, instead of Cartesian map projection coordinates. Numerical results of this research are reported in two categories: with or without (1) GCPs and (2) map projection step. Experiments on two Cartosat-1 stereo pairs show in category (1), improvement in geopositioning accuracy from 399.2 m and 124.0 m to 7.6 m and 2.6 m, respectively. In category (2), we observed RMS improvement in both datasets and in all components up to 3.1 m in longitude, 6.8 m in latitude and 1.2 m in height.

Appendices

Appendix 1

Substitution of Eq. (6) in Eq. (4) and neglecting the last term corresponding to scale parameter m results in:

$$ {h}_a-h=-\left(\frac{\partial {h}_a}{\partial \lambda}\frac{\partial \lambda }{\partial X}\right)R\updelta \lambda \cos \varphi -\left(\frac{\partial {h}_a}{\partial \varphi}\frac{\partial \varphi }{\partial Y}\right)R\updelta \varphi +\Delta h+\left(R\left(\varphi -{\varphi}_c\right)+\left(\frac{\partial {h}_a}{\partial \varphi}\frac{\partial \varphi }{\partial Y}\right)h\right)\varOmega +\left(-R\left(\lambda -{\lambda}_c\right)\cos \varphi -\left(\frac{\partial {h}_a}{\partial \lambda}\frac{\partial \lambda }{\partial X}\right)h\right)\varPhi +\left(\left(\frac{\partial {h}_a}{\partial \lambda}\frac{\partial \lambda }{\partial X}\right)R\left(\varphi -{\varphi}_c\right)-\left(\frac{\partial {h}_a}{\partial \varphi}\frac{\partial \varphi }{\partial Y}\right)R\left(\lambda -{\lambda}_c\right)\cos \varphi \right)K $$

(A.1)

Noting that

$$ \frac{\partial \lambda }{\partial X}=\frac{1}{R\cos \varphi },\kern0.5em \frac{\partial \varphi }{\partial Y}=\frac{1}{R} $$

(A.2)

and denoting $\frac{\partial {h}_a}{\partial \lambda }={S}_{\lambda }$ and $\frac{\partial {h}_a}{\partial \varphi }={S}_{\varphi }$ we obtain:

$$ {h}_a-h=-{S}_{\lambda}\left(\frac{1}{R\cos \varphi}\right)R\updelta \lambda \cos \varphi -{S}_{\varphi}\left(\frac{1}{R}\right)R\updelta \varphi +\Delta h+\left(R\left(\varphi -{\varphi}_c\right)+{S}_{\varphi}\left(\frac{1}{R}\right)h\right)\varOmega +\left(-R\left(\lambda -{\lambda}_c\right)\cos \varphi -{S}_{\lambda}\left(\frac{1}{R\cos \varphi}\right)h\right)\varPhi +\left({S}_{\lambda}\left(\frac{1}{R\cos \varphi}\right)R\left(\varphi -{\varphi}_c\right)-{S}_{\varphi}\left(\frac{1}{R}\right)R\left(\lambda -{\lambda}_c\right)\cos \varphi \right)K $$

(A.3)

Two terms ${S}_{\varphi}\left(\frac{1}{R}\right) h\varOmega$ and ${S}_{\lambda}\left(\frac{1}{R\cos \varphi}\right) h\varPhi$ are negligible because 1/R is a small real number (R ≈ 6371000m) and its multiplication by Ω and Φ which are themselves already small numbers becomes nearly zero. After some simplifications we obtain:

$$ {h}_a-h=-{S}_{\lambda}\updelta \lambda -{S}_{\varphi}\updelta \varphi +\Delta h+\left(\varphi -{\varphi}_c\right) R\varOmega -\left(\lambda -{\lambda}_c\right)\cos \varphi R\varPhi +\left({S}_{\lambda}\frac{\varphi -{\varphi}_c}{\cos \varphi }-{S}_{\varphi}\left(\lambda -{\lambda}_c\right)\cos \varphi \right)K $$

(A.4)

Denoting Q₁ = RΩ and Q₂ = RΦ we obtain the same equation as Eq. (8).

Appendix 2

Updating the horizontal location of IDEM after jth (passing from jth to (j + 1)th) iteration using Eq. (12) is done by:

$$ \left[\begin{array}{c}{\lambda}^{\left(j+1\right)}-{\lambda}_c\\ {}{\varphi}^{\left(j+1\right)}-{\varphi}_c\end{array}\right]=\left[\begin{array}{cc}1& \frac{\kappa^{(j)}}{\cos \varphi}\\ {}-{\kappa}^{(j)}\cos \varphi & 1\end{array}\right]\left[\begin{array}{c}{\lambda}^{(j)}-{\lambda}_c\\ {}{\varphi}^{(j)}-{\varphi}_c\end{array}\right]+\left[\begin{array}{c}\delta {\lambda}^{(j)}\\ {}\delta {\varphi}^{(j)}\end{array}\right] $$

(B.1)

Passing from jth to (j + 2)th iteration gives:

$$ \left[\begin{array}{c}{\lambda}^{\left(j+2\right)}-{\lambda}_c\\ {}{\varphi}^{\left(j+2\right)}-{\varphi}_c\end{array}\right]=\left[\begin{array}{cc}1& \frac{\kappa^{\left(j+1\right)}}{\cos \varphi}\\ {}-{\kappa}^{\left(j+1\right)}\cos \varphi & 1\end{array}\right]\left(\left[\begin{array}{cc}1& \frac{\kappa^{(j)}}{\cos \varphi}\\ {}-{\kappa}^{(j)}\cos \varphi & 1\end{array}\right]\left[\begin{array}{c}{\lambda}^{(j)}-{\lambda}_c\\ {}{\varphi}^{(j)}-{\varphi}_c\end{array}\right]+\left[\begin{array}{c}\delta {\lambda}^{(j)}\\ {}\delta {\varphi}^{(j)}\end{array}\right]\right)+\left[\begin{array}{c}\delta {\lambda}^{\left(j+1\right)}\\ {}\delta {\varphi}^{\left(j+1\right)}\end{array}\right] $$

(B.2)

In the same way, passing from jth to (j + 3)th iteration gives:

$$ \left[\begin{array}{c}{\lambda}^{\left(j+3\right)}-{\lambda}_c\\ {}{\varphi}^{\left(j+3\right)}-{\varphi}_c\end{array}\right]=\left[\begin{array}{cc}1& \frac{\kappa^{\left(j+2\right)}}{\cos \varphi}\\ {}-{\kappa}^{\left(j+2\right)}\cos \varphi & 1\end{array}\right]\left\{\left[\begin{array}{cc}1& \frac{\kappa^{\left(j+1\right)}}{\cos \varphi}\\ {}-{\kappa}^{\left(j+1\right)}\cos \varphi & 1\end{array}\right]\left(\left[\begin{array}{cc}1& \frac{\kappa^{(j)}}{\cos \varphi}\\ {}-{\kappa}^{(j)}\cos \varphi & 1\end{array}\right]\left[\begin{array}{c}{\lambda}^{(j)}-{\lambda}_c\\ {}{\varphi}^{(j)}-{\varphi}_c\end{array}\right]+\left[\begin{array}{c}\delta {\lambda}^{(j)}\\ {}\delta {\varphi}^{(j)}\end{array}\right]\right)+\left[\begin{array}{c}\delta {\lambda}^{\left(j+1\right)}\\ {}\delta {\varphi}^{\left(j+1\right)}\end{array}\right]\right\}+\left[\begin{array}{c}\delta {\lambda}^{\left(j+2\right)}\\ {}\delta {\varphi}^{\left(j+2\right)}\end{array}\right] $$

(B.3)

After multiplications and neglection of two multiplied azimuthal rotation angles κ, we obtain:

$$ {\lambda}^u-{\lambda}_c=\lambda -{\lambda}_c+\frac{\varphi -{\varphi}_c}{\cos \varphi }{\sum}_{j=1}^J{\kappa}^{(j)}+{\sum}_{j=1}^J\delta {\lambda}^{(j)}+\frac{1}{\cos \varphi}\left({\sum}_{j=1}^{J-1}\delta {\varphi}^{(j)}\right)\left({\sum}_{i=1}^{J-j}{\kappa}^{\left(i+j\right)}\right) $$

(B.4)

$$ {\varphi}^u-{\varphi}_c=\varphi -{\varphi}_c-\left(\lambda -{\lambda}_c\right)\cos \varphi {\sum}_{j=1}^J{\kappa}^{(j)}+{\sum}_{j=1}^J\delta {\varphi}^{(j)}-\cos \varphi \left({\sum}_{j=1}^{J-1}\delta {\lambda}^{(j)}\right)\left({\sum}_{i=1}^{J-j}{\kappa}^{\left(i+j\right)}\right) $$

(B.5)

where (λ^u, φ^u) are geodetic coordinates after the last iteration. Regarding to Eq. (B.4) and Eq. (B.5), the last terms in both equations are considerably small and one can neglect them. However, in Eq. (B.4) we pay attention that it is not valid for near-polar regions where cosφ_c goes to zero. Substitution from Eq. (13) to Eq. (B.4) and Eq. (B.5) gives:

$$ {\lambda}^u-{\lambda}_c=\lambda -{\lambda}_c+\frac{\varphi -{\varphi}_c}{\cos \varphi }K+\Delta \lambda $$

(B.6)

$$ {\varphi}^u-{\varphi}_c=\varphi -{\varphi}_c-\left(\lambda -{\lambda}_c\right)\cos \varphi K+\Delta \varphi $$

(B.7)

Taking into account the leveling parameters, estimated from Eq. (14), and combining with Eq. (B.6) and Eq. (B.7) the final matrix form of proposed BCM as Eq. (16) is obtained.