975

Sensors and Materials, Vol. 35, No. 3 (2023) 975–987
MYU Tokyo
S & M 3222

Robust Co-registration of External Laser Altimetry Points
and Stereo Images
Ruyan Zhou, Jinmeng Gao, Zhonghua Hong,* Haiyan Pan,
Yun Zhang,Yanling Han, Jing Wang, and Shuhu Yang
College of Information Technology, Shanghai Ocean University,
999 Huchenghuan Road, Pudong New District, Shanghai, 201306, China(Received December 28, 2022; accepted February 16, 2023)

Keywords: Gaofen-7, laser altimetry registration, combined adjustment, elevation positioning accuracy

Gaofen-7 (GF-7) is China’s first submeter stereo mapping satellite, and the laser altimetry
system equipped on it is used to improve its elevation positioning accuracy. The elevation
positioning accuracy of GF-7 may be limited by the absence of a laser altimetry system.
Therefore, it is necessary to study the combined adjustment of external laser altimetry points(LAPs) and GF-7 stereo images. In this study, we proposed a method consisting of registering
LAPs to GF-7 stereo images followed by combined adjustment. The registration method adopts acoarse-to-fine scheme: the LAPs are reprojected to stereo images to obtain coarse coordinates,
then fine coordinates are acquired by least-squares matching and using the epipolar geometry
constraint. The proposed method can effectively improve the registration accuracy: the root
mean square error (RMSE) of the registration is subpixel, compared with about three pixels
when only random sample consensus (RANSAC) is applied. At the same time, the number of
LAPs is also increased tenfold when using the proposed method. In the subsequent combined
adjustment, the RMSE of the elevation positioning is decreased from about 5.5 to 1.3 m.

1.

Introduction

As China’s first submeter stereo satellite, Gaofen-7 (GF-7) has a wide range of applications
such as natural hazard assessment, mapping, and so forth.(1,2) The elevation positioning accuracy
can be improved using its equipped laser altimetry and footprint camera.(3–5) However, theground resolution of its laser altimetry is relatively low, reaching 2.36 km in along-trackscanning and 12.25 km(6) in cross-track scanning. Moreover, the altimetry echo wave may beinvalid owing to clouds. As a result, the distribution of GF-7’s laser altimetry points (LAPs) is
not uniform, which has an adverse effect on its elevation positioning accuracy. Therefore, it isnecessary to study the use of GF-7 stereo images in combination with external LAPs such as
those obtained from the Ice, Cloud, and Land Elevation Satellites (ICESat-1, ICESat-2). External
LAPs have also been used to improve the elevation positioning accuracy of stereo images. Li etal.(7) and Qv et al.(8) improved the elevation positioning accuracy of Ziyuan-3 (ZY-3) to 3 musing ICESat-1; the elevation positioning accuracy of ZY-3-02 was about 3.0 m after its*Corresponding author: e-mail: zhhong@shou.edu.cn

https://doi.org/10.18494/SAM4288

ISSN 0914-4935 © MYU K.K.
https://myukk.org/

976

Sensors and Materials, Vol. 35, No. 3 (2023)

experimental laser altimetry data was used for adjustment,(9) and that of ZY-3-03 reached 2.58 m
after combined adjustment using its synchronous laser altimetry data.(10)However, ICESat-1 has a low spatial resolution, and some areas are not covered; thus, there
are limitations in improving the elevation positioning accuracy of stereo images. ICESat-2 has ahigher spatial resolution than ICESat-1; thus, the laser points of ICESat-2 are distributed more
uniformly and densely than ICESat-1. However, ICESat-2 has a smaller diameter (17.5 m(11)) than
ICESat-1 (70 m(12)), which means that ICESat-2 is more likely to be affected by the reprojection
error of stereo images than ICESat-1. The image points obtained by directly reprojecting external
LAPs onto stereo images using the rational function model (RFM)(13,14) are not corresponding
points. They cannot be used in the subsequent steps because their coordinates in image space aredifferent, which would lead to a considerable loss of positioning accuracy. Therefore, the main
problem in combined adjustment using external LAPs and stereo images is the registration of
LAPs.
The registration of external LAPs can be divided into two classes: registration based on
topographic products and registration based on image matching. Registration based on
topographic products such as a digital surface model (DSM) or a digital orthophoto map (DOM)
is mainly performed to calculate the local minimum error between LAPs and the DSM(15,16) orDOM.(17) The inconsistency between LAPs and stereo images still exists if the DSM is not
generated from stereo images to be used in the subsequent combined adjustment, which will
introduce more sources of errors. It will also result in failure or misregistration if there is a verylarge difference in ground sample distance between the stereo images and the DSM. Image
matching methods such as phase correlation(18–20) and least-squares matching (LSM)(21–23) are
widely applied in the registration of images. However, image matching methods are easily
affected by the quality of stereo images. For example, phase correlation is insensitive to the noise
and illumination of stereo images, but it does not perform well in plain areas or areas withmultiple similar peaks. Similarly, differences in the angle or level of radiation during imaging
will lead to the failure of LSM. Moreover, the size of the search window in LSM has a significant
impact on the registration result. Applying the RFM can reduce the search window size when
LSM is conducted. However, registration errors inevitably remain because of the difference inthe level of radiation between two stereo images with different imaging angles. Algorithms of
elimination such as random sample consensus (RANSAC)(24) can be used to divide extracted
feature points into inlier and outlier points to filter points containing registration errors.
RANSAC is a robust algorithm that can obtain points with high accuracy. However, RANSAC
often filters a large proportion of the points, which may result in an insufficient number of points
for the following steps. Therefore, it is necessary to further study the registration methods of
LAPs for stereo images.
To solve the above problems, in this paper, we propose a method of combining the adjustmentof external LAPs with that of stereo images to improve the elevation positioning accuracy. First,
the LAPs are registered to the stereo images. Then, based on the RFM of the stereo images,
different types of points including tie points, LAPs, and virtual control points are used toconstruct error equations. Finally, the RFM is compensated, and its elevation positioning
accuracy is improved.

Sensors and Materials, Vol. 35, No. 3 (2023)

977

The datasets of LAPs and stereo images and methods for the coarse-to-fine registration of
external LAPs and combined adjustment are described in detail in Sect. 2, the results ofregistration and combined adjustment and several factors contributing to their performance arepresented in Sect. 3, and conclusions on registration and combined adjustment are given in
Sect. 4.

2.

Materials and Methods

2.1

Materials

The data used in this study comprise GF-7 stereo images acquired in Luding, Sichuan, China,
in November 2020, including four pairs of stereo images, i.e., eight stereo images. The latitude
and longitude ranges are 29.348–30.22° and 101.96–102.394°, respectively. The resolutions of theforward (FWD) and backward (BWD) stereo images are 31268 × 31000 and 35864 × 40000
pixels, and their ground sample distances are about 0.8 and 0.65 m/pixel, respectively.
The LAPs used in this study were 668 points obtained from ICESat-2 in October 2018. To
achieve uniformly distributed LAPs, half of them were used as elevation control points (ECPs),
which were filtered by fixed longitude and latitude intervals, and the remaining LAPs were usedas check points (CKPs).
The distribution of the ECPs and CKPs of GF-7 is shown in Fig. 1, and the date, number,spatial resolution, and coverage area of the datasets are presented in Table 1.

Fig. 1. (Color online) Distribution of GF-7 stereo images and external LAPs. ECPs: elevation control points;
CKPs: check points.

978Sensors and Materials, Vol. 35, No. 3 (2023)

Table 1
Data used in this paper.
DatasetDate

NumberSpatial resolution (m)May, June 2020

8

0.65–0.8

ICESat-2 ECPs

Oct. 2018

300

–100

ICESat-2 CKPs

Oct. 2018

368

–100

GF-7 stereo images

2.2

Coverage area
29.348–30.22N
101.96–102.394E
29.348–30.22N
101.96–102.394E
29.348–30.22N
101.96–102.394E

Methods

The improvement of the elevation positioning accuracy of stereo images is divided into two
steps. As shown in Fig. 2, the LAPs are first registered to stereo images. The coarse-to-fineregistration consists of three steps. First, LAPs are reprojected to two stereo images, with one
treated as having fixed coordinates and the other treated as having coarse coordinates. Second,
LSM is applied between the fixed and coarse coordinates. A homography matrix is built using
the inlier points, and the best matrix is chosen as the epipolar geometry constraint, which isapplied to constrain the outlier points. Third, combined adjustment is performed on the stereo
images. The combined adjustment consists of three parts: the construction of compensation
models, the construction of partial derivative terms based on the RFM, and the solution of error
equations. The compensation model is constructed in a general affine manner, which has beenfound(25) to be effective for compensating the RFM. After the compensation model has been
appended to the RFM, a Taylor expansion is applied in the compensation of the RFM. Partial
derivatives concerning affine coefficients and coordinates of tie points in object space areconstructed. Finally, error equations are constructed then solved, after which refined RFMs are
generated.
2.2.1 Registration of LAPs
LAPs have high elevation positioning accuracy, but they cannot be directly used in stereo
images because of the inconsistency between LAPs and stereo images. However, accurate
coordinates are necessary, not only in object space but also in image space. To solve thisproblem, we propose a method of registering LAPs to stereo images. As shown in Fig. 3, thecoarse-to-fine registration method is described as follows:1)	
Reproject LAPs to stereo images to obtain coarse image coordinates (x0, y0), (x1, y1).
Although both coordinates are inconsistent with the object of LAPs, it is feasible andreasonable to deem (x0, y0) as true and (x1, y1) as coordinates that contain errors.
2)	Calculate the overlapping region of stereo images using the RFM and divide them into blocks
to make it possible to apply LSM in each block.
3)	Conduct LSM. Taking (x0, y0) as the center and 100 × 100 pixels as a block, slide the windowthat takes (x1, y1) as the center with 500 × 500 pixels as its area. The point (x1, y1) with the
maximum correlation coefficient is taken as relatively fine image coordinates of LAPs. The
formula used to calculate the correlation coefficient(26) isSensors and Materials, Vol. 35, No. 3 (2023)

979

Fig. 2. (Color online) Framework of external LAP registration and improvement of elevation positioning accuracyof stereo images.

Fig. 3.

(Color online) Registration of external LAPs.

h2

ρ( x , y ) =
2

2

∑ ∑

w
2

(g

h
w
( x0 + i , y0 + j )
−
−
i=
j=
2
2

h
2

∑ ∑

w
2

(g

h
w
( x0 + i , y0 + j )
−
−
i=
j=
2
2

−g

−g

) ( g'(

h
2

w
2

)∑
2

∑

x +i , y + j )

− g'

( g'

)

h
w
( x +i , y + j )
−
−
i=
j=
2
2

− g'

)

2

,

(1)

	where w and h are the width and height of the search window, g and g' are the gray values of
the windows on the BWD and FWD stereo images, and g and g ′ are the average grayvalues of the windows on the BWD and FWD stereo images, respectively.

980

Sensors and Materials, Vol. 35, No. 3 (2023)4)	However, image coordinates of LAPs inevitably contain errors after LSM because of thedifferences in the level of shadow or radiation in different stereo images. To solve thisproblem, we propose a filter using the RFM of stereo images. The filter consists of two parts:
residual calculation and the construction of an epipolar geometry constraint. We calculate theresidual from forward intersection and we reproject and reject residuals that are larger than
three pixels. Then, we construct an epipolar geometry constraint to fit outlier LAPs. The
forward intersection formula using the RFM is described as

 1 Nums ( Pn , Ln , H n ) cn =
Dens ( Pn , Ln , H n ) r1 = NumL ( Pn , Ln , H n ) nDenL ( Pn , Ln , H n ),' 2 Num s ( Pn , Ln , H n ) cn = Den' ( P , L , H )
s
n
n
n

'

Num L ( Pn , Ln , H n ) rn2 =
Den' L ( Pn , Ln , H n )(2)	where (Pn, Ln, Hn) and (cn, rn) are the normalized object and image coordinates of LAPs,
respectively.
	Then, we calculate the image coordinates (c1′, r1′) and (c2′, r2′) using (Pn, Ln, Hn) and theRFM. The residuals are obtained as
c1' − c1
 r=
c
,

'
r
=
r
−
r

 r 1 1(3)	where rc and rr are the residuals of the columns and rows, respectively. The values larger than
three pixels are rejected.
5)	To further decrease the number of mismatched points, we apply the epipolar geometry
constraint to the image coordinates of LAPs. The epipolar geometry constraint, such as thehomography matrix, is described as
 x1   h11
 y  = h
 1   21
 1   h31

h12
h22
h32

h13   x2 
h23   y2  ,h33   1 

(4)	where h11–h33 are the coefficients of the homography matrix and can be calculated from theimage coordinates of the LAPs. This method is used to update the homography matrix until itfits the largest number of image coordinates.

Sensors and Materials, Vol. 35, No. 3 (2023)

9816)	Finally, the LAPs that contain errors are corrected by the epipolar geometry constraint. The
fine and corrected LAPs are applied in the following adjustment.
2.2.2 Combined adjustment
The RFM is a mathematical fitting form of a rigorous imaging model(13,14,27) given by Eq. (2),where rn, cn, Pn, Ln, and Hn represent the normalized stereo image row, column, longitude,
latitude, and height calculated from the offset and scale, respectively, which can be easilyacquired from a rational polynomial coefficient (RPC) file. NumL, DenL, Nums, and Dens
represent the fitting form constructed from 20 coefficients of rows or columns, which can befound in the RPC file.
To reduce the errors generated in the push and sweep process of the satellite, different
compensation models could be applied in the RFM. However, some researchers havedemonstrated the effectiveness of the affine compensation model.(25) The affine compensation
model is expressed as∆r = b0 + b1c + b2 r,
∆c = a0 + a1c + a2 r

(5)

where Δr and Δc represent the compensation terms of the rows and columns of stereo images,
respectively. Terms with the subscript 0 represent the shifting errors, while terms with thesubscripts 1 and 2 represent the rotation errors for columns and rows, respectively. Equation (5)was used to compensate Eq. (2) to obtainNums ( Pn , Ln , H n )
=
+ ∆c
 cn
Dens ( Pn , Ln , H n ).

NumL ( Pn , Ln , H n )

r
=
+ ∆r
 n Den ( P , L , H )
L
n
n
n


(6)

To construct error equations, the Taylor expansion was applied to Eq. (6). Affine coefficientsand the object of tie points were taken as the partial derivatives with respect to columns androws. Terms with respect to longitude, latitude, and height comprise matrix A, and terms withrespect to coordinates of points in object space comprise matrix B. Then, the Taylor expansion
was appended to Eq. (6):V = AX + BY − L,(7)

where V represents the errors, X represents the compensation item of the affine coefficient, Y
represents the compensation item of the object of points, A represents the partial derivatives of
the compensation coefficient, and B represents the partial derivatives of the object of points. A,
B, X, and Y are described as982

Sensors and Materials, Vol. 35, No. 3 (2023)

0 0 0


A =  ∂c∂c∂c
 ∂a
 0 ∂a1 ∂a2∂r ∂b2  ,


0 0 0


(8)

∂r
∂L
∂c
∂L

∂r 
∂H 
,
∂c 

∂H 

(9)

 ∆b0 
 ∆b1  ∆b2 X ==Y
 ∆ a0  ∆a 
 1 
 ∆ a2  ∆P 
 ∆L  .
 ∆Η 



(10)

 ∂r
 ∂P
B=
 ∂c

 ∂P

∂r
∂b0

∂r
∂b1

Note that the longitude and latitude of LAPs must be compensated when the height isconstrained as a fixed value.(27)
The following error equations are used in this paper:
 Vtp = Atp X aff + Btp X tp − Ltp , Ptp
V = A X + B X − L , P
vcp
aff
vcp
vcp
vcp
vcp ,
 vcp
 V = A X +B X −L ,P
las
aff
las
las
las
las
 las

(11)

The terms with subscripts tp, vcp, and las denote the tie points, virtual control points, and
LAPs, respectively. The compensation item must be updated when the maximum value of X and
Y is greater than 10−8.

3.

Results and Discussion

We applied the proposed method to register external LAPs, which were used to improve theelevation positioning accuracy and decrease the inconsistency among stereo images.
Experiments were divided into four image combinations, (1)–(4), as shown in Fig. 1.
To register the LAPs, the experiment was designed as follows: we calculated the matrix
between the two sets of image coordinates, and we also calculated the residual by subtracting thecoordinates calculated using the matrix from the coordinates in set 2.
The results of applying LSM only, RANSAC, and the proposed method are presented inFig. 4. There are obvious errors in Fig. 4(a) due to the differences in the angle and level ofradiation between the BWD and FWD stereo images, and the differences are too large to applySensors and Materials, Vol. 35, No. 3 (2023)

983

(a)

(b)

(c)
Fig. 4.

(Color online) Results of LAP registration: (a) applied LSM only, (b) RANSAC, and (c) proposed method.

LAPs in the subsequent combined adjustment. The number of LAPs significantly decreased
when RANSAC was used, as shown in Fig. 4(b). After performing the coarse-to-fine registration
proposed in this paper, the LAPs with better matching were saved, as shown in Fig. 4(c).
As can be seen from Table 2, when only LSM was applied to the four stereo images, all ofthem showed unsatisfactory experiment results: although all points were reserved, the root meansquare error (RMSE) of two of the images reached a value above 840 pixels, one image had an
RMSE of 146 pixels, and one image had an RMSE of 416 pixels, which were unacceptable errors
for the subsequent steps. After RANSAC, the RMSE of the image coordinates of the LAPs
significantly decreased to about four pixels. However, the number of LAPs also increased toabout 20, which means that the retention rate was only 1%. A fatal flaw was that too few LAPs
(less than 10) were reserved for the subsequent steps. In contrast, after filtering using the filter
constructed using the RFM and epipolar geometry constraint proposed in this study, the RMSEsof the image coordinates of the LAPs decreased to less than one pixel, which means that the
LAPs were sufficiently accurate to be applied in combined adjustment. At the same time, theretention rate reached 15%, ten times that obtained using RANSAC.
The LSM exhibited poor performance for two reasons. First, the stereo images used in thisstudy mainly contained mountainous areas, making them easily affected by the inconsistency inthe shadows resulting from the imaging angle when applying LSM. Second, the texture in
mountainous areas showed weak and similar features, which is fatal for the gray-value-basedmatching method.

984

Table 2
Results of different LAP registration methods.
Image
LSM only
Numbercombination(pixels)
RMSE
REST
(1)
2800
416.166
2800
(2)
1266
951.804
1266
(3)
1216
146.8
1216
(4)
1931
849.247
1931

Sensors and Materials, Vol. 35, No. 3 (2023)

RANSAC(pixels)
RMSE
REST
4.0522120
1.808815
4.2716118
3.3073424

Proposed method (pixels)
RMSE
0.649687
0.857569
0.225993
0.62018

REST
164
189
129
280

Table 3 shows the number of ECPs used in the experiment, the elevation accuracy, and theresidual of the stereo images, including the RMSE, mid, and max values for different adjustment
schemes. The table shows significant improvements in the elevation accuracy and residual ofstereo images after the combined adjustment of LAPs and stereo images.
Generally, the RMSE elevation accuracy of stereo images improved from 5.5 to 1.3 m. The
RMSE elevation accuracies of image combinations (1)–(4) increased from 5.45 to 1.37 m, from5.52 to 1.58 m, from 5.04 to 1.26 m, and from 5.86 to 1.54 m, respectively.
Note that the elevation positioning accuracy of GF-7 is about 1 m after the combined
adjustment, but the experimental results showed lower elevation positioning accuracy. This isbecause of the terrain in the stereo images: 90% of their area consisted of mountains, meaningthat the LAPs were easily affected by the inconsistency between the LAPs and the stereo images.
Therefore, small errors in the image coordinates (larger than one pixel) will cause very large
losses in elevation positioning accuracy.
Before the combined adjustment, all image combinations showed similar elevation
positioning accuracies, meaning that the four stereo image combinations showed an almost thesame quality. After free network adjustment, the elevation positioning accuracy showed noapparent improvement, as expected. However, the different stereo image combinations showed
different elevation accuracies, with stereo image combination (3) having the highest elevation
accuracy and stereo image combination (2) having the lowest accuracy, with the other stereo
image combinations showing similar elevation accuracies. Several factors contributed to thisresult, such as the LAPs used in the adjustment, the terrain, and the elevation positioning
accuracy of the stereo images.
Although we have used the same number of ECPs in the experiment, other variables such as
the co-registration accuracy, quality, and distribution of ECPs are difficult to control to bestrictly consistent. The co-registration accuracy of LAPs is easily affected by the RFM of stereo
images and the gray values of LAPs located in them. As a result, there is still some difference inthe co-registration of LAPs, which will result in the difference in the elevation positioning
accuracy of stereo images. Moreover, the distribution and quality of LAPs, which affect theelevation positioning accuracy of stereo images, cannot be changed. When the ECPs are
distributed more uniformly, the improvement in the elevation positioning accuracy of stereo
images generally shows more consistency. Finally, the quality of LAPs slightly differs even if
they are filtered, which will further contribute to the nonconsistency in the experiments on thefour image combinations.

Sensors and Materials, Vol. 35, No. 3 (2023)

985

Table 3Results of different adjustment schemes.
Imagecombination

(1)

(2)

(3)

(4)

—

MID
3.89

Height (m)
RMSE
5.45

MAX
9.79

MID
1.75

—

3.88

5.44

9.78

0.48

0.62

0.82

75

1.30

1.37

4.00

0.48

0.63

0.82

—

4.69

5.52

9.75

3.56

3.60

5.23

—

4.83

5.53

9.4

1.28

1.29

4.07

75

1.36

1.58

1.76

1.00

1.25

2.04

—

3.73

5.04

9.81

4.38

4.59

5.72

—

8.10

4.60

9.40

1.01

1.18

1.76

75

1.04

1.26

2.86

0.98

0.96

1.76

—

8.14

5.86

10.39

3.73

3.46

4.52

—

9.33

5.28

10.84

0.29

0.39

0.90

75

2.28

1.54

4.54

0.23

0.39

0.90

Adjustment scheme

No of ECPs

Before adjustment
Free network
adjustment
Adjustment with
ICESat-2
Before adjustment
Free network
adjustment
Adjustment with
ICESat-2
Before adjustment
Free network
adjustment
Adjustment with
ICESat-2
Before adjustment
Free network
adjustment
Adjustment with
ICESat-2

Residual (pixel)
RMSE
MAX
3.00
5.84

As mentioned above, the terrain and elevation positioning accuracy of stereo images areimportant factors in the results. The more mountains or hills located in the stereo images, thegreater the uncertainty in co-registration accuracy. The elevation positioning accuracy of stereo
images under adjustment with or without LAPs exhibits the following relationship: image
combinations with higher accuracies under free network adjustment are more likely to show
higher accuracies after adjustment with LAPs. For example, among the four image combinations,image combination (4) showed the highest elevation positioning accuracy under the free network
adjustment and also under the adjustment with ICESat-2.
There are also significant improvements in the residuals of the stereo images after combined
adjustment: the RMSE residuals of image combinations (1)–(4) decreased from 3.00 to 0.63
pixels, from 3.60 to 1.25 pixels, from 4.59 to 0.96 pixels, and from 3.46 to 0.39 pixels,
respectively. The decrease in stereo image residual means that the consistency between theimage and object spaces of the stereo images is improved. The quality of the stereo image
residual after adjustment depends on the tie points used in the experiment: the more uniformly
the tie points are distributed, the greater the decrease in the residual obtained.
Several improvements remain to be made, such as those in the registration of LAPs, theretention rate of LAPs, and the elevation positioning accuracy. For the LSM, the template andsearch windows affect the registration, and the size of the windows should be adaptive to
specific applications. The retention is low because the points acquired by the LSM inevitably
contain errors owing to the quality or features of stereo images. The elevation positioning
accuracy is affected by many factors, such as the quality of tie points, LAPs, and stereo images,
which have a strict standard for the data accessed.

986

4.

Sensors and Materials, Vol. 35, No. 3 (2023)

Conclusions

In this study, a new method of registering external LAPs to stereo images is proposed andused in the subsequent combined adjustment. In the method, ICESat-2 LAPs are registered to
GF-7 stereo images. We performed experiments to demonstrate the effectiveness of the proposed
method and obtained the following results:(1)	The proposed method significantly improved the registration accuracy, with the RMSE being
reduced from about three pixels for RANSAC to less than one pixel. In addition, the proposed
method reserved ten times the number of points as that obtained by the former method.
(2)	The LAPs were then used to improve the elevation positioning accuracy of GF-7 stereo
images. In the four stereo image combinations in this study, the RMSE of the elevation
positioning error decreased from about 5.5 to 1.3 m, and that of the stereo image residual
decreased from about four pixels to less than one pixel.

Acknowledgments
This work was supported in part by the National Natural Science Foundation of China underGrant 41871325.

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