Geolocation error tracking of ZY-3 three line cameras

ISPRS Journal of Photogrammetry and Remote Sensing 123 (2017) 62–74

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Geolocation error tracking of ZY-3 three line cameras Hongbo Pan School of Geosciences and Info-Physics, Central South University, #932 Lunan Road, Changsha, Hunan Province 410083, PR China

a r t i c l e

i n f o

Article history: Received 10 June 2016 Received in revised form 17 November 2016 Accepted 18 November 2016

Keywords: Bundle adjustment ZY-3 three line camera Error propagation Geolocation error Relative orientation

a b s t r a c t The high-accuracy geolocation of high-resolution satellite images (HRSIs) is a key issue for mapping and integrating multi-temporal, multi-sensor images. In this manuscript, we propose a new geometric frame for analysing the geometric error of a stereo HRSI, in which the geolocation error can be divided into three parts: the epipolar direction, cross base direction, and height direction. With this frame, we proved that the height error of three line cameras (TLCs) is independent of nadir images, and that the terrain effect has a limited impact on the geolocation errors. For ZY-3 error sources, the drift error in both the pitch and roll angle and its influence on the geolocation accuracy are analysed. Epipolar and common tiepoint constraints are proposed to study the bundle adjustment of HRSIs. Epipolar constraints explain that the relative orientation can reduce the number of compensation parameters in the cross base direction and have a limited impact on the height accuracy. The common tie points adjust the pitch-angle errors to be consistent with each other for TLCs. Therefore, free-net bundle adjustment of a single strip cannot significantly improve the geolocation accuracy. Furthermore, the epipolar and common tie-point constraints cause the error to propagate into the adjacent strip when multiple strips are involved in the bundle adjustment, which results in the same attitude uncertainty throughout the whole block. Two adjacent strips—Orbit 305 and Orbit 381, covering 7 and 12 standard scenes separately—and 308 ground control points (GCPs) were used for the experiments. The experiments validate the aforementioned theory. The planimetric and height root mean square errors were 2.09 and 1.28 m, respectively, when two GCPs were settled at the beginning and end of the block. Ó 2016 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.

1. Introduction High-accuracy geolocation is a key factor for integrating multitemporal multi-sensor high-resolution satellite images (HRSIs) for national and even global monitoring and mapping. Even though the geolocation accuracy has been improved to within several meters, ground control points (GCPs) are required for image orientation. Two different strategies are used for orientation in photogrammetry: directly estimating the sensor model with GCPs and compensating the integrated sensor error (Di et al., 2014; Heipke, 1997; Heipke et al., 2002; Muller et al., 2012). Directly estimating the sensor model utilizes GCPs to estimate the orientation parameters with a given sensor model, while the position and attitude data are used to estimate the initial value. Owing to the innovative orbit and attitude-determination techniques, methods for compensating the integrated sensor orientation are widely used for HRSI orientation with both the rigorous sensor model (RSM) and the rational function model (RFM), and these methods achieve

E-mail address: [email protected]

remarkable accuracy with fewer GCPs (Fraser and Ravanbakhsh, 2009; Grodecki and Dial, 2003; Li et al., 2011; Teo, 2011). In this paper, we mainly focus on the method of compensating the integrated sensor error. The geolocation error of HRSIs depends on the exterior orientation parameter (EOP) errors, interior orientation parameter (IOP) errors, and imaging geometry of the stereo camera. The variancecovariance matrix is a powerful tool to assess the accuracy of the geolocation. Topan and Kutoglu (2009) used the figure condition method to assess the georeferencing accuracy of a single scene with affine projection. Li et al. (2009) showed that the geolocation error depends on the convergence angle of stereo images. In addition to the convergence angle, the bisector elevation and asymmetry angle are important considerations for interpreting the stereo geometry (Dolloff and Theiss, 2012). To quantitatively estimate the positional accuracy of satellite stereo pairs, the modified convergence, bisector elevation, and asymmetry angles were investigated by Jeong and Kim (2016). Tang et al. (2015b) analysed comprehensive planar and vertical errors with independent error sources for ZY-3. In theory, the error ellipsoids statistically represent the geolocation error. However, the physical explanation of

http://dx.doi.org/10.1016/j.isprsjprs.2016.11.007 0924-2716/Ó 2016 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.

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H. Pan / ISPRS Journal of Photogrammetry and Remote Sensing 123 (2017) 62–74

geolocation uncertainty is still ambiguous for stereo pairs. In this paper, we will build a quantitative framework for analysing the geolocation error of a stereo pair, in which three-axes of the error ellipsoid are determined with the line-of-sight (LoS) uncertainty in the orthogonal coordinates. This contribution will help to understand the specifications of geolocation accuracy, typically in terms of CE90 (circular error at 90% confidence) or LE90 (linear error at 90% confidence). Owing to the available EOPs, the direct orientation is a possible method to obtain the object coordinates with conjugate points. However, the EOP errors cause the LoS to be biased from the true position and direction. To compensate the EOP errors, bundle adjustment, with or without GCPs, is the most powerful tool. In the airborne case, the free-net bundle adjustment could improve the accuracy in image space with tie points for integrated sensor orientation (Heipke et al., 2002; Khoshelham, 2009). However, the EOPs of HRSIs are correlated because of the very narrow field of view (FoV) and stably dynamic imaging system. The change of EOP errors of HRSIs is still unknown when bundle adjustment without GCPs, also named as free-net bundle adjustment in this paper, is performed. In compensating the integrated sensor error perspective, the free-net bundle adjustment would adjust the compensation parameters to make all corresponding rays intersect with each other. It is equivalent to the photogrammetric relative orientation with the initial value of ‘seven parameters,’ because there are no GCPs involved and the rays are still biased. In this paper, we will investigate the degree of freedom of the free-net bundle adjustment for HRSIs. Compared to frame cameras, HRSIs suffer from an illconditioned normal equation for block bundle adjustment, because the very narrow FoV and dynamic imaging system causes images between the strips to have a very small convergence angle and the images in the same strip to have parallel rays. To utilize the stability of HRSI, the images in the same strip usually are treated as a unit (Fraser and Ravanbakhsh, 2011; Pan et al., 2016a). However, the constraint between the strips is rarely studied with a rigorous sensor model. Lutes and Grodecki (2004) used an azimuthelevation block adjustment model with only two offset parameters to analyse the error propagation of IKONOS Geo products. The geolocation error of adjacent strips have large discrepancies as shown in ALOS (Ravanbakhsh et al., 2012), IRS-P5 (d’Angelo and Reinartz, 2012), and ZY-3 (Zhang et al., 2015). With analysis of the error propagation, we will prove that the roll and pitch angle compensation model could reduce the discrepancy, and the result is independent to the overlap between the adjacent strips. To pursue higher geolocation accuracy for ZY-3, we track the geolocation error with different orientation strategies, including direct orientation, free-net bundle adjustment, and bundle adjustment with GCPs. A quantitative frame is proposed to analyse the impact of EOP errors on the geolocation accuracy. To understand the error compensation procedure, the epipolar constraint is studied with relative orientation. After that, the bundle adjustment is used to further reduce the EOP errors, and we elaborate on how error propagation affects three line cameras (TLCs) and the constraints between adjacent strips. Owing to the high geolocation of ZY-3,
10 m, the yaw angle is not significant compared with the roll and pitch angle, and the major errors could be modelled with the shift and drift compensation model on the pitch and roll angles (Pan et al., 2016a). Therefore, the drift compensation model is used as the EOP error model, and two adjacent strips of ZY-3 with 308 GCPs of field survey data are used for the experiments. This paper is structured as follows. In Section 2, the geolocation error frame of a HRSI is proposed. Two simplified rays with nadir viewing are first introduced, following the general condition. The error compensation is presented in Section 3, in which the relative orientation, bundle adjustment, and error propagation are

explained. In Section 4, experiments are described. Finally, the summary and concluding remarks are presented in Section 5. 2. Geolocation error of HRSIs stereo The geolocation error of HRSIs depends on the EOP errors, IOP errors, picking-point (matching) errors, and imaging geometry. The errors can be simplified as the LoS error, as they can be modelled by the change in the LoS. Owing to the precise orbit determination, a centimetre-level position error is negligible for photogrammetric applications with a meter-level ground sample distance. The routine geometric calibration reduces the IOP error to sub pixels, which is the same level as the picking-point errors. However, the attitude uncertainty might be introduced by heat effects during the imaging process (Pan et al., 2016a). In this section, the influence of the LoS uncertainty is used to study the geolocation error. 2.1. Geolocation error in the epipolar coordinate system To simplify the derivation, a stereo with two rays is analysed. The two rays S1P and S2P meet at P with angles h1 and h2 , where S1 and S2 are the imaging positions. According to the stereo geometry, the epipolar coordinate system is built, whose X-axis is the baseline, the Z-axis passes though the ground point P and perpendicular to the baseline in the S1S2P plane, and the Y-axis is determined by the right-hand rule. The angle between the ray and the perpendicular of the baseline is within ½p=2; p=2, and the flight direction is positive. Therefore, the h1 of the forward ray is positive, and the h2 of the backward ray is negative, as shown in Fig. 1. The LoS uncertainty could be divided into two directions: the base direction Da and the cross base direction Db. Owing to the LoS uncertainty, the two rays, S1P1 and S1P2, may not intersect in three-dimensional space. In photogrammetry, a least squares estimation is used to calculate the position with minimum square of residual in the image space (Mikhail et al., 2001). In this paper, we simplify this solution to find the height H, in which the distance between P1 of S1P and P2 of S2P is minimum. The calculated object coordinate is then equal to the mean value of the two points, if they are of equal weight, as P0 in the larger version of Fig. 1. Then, the calculated ground point P0 is biased from P (0, 0, h) in three directions ½ dX dY dZ . The ray from S1 would intersect the Z-plane H at P1, whose coordinate is (X1, Y1, H). At the same time, the ray from S2 meets Z-plane Y

Z

S2 (XS2,0,0)

O

S1 (XS1,0,0)

θ2 θ1 Δβ1

h

Δα1

Δβ 2

Z

X

Y X

δX

δY

Δα 2

P

δZ P

H P1 (X1,Y1,H)

P

P (0,0,h) P2(X2,Y2,H)

Fig. 1. Schematic diagram of geolocation error in the epipolar coordinate system. Owing to LoS uncertainty, the calculated object point P0 is biased from the true position P in three directions.

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H. Pan / ISPRS Journal of Photogrammetry and Remote Sensing 123 (2017) 62–74

H at P2 (X2, Y2, H). Then, the square of the distance between P1 and P2, d(H), is as follows:

dðHÞ ¼ ðX 1  X 2 Þ2 þ ðY 1  Y 2 Þ2 :

ð1Þ

Without loss of generality, we will hereinafter suppose that the LoS uncertainty in the cross base direction, Da, had been engaged into the viewing angle h. According to the geometry model, the planimetric coordinate of P1 is




X 1 ¼ ðH  hÞ  tan h1 Y 1 ¼ H  sec h1  tan Db1

;

ð2Þ

 sec2 h1  tan h2  Da1 þ sec2 h2  tan h1  Da2

ðtan h1  tan h2 Þ2  sec2 h1  Da1 þ sec2 h2  Da2 dZ ¼ b ðtan h1  tan h2 Þ2

b

ð9Þ ð10Þ

Following from (6) and (7), the geolocation error in Y direction dY is given by

dY ¼

sec h1  Db1 þ sec h2  Db2  b: 2ðtan h1  tan h2 Þ

ð11Þ

If the LoS uncertainty is independent in both base and cross base direction, the geolocation error ellipsoid could be calculated with (9)–(11).

and the planimetric coordinate of P2 is:




dX ¼

X 2 ¼ ðH  hÞ  tan h2 : Y 2 ¼ H  sec h2  tan Db2

ð3Þ 2.2. Geolocation error in the geodetic system

If (2) and (3) are substituted into (1), then 2

dðHÞ ¼ ðH  hÞ ðtan h1  tan h2 Þ2 þ H2 ðsec h1  tan Db1  sec h2  tan Db2 Þ2 :

ð4Þ

To minimize d(H), H needs to fit the following condition:

H ¼h

ðtan h1  tan h2 Þ2 ðtan h1  tan h2 Þ þ ðsec h1  tan Db1  sec h2  tan Db2 Þ2 ð5Þ

For HRSI, tan h1  tan h2 is equal to the base to height ratio, which is usually designed to be 0.6–0.8. Meanwhile, the LoS uncertainty Db is a very small value because the geolocation of a HRSI is generally tens of meters. Therefore, Db2  h  tan h1  tan h2 . For HRSI, the LoS uncertainty Db would not introduce geolocation errors in the X and Z directions. With H ¼ h in (2) and (3), the dY is

sec h1  tan Db1 þ sec h2  tan Db2 : 2

ð6Þ

The LoS uncertainty in the epipolar direction would introduce a geolocation error in both the X and Z directions, as shown in Fig. 2. We define the length of the stereo baseline as b. The height h of the triangle PS1S2 is

h¼

b : tan h1  tan h2

2 3 3 dX DE 7 6 7 g 6 4 DN 5 ¼ Rl  4 dY 5; dZ DH 2

2

dY ¼ h 

The geolocation accuracy is evaluated in the geodetic system, rather than a local system. Therefore, the epipolar coordinate system should be transformed into the geodetic system. This system can be derived from the local geodetic system O-NEH with three rotations: j about the H-axis, u about the N-axis, and x about the E-axis, as shown in Fig. 3. The geodetic geolocation error is

ð7Þ

ð12Þ

where Rlg is defined as:

2

Rlg

3 2 3 cos u 0  sin u cos j  sin j 0 4 5 4 ¼ sin j cos j 0  0 1 0 5 0 0 1 sin u 0 cos u 2 3 1 0 0  4 0 cos x  sin x 5: 0 sin x cos x

ð13Þ

In practice, Rlg can be calculated using the geodetic coordinates of the points P, S1, and S2, according to the definition of the local system. It should be noted that u, x, and j do not correspond to the roll, pitch, and yaw angles of the images in the orbit coordinate system but can be approximated with the orbit direction and view angle of the camera.

The distance d between the foot point and S1 is

d¼

tan h1  b: tan h1  tan h2

ð8Þ

Hϕ

If there are LoS errors Da1 and Da2 of the rays from S1 and S2, respectively, the rays intersect at point Q with an X error dX and Z error dZ.

ω

θ1

Δα1

θ2

H

N

X

θ2

H

Nκ

ϕ κ

N

Eκ −ϕ

ϕ

κ

Q

δX

ω

Hϕ

Δα 2

h

Nκ

θ1

S2

S1

Y

δZ P

Fig. 2. Sketch of the intersection of two rays in the epipolar plane. The LoS uncertainty Da1 of S1P and Da2 of S2P cause geolocation errors dX and dZ.

O

P

Eκ

E

E

Fig. 3. Sketch of 3D geolocation error. Transformation from the geodetic system to epipolar coordinates is performed by three rotations about three axes-j about the H-axis, u about the N-axis, and x about the E-axis, which correspond to the Z-Y-X axes, respectively.

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H. Pan / ISPRS Journal of Photogrammetry and Remote Sensing 123 (2017) 62–74

2.3. Geolocation error of ZY-3 TLCs In the case of the ZY-3 forward and backward cameras, h1 ¼ h2 when the satellite has the nadir viewing angle. Substituting h ¼ h1 into (9)(11), the geolocation error of rays S1P1 and S2P2 is

2

3

dX 12 6 7 4 dY 12 5 ¼ dZ 12

2

3

Da1 þDa2 b 2 sin 2h 6 Db1 þDb2 7 6 7 4 4 sin h  b 5: Da1 þDa2 b 4 sin2 h

ð14Þ

The geodetic geolocation error depends on the geodetic coordinate system. For the nadir viewing angle, dX 12 and dY 12 represent the planimetric errors and dZ 12 is the height error. If the attitude errors of the both cameras are equal, the height error is 0, and the planimetric error is Da1  b= sin 2h. This theory may explain the temporal correlation of commercial satellites, which have a better height accuracy than horizontal accuracy (Dolloff and Theiss, 2012). On the contrary, when Da1 ¼ Da2 , the planimetric 2

error is 0, and the height error is Da1  b=2 sin h. For TLCs, the epipolar planes of multi-stereo images are coplanar. The geolocation of multiple rays could be simplified as a fusion of multiple stereo images. If there are LoS uncertainties, Da1 , Da2 , and Da3 , of TLCs in the base direction, then there are three intersections, P12, P23, and P13, as depicted in Fig. 4. The geolocation errors of rays S1P1 and S2P2 have been studied as (14). However, the incline of the baseline should be taken into consideration for the geolocation error of rays S1P1 and S3P3. The triangle S1PS3 could be determined with the imaging geometry, because the S3P is equal to orbit height. Let \S2 S1 S3 ¼ u, then h1 ¼ h  u, h2 ¼ u, and b13 ¼ b=2  sec u. Using (9)–(11), the geolocation errors of rays S1P and S3P is

2

dX 13

3

2 sec2 ðhuÞtan uDa

6 6 7 6 4 dY 13 5 ¼ 6 4 dZ 13

2 1 þtanðhuÞsec uDa3 2ðtanðhuÞþtan uÞ2

secðhuÞDb1 þsec uDb3 4ðtanðhuÞþtan uÞ

 b  sec u

 b  sec u

 sec2 ðhuÞDa1 þsec2 uDa3 2ðtanðhuÞþtan uÞ2

3 7 7 7: 5

ð15Þ

 b  sec u

According to (15), the geolocation of P23 is

2

dX 32

3

2 sec2 utanðhuÞDa

6 6 7 6 4 dY 32 5 ¼ 6 4 dZ 32

2 3 þsec ðhuÞtan uDa2 2ðtanðhuÞþtan uÞ2

sec uDb3 þsecðhuÞDb2 4ðtanðhuÞþtan uÞ

 b  sec u

 b  sec u

 sec2 uDa3 þsec2 ðhuÞDa2 2ðtanðhuÞþtan uÞ2

3 7 7 7: 5

ð16Þ

 b  sec u

It should be noticed that (15) and (16) describe the geolocation error in the epipolar coordinate system. To integrate three geolocation errors, (15) and (16) should be substituted into (13), while

S3 S1

S2

ϕ θ

Δα 3

Δα1

Δα 2

θ P13

P12

P23 P Fig. 4. Sketch of the geolocation error for TLCs. The LoS uncertainties cause the three rays to intersect at three different positions P12, P23, and P13.

only the rotation angle of the Y-axis is taken into consideration. However, the triangles S1PS3 and S2PS3 have the same angle value u but with different directions. Therefore, the geolocation error of TLCs is

82 3 2 3 dX 13 > < cos u 0  sin u 1 6 7 6 7 6 7 1 0 5  4 dY 13 5 4 dY 5 ¼  4 0 3 > : dZ dZ 13 sin u 0 cos u 2 3 2 3 2 39 dX 32 dX 12 > cos u 0 sin u = 6 7 6 7 6 7 þ4 0 1 0 5  4 dY 32 5 þ 4 dY 12 5 : > ; dZ 32 dZ 12  sin u 0 cos u 2

dX

3

ð17Þ

With (14)–(17), we find that the height error only depends on the LoS error of the forward and backward cameras, and the nadir view camera has little effect on improving the height accuracy:

" # 1 sec2 ðh  uÞ  ð1  tan2 uÞ 1 dh ¼ þ  b  ðDa2 2 3 2ðtanðh  uÞ þ tan uÞ2 4 sin h  Da1 Þ:

ð18Þ

For a low-orbit satellite, u is a small value, and is approximately 0.91° for ZY-3. If u ¼ 0, then

2

dX

3

2 Da

6 7 6 4 dY 5 ¼ 6 4 dZ

2 1 þDa2 þ4 cos

hDa3 6 sin 2h Db1 þDb2 þcos hDb3  6 sin h Da1 þDa2  b 4 sin2 h

b

3

7 b 7 5:

ð19Þ

Therefore, the height error of TLCs is almost equal to two rays of the forward and backward cameras. However, the planimetry error of TLCs depends on the LoS uncertainty of all three cameras, as depicted in Fig. 4. The LoS uncertainty of the nadir camera has a significant impact on the planimetric error in the base direction. Owing to the nearly circular orbit, the baseline changes slightly in the strip if the attitude orbit control system (AOCS) maintains the same off-nadir angle. In this situation, the height difference of the target has a limited impact on the geolocation errors. According to (7), the height difference causes a change in the baseline. However, compared with the orbit height, even a 5-km height difference is very small and introduces a
1% change in the baseline length b, yielding a 1% change in the geolocation error. Therefore, the terrain has a limited impact on the geolocation accuracy of the stereo images.

3. Error compensation Aerotriangulation could determinate orientation elements for images of multiple strips simultaneously. However, the block adjustment of HRSIs suffers from large discrepancies between adjacent stereo models (Passini and Jacobsen, 2006; Zhang et al., 2015). To connect the scenes in the strip, the whole strip is used as one unit for bundle adjustment (Fraser and Ravanbakhsh, 2011; Zhang et al., 2014). However, the reasons for discrepancies between the strips are still unclear. Compensating EOP errors is a major task for block bundle adjustment of HRSIs. In this context, it is necessary to understand the change of EOP errors with or without GCPs becoming involved into block adjustment. Therefore, we discuss the relative orientation of HRSIs to determine the epipolar constraint between images. As a previous study proved that the yaw angle is not as significant as pitch and yaw angle for HRSIs, the major error of the ZY-3 TLCs is the attitude drift in both pitch and roll angles with different drift rates (Pan et al., 2016a). In this section, we prove that ignoring yaw angle could eliminate the discrepancy between the strips with error propagation.

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H. Pan / ISPRS Journal of Photogrammetry and Remote Sensing 123 (2017) 62–74

3.1. Relative orientation In contrast to the frame cameras, push broom cameras have one projection centre for each scan line. Therefore, the relative orientation of the frame images is different from HRSIs. The relative orientation means that two homologous rays in a pair intersect each other (Wang, 1990). For the intersection of two rays in three dimensions, it is required that both rays are in the same plane, which is an epipolar plane. After the relative orientation, the errors perpendicular to the epipolar plane are equal. Owing to the attitude errors, two rays S1P1 and S2P2 are biased from the true ground point P, as illustrated in Fig. 5. The relative orientation modifies the attitude error in the cross base direction to make both rays intersect in the plane S1S2P0 . In this plane, the geolocation errors are the same as those in Fig. 2. The epipolar direction depends on the stereo coverage. For along-track stereo images, the epipolar direction is close to the flight direction. Therefore, relative orientation causes the attitude errors of the roll angle to be the same for both rays; however, it has little impact on the attitude error of the pitch angle and yaw angle. For an across-track stereo images, the epipolar direction is close to the scan direction, and the compensation of the pitch and yaw angle error are the same as the roll angle of the alongtrack stereos. In the case of ZY-3, the unknowns of two rays are eight independent parameters, which are defined as:

8 Da1 ¼ a1;0 þ a1;1  t > > > < Db ¼ b þ b  t 1 1;0 1;1 : > > Da2 ¼ a2;0 þ a2;1  t > : Db2 ¼ b2;0 þ b2;1  t

ð20Þ

Relative orientation would make two rays of the along-track stereo have the same error in the cross base direction. That is, the relative orientation parameters of along-track stereo are x0 ; x1 of the second image. The coplanarity condition could be used to calculate the elements. After relative orientation, the attitude error changes to

8 > < Da1 ¼ a1;0 þ a1;1  t Da2 ¼ a2;0 þ a2;1  t : > : Db ¼ b0 þ b1  t

If the attitude error of both cameras is time-dependent, the geolocation error is time-dependent. After substituting (21) into (14), we obtain

3 1;0 þa2;0 þða1;1 þa2;1 Þt b 2 sin 2h 7 6 7 6 7 6 b0 þb1 t b 7: 4 dY 5 ¼ 6 2 sin h 5 4 a2;0 a1;0 þða2;1 a1;1 Þt dZ b 2 2

dX

2a

ð22Þ

4sin h

If the attitude error is constant during image capturing, the geolocation error of the whole scene is the same. The height error is invariable and is related to the initial attitude errors a1;0 and a2;0 when both cameras have the same drift rate. 3.2. Bundle adjustment Estimating the object coordinates of tie points and compensation parameters is the major task of image orientation. Owing to the physical meaning, the RSM is preferred. The observation equation of bundle adjustment is:

8 AX þ BY ¼ L1 ; P1 > < X ¼ L2 ; P2 ; > : Y ¼ L3 ; P3

ð23Þ

where X is the correction value of the compensation parameters; Y is the correction value of the object coordinates; A is the design matrix of X; B is the design matrix of the object coordinates; L1 ; L2 ; L3 are the corresponding differences between the measured value and the estimated value; and P1 ; P2 ; P3 is the corresponding weight matrix. In practice, it is relative orientation when only tie points are used to estimate all compensation parameters via free-net adjustment. However, overparametrization makes the normal equation ill-conditioned. Pseudo observations are used to estimate the correction parameters with priori information. The weight of pseudo observation is defined as:

p¼ ð21Þ

3

r20 ; r2

ð24Þ

where r20 is the error of the unit weight, and r are the priori errors of pseudo observations (Cooper and Cross, 1988). A smaller weight means that the pseudo observation has a larger deviation from the real value. The first equation of (23) is the partial differential function of the collinearity equation, whose weight matrix P1 corresponds to the measurement accuracy of the image coordinates. Owing to the outstanding points, the error of the picking points is assumed as one-third of a pixel. The horizontal and cross base root mean square errors (RMSEs) of ZY-3 TLCs are approximately 10 and 5 m, respectively, for block adjustment without GCPs (Tang et al., 2015a). The initial value of X is assigned as 0, as the horizontal and cross base errors are irregular. The error of a0 ; b0 is assigned as 3 pixels, and the drift error of both angles is assumed as 0.2 pixels per second. Thus, the weight matrix P2 is defined. The third equation of Eq. (23) can be divided into two portions: the GCPs and tie points. The weights of the GCPs are very large because of the high measurement accuracy. The weights of the tie points depend on the methods for estimating the initial values of the ground coordinates. If the convergence angle is sufficient, this equation is not necessary for the tie points. 3.3. Error propagation

Fig. 5. Relative orientation of HRSIs cause the LoS uncertainty in the cross base direction to be consistent with corresponding rays intersecting in the base direction.

Using the observation equation, the error equation can obtained as:

H. Pan / ISPRS Journal of Photogrammetry and Remote Sensing 123 (2017) 62–74

8 > < V1 ¼ AX þ BY  L1 ; P1 V2 ¼ X  L2 ; P2 : > : V3 ¼ Y  L3 ; P3

ð25Þ

The ground coordinates can be determined by a weighted leastsquares estimation:

h

i  AT PB  Y ¼  : ð1Þ BT P1 L1 þ P3 L3  BT P1 A  ðAT P1 A þ P2 Þ  AT PL1 þ P2 L2 BT P1 B þ P3  BT P1 A  ðAT P1 A þ P2 Þ

ð1Þ

ð26Þ ð1Þ

If the matrix BT P1 B þ P3  BT P1 A  ðAT P1 A þ P2 Þ  AT PB is invertible, there is a unique solution Y. In the case of GCPs, the Y is approximately equal to L3 , because the weight P3 is assigned a large value. For tie points, the solution depends on many factors, including the priori information P2 , P3 and the convergent angle. The weighted least-squares estimation minimizes the weighted residual:

min

VT1 P1 V1 þ VT2 P2 V2 þ VT3 P3 V3 :

treat these points as GCPs with inaccurate object space coordinates. Therefore, both roll and pitch angle would be adjusted for TLCs to minimize the target function (27). When two strips are involved in the bundle adjustment, the common tie points are distributed in the along-track direction, as TP1, TP2, and TP3 in Fig. 6. If the yaw angle is taken into consideration, the four GCPs in the first strip are sufficient to calculate the compensation models for the first strip. After that, the common tie points should achieve reasonable accuracy. Hence, TP1, TP2, and TP3 could be treated as the GCPs for the second strip. However, other tie points TP4, TP5, and TP6 prefer to stay at their previous inaccurate position because of the priori observation functions. The target function min VT1 P1 V1 þ VT2 P2 V2 would achieve a minimum residual after mistakenly adjusting the yaw angle. Larger overlap would help to obtain a more accurate yaw angle because

ð27Þ

After iterative computation, the solution X fits all constraints. There are two constraints in the bundle adjustment of HRSI: the epipolar constraint and the common tie-points constraint. The epipolar constraint causes the roll-angle errors of the along-track stereo to be the same, as illustrated in Section 3.1. Owing to the very narrow FoV, the tie points between the adjacent strips have small convergence angle, which cause the BT P1 B rank defect. To distinguish the traditional tie points, the common tie points represent the conjugate points, which could be identified from all corresponding images. The common tie-points constraint plays a very important role in the bundle adjustment of the pitch angle for TLCs, even though the pitch-angle error does not change during the relative orientation of two rays. The pitch-angle uncertainty of the redundancy observation should be consistent with others, if we

Fig. 6. Bundle adjustment of multiple strips with the common tie points constraints. The block is composed of two strips; each has a forward and a backward image. There are several common tie points in the along-track direction and GCPs are settled in the first strip only.

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Fig. 7. Distribution of the GCPs and boundaries of strips.

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Fig. 8. Time-related normalized residuals of the direct orientation in the image space for (a) Orbit 305 and (b) Orbit 381.

Fig. 9. Geolocation error of the direct orientation for (a) Orbit 305 and (b) Orbit 381.

the yaw angle error would introduce bias, which is related to the sample coordinates. Unfortunately, large overlap means large redundancy, and it is uneconomic and has a critical requirement for the satellite. The great benefit of omitting yaw angle is that the bundle adjustment is independent of the overlap area between the adjacent strips and can extend to the block area even when the GCPs are only in one strip. The time-related compensation model requires the GCPs to be distributed at the beginning and end of the strips. Because the object space coordinate errors of the common tie points are the same, the uncertainty of the attitude in both the roll and pitch angles should be the same after relative orientation. The epipolar constraint between adjacent strips causes the pitch-angle errors to be the same, which are similar to the error of the roll angle of the along-track stereo. The bundle adjustment of multiple temporal images yields a higher accuracy in the absence of ground control when the attitude uncertainty is subject to Gaussian distribution with mean 0, and

multiple strips are involved. Routine geometric calibration is the key issue. Even though a long-period geolocation error might exist, the attitude uncertainty of multiple temporal strips is approximately random in a very long time period, such as several years. If GCPs are used for bundle adjustment, the random error of picking points, or matching points, must be considered, especially if there are no GCPs in the strip. Therefore, GCPs with several strip intervals are required.

4. Experimental results and analyses 4.1. Datasets Because of the limited available datasets, only two adjacent strips covering Taihang Mountain were used for the experiments. The shorter strip, named Orbit 305, was captured on January 29, 2012, covering seven standard scenes. The other strip, named Orbit

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Fig. 10. Time-related normalized residual of the free-net bundle adjustment applied independently in the image space for (a) Orbit 305 and (b) Orbit 381.

Fig. 11. Geolocation error of the relative orientation for (a) Orbit 305 and (b) Orbit 381.

381, was acquired on February 3, 2012, covering 12 standard scenes. The elevation range of the experiments area was 64– 2705 m, as illustrated in Fig. 7. There were 308 GCPs distributed evenly throughout the dataset. These were measured primarily using a static global positioning system, which could achieve an accuracy of 0.1 m. Most of the GCPs were placed in sections of roads and picked manually. There were 159 and 221 GCPs in Orbit 305 and Orbit 381, respectively, and 72 GCPs were common points, as shown in Fig. 7. Depending on the surveyor, the GCPs of Orbit 305 were road centre-line intersections, while many of the Orbit 381 GCPs were intersections between edges and centre lines, which were more difficult to identify in the images. 4.2. Experiments The major error source of ZY-3 TLCs is the attitude drift in both the roll and pitch angle with different drift rates, which can be compensated with a drift-compensation model using the RSM or

drift model in the image space. The errors in the image space are used to illustrate the attitude errors, as the attitude error of narrow field of view cameras can be absorbed with the drift-compensation parameters in the image space. The picking-points and attitude oscillation (Pan et al., 2016b) have random errors that are ignored in the experiments. The attitude error Dh is calculated with the image error Dp, pixel size l, and principal distance f using the following equation.

Dh ¼

Dp  l f

ð28Þ

The principal distances of ZY-3 TLCs are
1.7 m. The pixel size of the forward and backward cameras are 10 lm, whereas that of the nadir camera is 7 lm. Therefore, the error in the columns Dx is equalized with respect to the roll-angle error, and the error in the rows Dy is equalized with respect to the pitch-angle error. The unit pixel of TLCs has a different attitude error. In the context, the normalized error of a nadir camera in the image space is used. An error of 1 pixel corresponds to an attitude error of 1.2 arcsec.

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Fig. 12. Time-related normalized residual of the free-net bundle adjustment in the image space for (a) Orbit 305 and (b) Orbit 381.

3 2

Difference (m)

1 0 -1

1 3 5 7 9 1113151719212325272931333537394143454749515355575961636567697173

-2 -3 -4 -5 X

Y

Z

Fig. 13. Coordinate difference of the common tie points between Orbit 381 and Orbit 305.

To illustrate the attitude error, all the GCPs are used to calculate the normalized residual in the image space directly. Owing to the attitude drift error, the residual is time-related, as shown in Fig. 8. There is a significant linear tendency for the TLCs of Orbit 305, with a random error of approximately 1 pixel. The TLCs have the similar roll-angle drift rates but slightly different biases. The drift rates and biases of the pitch angle of the TLCs differ. In the case of Orbit 381, there is a significant drift effect for both the roll angle and the pitch angle of the forward camera. However, the attitude uncertainty of the other two cameras is more stable. 4.2.1. Direct orientation Least-squares estimation is used for the intersection when more than two rays are used. In this section, two strips are used to estimate the geolocation error independently. For ZY-3, the baseline is
443 km when the average orbit height is
505 km. To evaluate the planimetric error, all GCPs are transformed to the Universal Transverse Mercator coordinate system. Taking point 100015 of Orbit 305 for example, the planimetric and height error of direct orientation with TLCs are 7.85 m and 6.49 m, respectively, while the normalized residuals of TLCs in sample Dx and line Dy are 0.60 pixels and 2.86 pixels (BWD, backward image), 1.84 pixels and 1.24 pixels (FWD, forward image), and 1.13 pixels and 2.15 pixels (NAD, nadir image). Using (18) and (28), the predicted height error is 6.46 m, which is very close

to the actual height error 6.49 m. The height error difference between two rays and TLCs is less than 1 cm because u is very small. The predicted planimetric error of two rays in epipolar coordinates is 7.22 m (DX) and 3.93 m (DY). And the total predicted planimetric error is 8.22 m, while the planimetric error of two rays is 8.21 m. The predicted planimetric error become 7.67 m when the NAD image is involved. This small difference might be introduced by the imprecise transformation between the pixels to angle in the line direction. The height error of Orbit 305 is within 6 and 2 m when the pitch-angle error difference is between 2 and 0.7 pixels, as shown in Fig. 8(a). This result coincides with the direct-orientation result, as illustrated in Fig. 9(a). The planimetric error in the east-west direction changes with time, which agrees with the roll-angle errors, even though the along-track direction deviated from north slightly. The geolocation in the north-south direction is more stable because the pitch-angle error of the BWD camera had less change. Because the attitude errors of the FWD and BWD cameras have almost equal value but opposite symbols, the planimetric RMSE of Orbit 381 is within 3 m, as shown in Fig. 9(b). However, there is a large pitch-angle difference, which introduces a height error of over 10 m. 4.2.2. Bundle adjustment without GCPs In the geometric processing of HRSIs, the free-net bundle adjustment is applied when no GCPs are involved in the image orientation. As clarified in Section 3, the free-net bundle adjustment of triplet stereo images utilizes the epipolar constraint and common tie-points constraint, which results in a coincident attitude uncertainty. First, the free-net bundle adjustment is applied to Orbit 305 and Orbit 381, independently. The RMSE in the image space is 0.14 pixels in the sample direction and 0.02 pixels in the line direction when only FWD and BWD images of Orbit 305 are involved in the bundle adjustment. The respective RMSEs of Orbit 381 in the image space are 0.18 and 0.01 pixels. The reason for the nearzero RMSE in the line direction is that only the epipolar constraint is adopted in the free-net bundle adjustment of two rays. The common tie points contribute to the bundle adjustment of triplet stereo images. In such a case, the RMSEs of Orbit 305 in the image space are 0.21 pixels in the sample direction and 0.19 pixels

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Fig. 14. Residual errors after bundle adjustment. (a) Pattern A, (b) pattern B. Red arrows indicate plane errors; blue arrows represent height errors (upward is positive and downward is negative). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

in the line direction, and the RMSEs of Orbit 381 in the image space are 0.26 pixels in the sample direction and 0.16 pixels in the line direction. The normalized residuals of the TLCs are illustrated in Fig. 10. The roll-angle error of the TLCs is almost equal for Orbit 305 and Orbit 381, separately. The pitch-angle error of the nadir camera is almost equal to the mean value of the FWD and BWD cameras. The pitch-angle drift rate of the FWD camera in Orbit 381 becomes stable because the BWD and NAD have little drift, as shown in Fig. 8. However, relative orientation does not guarantee a higher accuracy when only one strip is used for bundle adjustment. Owing to

the larger pitch-angle difference for relative orientation than direct orientation, Orbit 381 has a larger height error—approximately 20 m—as illustrated in Fig. 11(b). In addition to the two constraints, the priori information and pseudo observation play a very important role in the free-net bundle adjustment. The attitude uncertainty should be the same when two strips are involved in bundle adjustment simultaneously. The residuals of the free-net bundle adjustment of Orbit 305 and Orbit 381 are illustrated in Fig. 12. Compared with Fig. 10, the attitude uncertainty differs. The attitude uncertainties of Orbit 305 and Orbit 381 have a similar drift rate. The slight difference might be intro-

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Fig. 15. Residual errors after bundle adjustment. (a) Pattern C, (b) pattern D.

duced by the random error of the picking points and controlled by the convergent conditions. After the free-net bundle adjustment of multiple strips, the RMSE of Orbit 305 and Orbit 381 are calculated separately. The planimetric and height RMSE errors of Orbit 305 are 4.44 and 7.83 m, respectively. However, the planimetric and height RMSE errors of Orbit 381 are 5.25 and 10.21 m, respectively. The accuracy difference is introduced by the different orbit lengths. The planimetric and height RMSE errors of the total 133 CKPs are 4.57 and 7.83 m in the first seven scenes, which are almost equal to those of Orbit 305. Furthermore, only common tie points are used to evaluate the difference, as shown in Fig. 13. The discrepancies of

X, Y, and Z are 0.69, 0.72, and 1.30 m, respectively, which are introduced by the picking-points errors. 4.2.3. Bundle adjustment with GCPs Only two GCPs are required for bundle adjustment, as the attitude uncertainty of adjacent strips are close to each other. In this section, four GCPs settings are investigated: (A) two GCPs in Orbit 305 only, (B) two GCPs in Orbit 381 only, (C) two GCPs in both Orbit 305 and Orbit 381, and (D) one GCP in Orbit 305 and one GCP in Orbit 381, as shown in Figs. 14 and 15. The error in the first seven scenes of Orbit 381 is compensated along with Orbit 305 when two GCPs are placed in Orbit 305, as

H. Pan / ISPRS Journal of Photogrammetry and Remote Sensing 123 (2017) 62–74 Table 1 RMSE in the object space (East, North, Planimetry, and Height) for four different GCP settings. Patterns of GCP setting

RMS-E (m)

RMS-N (m)

RMS-P (m)

RMS-H (m)

A B C D

2.12 1.47 2.33 1.47

1.71 1.48 1.54 1.46

2.72 2.09 2.79 2.07

1.54 1.27 1.90 1.28

shown in Fig. 14(a). However, there are growing errors in both the plane and height in the last five scenes of Orbit 381. The total RMSE of the plane is 2.72 m, and the RMSE of the height is 1.54 m. The RMSEs of the plane and height were reduced to 2.09 and 1.27 m, respectively, when two GCPs were settled in the first scene and last scene of Orbit 381. The error of Orbit 305 was compensated at the same time. The RMSE comparison of the four different GCPs settings is illustrated in Table 1. The drift model of the pitch and roll angle can establish a strong geometric constraint when multiple strips are involved in the bundle adjustment. The error is compensated even when the GCPs are placed only in a single strip. However, one GCP in the first scene and one GCP in the last scene of the longest strip are required to improve the drift parameters accuracy. The picking-points error is significant for the HRSI with subpixel orientation accuracy, and multiple GCPs are preferred to reduce the picking points errors. Therefore, the GCPs should be selected carefully. The picking-points error causes the difference between Pattern A and Pattern C. The RMSE is close to Pattern A when two GCPs are placed in the common area of Orbit 305 and Orbit 381. There is a symmetric error in the last five scenes, as shown in Fig. 15(a). The two GCPs compensate the error of the entire block if one is placed in the first scene of Orbit 305 and the other is in the last scene of Orbit 381. The accuracy of Pattern D is almost as the same as that of Pattern B. 5. Conclusions and discussions A generic geolocation-error analysis frame is proposed for stereo images. With this frame, the terrain has a limited impact on the geolocation error. The attitude error introduces a large planimetric error and no height error if the attitude errors of the stereo images are the same. On the contrary, a large height error and no planimetric error are introduced if the attitude errors of both images have the same value but opposite direction. The frame was validated by the geolocation accuracy of Orbit 305 and Orbit 381 with independent free-net bundle adjustment. Relative orientation is the simplest situation in the case of bundle adjustment. Epipolar constraints cause corresponding rays to intersect with each other, which adjusts the error of the stereo to satisfy the aforementioned condition. In the case of along-track stereo, the relative orientation causes the two rays to have the same roll-angle error. The pitch-angle errors are adjusted to be the same when an across-track stereo is involved. Hence, relative orientation can barely improve the height accuracy. The RMSEs in the image space of the along-track stereo have approximately 0.2 pixels in the sample direction, but almost 0 in the line direction, which reflects the epipolar constraint direction. When a triplet stereo is involved in the free-net bundle adjustment, the common tie-points constraint plays a very important role in adjusting the pitch-angle error. In the case of ZY-3, the pitch-angle error of NAD is approximately the mean value of FWD and BWD, as illustrated by Orbit 305 and Orbit 381. Therefore, the free-net bundle adjustment of the images of the TLCs cannot guarantee the improvement of the height accuracy, which depends on the error characteristic of the TLCs.

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The bundle adjustment of multiple strips utilizes both the epipolar and common tie-point constraints to cause the errors of the multiple strips to be the same for the drift-compensation model. Because the drift errors of both the roll and pitch angle are independent of the sample pixel, the slice of common tie points yields the same attitude uncertainty. Remarkable accuracy can be achieved when two GCPs are placed at the beginning and end of the strip, even within the same strip. Owing to the high geolocation accuracy and very narrow field of view, the drift-compensation model of the roll and pitch angles might be a generic model for HRSIs. As demonstrated in this paper, the linear drift-compensation model can be used for the bundle adjustment of multiple strips, and the accuracy—with planimetric and height RMSEs of 2.09 and 1.28 m, respectively—can be achieved even with only two GCPs. Acknowledgments The author would like to thank the anonymous reviewers for their comments and constructive suggestions to improve the paper. The author also would like to thank Wenchao Huang from Wuhan University for verification of the equations. This work was supported by the China Postdoctoral Science Foundationfunded project 2015M572268, the Fundamental Research Funds for Chinese Academy of Surveying & Mapping, and the Key Laboratory of Satellite Mapping Technology and Application, National Administration of Surveying, Mapping and Geoformation. References Cooper, M.A.R., Cross, P.A., 1988. Statistical concepts and their application in photogrammetry and surveying. Photogramm. Rec. 12, 637–663. http://dx.doi. org/10.1111/j.1477-9730.1988.tb00612.x. d’Angelo, P., Reinartz, P., 2012. DSM based orientation of large stereo satellite image blocks. Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci. XXXIX-B1, 209– 214. http://dx.doi.org/10.5194/isprsarchives-XXXIX-B1-209-2012. Di, K.C., Liu, Y.L., Liu, B., Peng, M., Hu, W.M., 2014. A self-calibration bundle adjustment method for photogrammetric processing of Chang’E-2 stereo lunar imagery. Ieee T Geosci Remote 52, 5432–5442. http://dx.doi.org/10.1109/ Tgrs.2013.2288932. Dolloff, J.T., Theiss, H.J., 2012. Temporal correlation of metadata errors for commercial satellite images: representation and effects on stereo extraction accuracy. Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci. XXXIX-B1, 215– 223. http://dx.doi.org/10.5194/isprsarchives-XXXIX-B1-215-2012. Fraser, C.S., Ravanbakhsh, M., 2009. Georeferencing accuracy of GeoEye-1 imagery. Photogramm. Eng. Rem. Sens. 75, 634–638. Fraser, C.S., Ravanbakhsh, M., 2011. Precise georefrencing of long strips of ALOS imagery. Photogramm. Eng. Rem. Sens. 77, 87–93. http://dx.doi.org/10.14358/ PERS.77.1.87. Grodecki, J., Dial, G., 2003. Block adjustment of high-resolution satellite images described by rational polynomials. Photogramm. Eng. Rem. Sens. 69, 59–68. http://dx.doi.org/10.14358/PERS.69.1.59. Heipke, C., 1997. Automation of interior, relative, and absolute orientation. ISPRS J. Photogramm. Rem. Sens. 52, 1–19. http://dx.doi.org/10.1016/S0924-2716(96) 00029-9. Heipke, C., Jacobsen, K., Wegmann, H., 2002. Analysis of the results of the OEEPE test ‘‘Integrated Sensor Orientation‘‘, OEEPE Integrated Sensor Orientation Test Report and Workshop Proceedings. Jeong, J., Kim, T., 2016. Quantitative estimation and validation of the effects of the convergence, bisector elevation, and asymmetry angles on the positioning accuracies of satellite stereo pairs. Photogramm. Eng. Rem. Sens. 82, 625–633. http://dx.doi.org/10.14358/PERS.82.8.625. Khoshelham, K., 2009. Role of tie points in integrated sensor orientation for photogrammetric map compilation. Photogramm. Eng. Rem. Sens. 75, 305–311. http://dx.doi.org/10.14358/PERS.75.3.305. Li, R., Hwangbo, J., Chen, Y., Di, K., 2011. Rigorous photogrammetric processing of hirise stereo imagery for Mars topographic mapping. Ieee Trans. Geosci. Rem. 49, 2558–2572. http://dx.doi.org/10.1109/tgrs.2011.2107522. Li, R.X., Niu, X.T., Liu, C., Wu, B., Deshpande, S., 2009. Impact of imaging geometry on 3d geopositioning accuracy of stereo ikonos imagery. Photogramm. Eng. Rem. Sens. 75, 1119–1125. http://dx.doi.org/10.14358/PERS.75.9.1119. Lutes, J., Grodecki, J., 2004. Error propagation in ikonos mapping blocks. Photogramm. Eng. Rem. Sens. 70, 947–955. http://dx.doi.org/10.14358/ PERS.70.8.947. Mikhail, E.M., Bethel, J.S., McGlone, J.C., 2001. Introduction to Modern Photogrammetry. Wiley.

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