Satellite Jitter Estimation and Validation Using Parallax Images

sensors Article

Satellite Jitter Estimation and Validation Using Parallax Images Jun Pan 1,2 , Chengbang Che 1 , Ying Zhu 1 and Mi Wang 1,2, * 1

2

*

The State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, 129 Luoyu Road, Wuhan 430079, China; [email protected] (J.P.); [email protected] (C.C.); [email protected] (Y.Z.) Collaborative Innovation Center for Geospatial Technology, 129 Luoyu Road, Wuhan 430079, China Correspondence: [email protected]; Tel./Fax: +86-27-6877-8969

Academic Editor: Hans Tømmervik Received: 7 October 2016; Accepted: 30 December 2016; Published: 2 January 2017

Abstract: Satellite jitter (SJ) is an important error source that affects the geometric accuracy of high resolution satellite imagery. In this paper, the quantitative relationship between the jitter displacement (image displacement caused by SJ) and relative registration error obtained from parallax images is deduced to be theoretical in detail, and the jitter displacement estimation model is built to estimate the jitter displacement. Then, a simulation experiment is carried out to validate the feasibility of using the built jitter displacement estimation model to estimate the jitter displacement. Finally, experiments with real images in DengFeng (China) including multispectral images of Ziyuan-3 (ZY-3) satellite and high resolution (HR) images of Ziyuan1-02C (ZY1-02C) satellite are used to validate the effectiveness of utilizing the built jitter displacement estimation model to do the jitter estimation. High accuracy ground reference data are further used to evaluate the accuracy of the estimation. Experimental results show that the average estimated error for jitter displacement of ZY-3 is 2.96% and 0.11% in amplitude and frequency respectively, and the estimated error for jitter displacement of ZY1-02C is 8.46% and 0.35% in amplitude and frequency, respectively. Keywords: satellite jitter; parallax images; jitter displacement; jitter estimation; jitter validation

1. Introduction With increased spatial resolution of satellite images, the influence of satellite jitter (SJ) on the imaging quality becomes more and more obvious, and seriously limits the application of satellite image products [1–5]. For instance, Ayoub et al. found that the unmodelled jitter of QucikBird produced an approximately 5 pixels (2.5 m) geometric distortion mainly around 1 Hz and an approximately 0.2 pixels (0.1 m) geometric distortion mainly around 4.3 Hz [6]. Takaku and Tadono encountered two kinds of systematic waving noise in some Digital Surface Model generated from ALOS PRISM processing and examined their relations to attitude fluctuations [7]. Moreover, SJ has a great influence on internal and external calibration of cameras during geometric preprocessing [8,9]. Therefore, the estimation of SJ is more essential and significant because it is the foundation of the compensation for image distortions caused by SJ. Much research on utilizing parallax images to estimate SJ has been performed under circumstances lacking high accuracy satellite attitude measuring instruments (star-tracker and gyro). Roques et al. proposed a method to identify micro-vibrations of satellite platforms in pitch, row and yaw—three directions—using stereo images [10]. Teshima and Iwasaki estimated attitude fluctuation between two different bands of ASTER short-wave infrared images and found a SJ with a frequency of about 1.5 Hz [11]. Mattson et al. reconstructed a jitter series for High Resolution Imaging Science Experiment (HiRISE) onboard the Mars Reconnaissance Orbiter (MRO) using staggered Charge-Coupled Devices Sensors 2017, 17, 83; doi:10.3390/s17010083

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(CCDs) [12]. During the in-flight commissioning period of PLEIDES-HR satellites, Amberg et al. estimated the attitude perturbances caused by SJ by exploiting multispectral CCD arrays [13]. Jiang et al. considered the impact of SJ on Ziyuan1-02C (ZY1-02C) imaging and used staggered CCDs to restore the high-frequency attitude based on a rigorous geometric model [14]. Sun et al. used eight staggered CCD arrays to detect the satellite jitter of Chinese mapping satellite-1 [15]. Tong et al. and Zhu et al. also presented several jitter detection methods based on multispectral images, stereo images and panchromatic images, and validated them by ZY-3 images [5,16–19]. In this work, a jitter displacement (image displacement caused by SJ) estimation model was built according to the deduced quantitative relationship between the jitter displacement and relative registration error obtained from parallax images. Both simulation experiments and real image experiments were carried out to validate the feasibility and effectiveness of utilizing the built jitter displacement estimation model to do the jitter estimation. 2. Methodology 2.1. Jitter Displacement Estimation Modelling The principle of jitter displacement estimation is based on the design of satellite CCD sensors with parallax observation systems. In the parallax observation system, each linear sensor scans the same ground feature at different times and an object will be imaged at the same location in different images but at different times. However, some exterior orientation elements of the camera used in the observation system might have some small changes due to SJ during the observing interval time, and thus the same object might be imaged at slightly different locations in different images [17,20]. Thus, SJ can be reflected by relative registration error obtained from parallax images and the jitter displacement might be estimated by the relative registration error. According to theory of Fourier transform, SJ can be converted into the attitude jitter component combined by several sinusoidal functions with different amplitudes, frequencies and phases [21], as shown in the following: ∞

ϕ(t) =

∑ Ai sin(2π fi t + ϕi )

(1)

i =1

where ϕ(t) is the jitter component of satellite attitude at imaging time t, and Ai , f i , ϕi are the amplitude, frequency and phase of the ith component, respectively. In order to simplify the derivation, it is assumed that the satellite platform contains only a single frequency jitter as follows: ϕ(t) = A sin(2π f t + ϕ)

(2)

Considering the angle of attitude fluctuation caused by SJ is small, the jitter displacement, d(t), can be calculated based on photogrammetric theory according to Equation (3): d(t)

= f c · ϕ(t)/p = Ad · sin(2π f t + ϕd )

(3)

where d(t) is jitter displacement, f c is the focal length of the camera, p is the pixel size in focal plane; Ad , f , ϕd are the amplitude, frequency and phase of jitter displacement and Ad = f c · A/p, ϕd = ϕ. According to the above-mentioned principle of jitter estimation using parallax images, relative registration error between parallax images with some imaging time interval can be denoted as r (t) = d(t + ∆t) − d(t)

(4)

where r (t) is the relative registration error. By substituting Equation (3) into Equation (4), there is r (t) = Ad · (sin(2π f (t + ∆t) + ϕd ) − sin(2π f t + ϕd ))

(5)

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SJ is time varying, so the relative registration error caused by SJ is also varying with the imaging time interval of parallax images. When ∆t = nT (n ∈ N, T is the minimum positive period of SJ and T = 1/ f ), the jitter displacement of two parallax images are same, which means that the relative registration error will be zero and the SJ cannot be estimated by it. When ∆t 6= nT, it is enough to just analyze r (t) in the minimum positive period, i.e., 0 < f ∆t = ∆t/T < 1. Thus, Equation (5) can be expanded as r (t)

= Ad · (sin(2π f t + ϕd + 2π f ∆t) − sin(2π f t + ϕd )) = Ad · (sin(2π f t + ϕd ) cos(2π f ∆t) + cos(2π f t + ϕd ) sin(2π f ∆t) − sin(2π f t + ϕd ))

(6)

Let sin(2π f ∆t) = a, cos(2π f ∆t) = b, so a2 + b2 = 1. According to the theory of trigonometric function, Equation (6) can be simplified as r (t)

= Ad · q ((b − 1) sin(2π f t + ϕd ) + a cos(2π f t + ϕd ))

= Ad ·

= Ad · where sin θ = √

√

( b − 1)2 + a2 ( √

= √

r (t) = Ad · p

sin(2π f t + ϕd ) + √

a ( b −1)2 + a2

cos(2π f t + ϕd ))

(7)

2 − 2b sin(2π f t + ϕd + θ )

a , cos θ ( b −1)2 + a2

Let Ar = Ad ·

b −1 ( b −1)2 + a2

b −1 . ( b −1)2 + a2

q

Substituting cos(2π f ∆t) = b into Equation (7), there is

2 − 2 cos(2π f ∆t) sin(2π f t + ϕd + θ )

(8)

2 − 2 cos(2π f ∆t), ϕr = ϕd + θ, there is r (t) = Ar · sin(2π f t + ϕr )

(9)

where Ar , ϕr is the amplitude and phase of relative registration error. According to Equations (8) and (9), the quantitative relationship between Ad and Ar is given as A d = Ar /

q

2 − 2 cos(2π f ∆t) = Ar /(2 sin(π f ∆t))

(10)

As to θ, there is sin θ

=

sin(2π f ∆t) q

(cos(2π f ∆t)−1)2 +sin2 (2π f ∆t)

=

2 sin(π f ∆t)·cos(π f ∆t)

√

4 sin2 (π f ∆t)

= cos(π f ∆t) cos θ

=

cos(2π f ∆t)−1 q

(cos(2π f ∆t)−1)2 +sin2 (2π f ∆t)

√2 sin 2(π f ∆t)) = −

(11)

2

4 sin (π f ∆t)

(12)

= − sin(π f ∆t) Let π f ∆t = α, then according to Equations (11) and (12), sin θ = cos(α), cos θ = − sin(α). Considering 0 < f ∆t = ∆t/T < 1, there is θ = π/2 + α according to the theory of trigonometric function. Then the phase of jitter displacement can be given as ϕd = ϕr − θ = ϕr − π/2 − π f ∆t

(13)

Therefore, the quantitative relationship between the jitter displacement and relative registration error obtained from parallax images is determined and the jitter displacement estimation model is also established. After the relative registration error is obtained, components of jitter displacement including amplitude Ad and phase ϕ can be estimated by Equations (10) and (13) respectively, and the

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frequency of jitter displacement is equal to the frequency of relative registration error. By substituting Equations (10) and (13) into Equation (3), the jitter displacement can be represented as d(t) =

Ar · sin(2π f t + ϕr − π/2 − π f ∆t) 2 sin(π f ∆t)

(14)

where Ar , f , ϕr and ∆t is the amplitude, frequency, phase and imaging time interval of the relative registration error between parallax images. If SJ contains multiple frequencies, each component of the jitter displacement can be estimated from the corresponding component of the relative registration error [19]. 2.2. Workflow of Jitter Displacement Estimation and Validation The jitter displacement estimation and validation are carried out based on dense matching between parallax images. Both parallax images and corresponding high accuracy ground reference data (Digital Orthophoto Map (DOM) or Digital Elevation Model (DEM)) are used to do the jitter displacement estimation and validation. The workflow is shown in Figure 1. (1)

Jitter displacement estimation Jitter displacement estimation is carried out based on the relative registration error between parallax images, where the relative registration error is computed by dense points matching. In matching, the correlation matching is used to get the initial values of corresponding points and the least squared matching is further used to obtain more precise positions of corresponding points. Then certain thresholds of coordinate differences for corresponding points are selected to filter out the mismatching points. In this paper, three times of root mean square error are chosen. After filtering mismatching points, coordinate differences of remaining corresponding points are fitted using sinusoidal function by the least squared method as the relative registration error curve. Finally, the estimated values of jitter displacement are estimated using the parameters of the fitted relative registration error curve including amplitude, frequency and phase, according to Equation (14).

(2)

Jitter displacement validation Jitter displacement validation is carried out based on the registration error between parallax image (panchromatic image or some band of multispectral image) and reference DOM data, i.e., absolute registration error. Because of the high accuracy of the reference DOM data, it can be considered as a true reflection of the ground scene, i.e., ground truth. The registration error between parallax image and reference DOM data is thus the absolute registration error. It is also an exact measurement of the jitter displacement since there should be no registration error between parallax image and reference DOM data if there is no satellite jitter. Therefore, the absolute registration error can reflect the jitter displacement in some imaging time interval of the satellite and is used as a validation of jitter displacement in this paper. The absolute registration error between parallax image and reference DOM data is also computed by dense points matching. Parallax image should be rectified first using rational function coefficients (RPCs) and high accuracy DEM data. Then a dense points matching is conducted between the rectified parallax image and reference DOM data. Similarly, mismatching points in obtained corresponding points are also filtered out. Coordinate differences of remaining corresponding points are also fitted using sinusoidal function by the least squared method. Parameters of the fitted curve including frequency, amplitude and phase are considered as the reference values of jitter displacement and are used to evaluate the accuracy of the estimated values of jitter displacement purely based on parallax images.

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Jitter displacement estimation Parallax images

Dense points matching

Relative registration error

Jitter displacement Jitter displacement (estimated values) estimation model Validation

DOM/ DEM

Dense points matching

Absolute registration error

Jitter displacement (reference values)

Jitter displacement validation

Figure Flow chart jitter estimation and validation. Figure 1. 1. Flow chart of of jitter estimation and validation.

Experiments and Discussion 3. 3. Experiments and Discussion Bothsimulation simulationexperiments experimentsand andreal realimage imageexperiments experimentswere werecarried carriedout outto tovalidate validatethethe Both feasibility and effectiveness of utilizing the built jitter displacement estimation model to jitter feasibility and effectiveness of utilizing the built jitter displacement estimation model to dodo thethe jitter estimation. Simulation experiments were carried out to analyze the change rule of jitter estimation. Simulation experiments were carried out to analyze the change rule of jitter displacement at different imagingand timetointervals and to verify the built jittermodel. displacement model. at displacement different imaging time intervals verify the built jitter displacement Real image Real image experiments were conducted according to the workflow in Figure 1 to further evaluate experiments were conducted according to the workflow in Figure 1 to further evaluate the performance of the estimation jitter displacement based on the built model. of the the performance jitter displacement based onestimation the built model. 3.1. Simulation Experiments 3.1. Simulation Experiments Simulation experiments verify the whether the between relationship between jitter Simulation experiments were were used toused verifytowhether relationship jitter displacement displacement curves and registration error curves satisfied the built model given by Equation (14) curves and registration error curves satisfied the built model given by Equation (14) if there was a if there was a jitterwith displacement with a single sin function. Simulation experiments out jitter displacement a single sin function. Simulation experiments were carried outwere with carried different 00 with different ( ) and time t was a range from 0 to 2.5 s. Assuming  t  0.1 T ,0.2 T ...0.9 T A  1'' ,  t ∆t (∆t = 0.1T, 0.2T...0.9T) and time t was a range from 0 to 2.5 s. Assuming A = 1 , f = 1 Hz, i.e., 5 − 5 length 2 m, sizeaccording m, then  d  0 frad fc is 10Equation T focal 1 s , length p is 2 to T =f 1 1 s, Hz ϕd ,=i.e., 0 rad, m, and the pixel size p isand 2 × the 10 pixel m, then c is ,2 focal (3),according d(t) is determined and A is 0.4848 pixel. According to Equation (4), if ∆t is determined, r (t)(4), is if to Equation (3), dd (t ) is determined and Ad is 0.4848 pixel. According to Equation also Therefore, can be verified whether the relationship between d(t) and t)satisfies is determined, determined. Therefore, it can be verified whether ther (relationship r (t ) isit also tdetermined. thebetween built model given by Equation (14) in the following three aspects: d (t ) and r (t ) satisfies the built model given by Equation (14) in the following three aspects: (1)(1) Whether the frequency ofof d(td) (ist )consistent withwith the frequency of r (of t); r (t ) ; Whether the frequency is consistent the frequency (2)(2) Whether r (t)rsatisfies the Equation (10);(10); Whetherthe therelationship relationshipbetween betweenamplitudes amplitudesofofd(dt)(tand the Equation ) and (t ) satisfies (3)(3) Whether the relationship between phases of d ( t ) and r ( t ) satisfies Equation (13). Whether the relationship between phases of d (t ) and r (t ) satisfies Equation (13). The simulated d(td) (and r (t) rwith different ∆t aretshown in Figure 2. It is2.obvious that r (that t) The simulated different are shown in Figure It is obvious t ) and (t ) with hasr (same frequency as d ( t ) . The amplitudes and phases of simulated r ( t ) are as shown in Tables 1 t ) has same frequency as d (t ) . The amplitudes and phases of simulated r (t ) are as shown in and 2 respectively. The letter ‘Y’ in Tables 1 and 2 means simulation results are satisfied with the Tables 1 and 2 respectively. The letter ‘Y’ in Tables 1 and 2 means simulation results are satisfied corresponding equations, i.e., Equations (10) and (13) respectively. In addition, Figure 3 shows the with the corresponding equations, i.e., Equations (10) and (13) respectively. In addition, Figure 3 simulated r (t) at different ∆t. It is obvious that r (t) has a local maximum when ∆t is equal to 0.5T. It is shows the simulated r (t ) at different t . It is obvious that r (t ) has a local maximum when t also easy to explain using Equation (10). All of these validate the feasibility of utilizing the built jitter is equal to 0.5T. It is also easytotodo explain using Equation (10). All of these validate the feasibility of displacement estimation model the jitter estimation. utilizing the built jitter displacement estimation model to do the jitter estimation. Table 1. The amplitudes of simulated jitter displacements and registration errors at different imaging Table 1. The amplitudes of simulated jitter displacements and registration errors at different time intervals. imaging time intervals. Simulation Simulation Ad Ad Ar Ar Ar = 2Ad sin(π f ∆t)

Ar  2 Ad sin(

0.1T 0.2T 0.3T 0.4T 0.5T 0.6T 0.7T 0.8T 0.9T 0.1T 0.2T 0.3T 0.4T 0.5T 0.6T 0.7T 0.8T 0.9T 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.4848 0.29960 0.56996 0.7844 0.92211 0.96960 0.9221 0.78440 0.56996 0.2996 0.29960 0.56996 0.7844 0.92211 0.96960 0.9221 0.78440 0.56996 0.2996 Y Y Y Y Y Y Y Y Y f t ) Y Y Y Y Y Y Y Y Y

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Table 2. The phases ofofsimulated simulated jitter displacements and registration errors atatdifferent Table Thephases phasesof simulatedjitter jitterdisplacements displacementsand andregistration registrationerrors errorsat differentimaging imaging Table 2. 2.The different imaging time intervals. time intervals. time intervals. Simulation Simulation Simulation

0.1T 0.2T 0.2T 0.3T 0.3T 0.4T 0.4T 0.1T 0.1T 0.2T 0.3T 0.4T

0.5T 0.5T 0.5T

0 0 0 0 0  d d ϕd  r 0 0 0 0 00 00 00 1.8850 2.1991 2.1991 2.5133 2.5133 2.8274 2.8274 3.1416 3.1416  r 1.8850 ϕr 1.8850 2.1991 2.5133 2.8274 3.1416  d r −rπ/2 r  −/ π 2 f ∆tf f tt Y Y Y Y Y Y Y Y Y ϕ YY Y YY Y  / 2 d d=ϕ

1 1t(s) 2 2

t(s)  = 0.7T t =t 0.7T

0 0

-2 -2 0 0

2 2

1 1t(s) 2 2

t(s)  = 0.8T t =t 0.8T

0 0

1 1t(s) 2 2

t(s)

-2 -2 0 0

d(t) d(t)

t(s) = 0.6T t=t 0.6T

2 2

r(t)(pixel) r(t)(pixel)

-2 -2 0 0

1 1t(s) 2 2

0 0

r(t)(pixel) r(t)(pixel)

r(t)(pixel) r(t)(pixel)

2 2

t(s) = 0.5T t=t 0.5T

0 0

1 1t(s) 2 2

-2 -2 0 0

1 1t(s) 2 2

-2 -2 0 0

1 1t(s) 2 2

t(s)  = 0.9T t =t 0.9T

2 2

r(t)(pixel) r(t)(pixel)

0 0

-2 -2 0 0

2 2

0 0

r(t)(pixel) r(t)(pixel)

t(s) = 0.4T t=t 0.4T

-2 -2 0 0

= 0.3T t=t 0.3T

2 2

0 0

r(t)(pixel) r(t)(pixel)

2 2

= 0.2T t=t 0.2T

r(t)(pixel) r(t)(pixel)

0 0

-2 -2 0 0

2 2

r(t)(pixel) r(t)(pixel)

= 0.1T t=t 0.1T

r(t)(pixel) r(t)(pixel)

2 2

0.6T 0.7T 0.7T 0.8T 0.8T 0.9T 0.9T 0.6T 0.7T 0.8T 0.9T 0 0 0 0 0 0 0 0 0 0 0 0 3.4558 3.7699 3.7699 4.0841 4.0841 4.3982 4.3982 3.4558 3.4558 3.7699 4.0841 4.3982 Y YY YY Y YYY YYY

0 0

1 1t(s) 2 2

t(s)

d(t+ d(t+ t)t)

-2 -2 0 0

1 1t(s) 2 2

t(s) r(t)=d(t+ t)-d(t) r(t)=d(t+ t)-d(t)

Figure The simulated jitter displacement curves and registration error curves at different imaging Figure 2. 2. The simulated jitter displacement curves imaging Figure 2. The simulated jitter displacement curves and and registration registration error error curves curves at at different different imaging time intervals. time intervals. time intervals.

22

22

Simulated registration error (pixel) Simulated registration error (pixel)

33

Simulated registration error (pixel) Simulated registration error (pixel)

33

11

11

00

00

-1 -1

-1 -1 -2 -2 -3 -3 -4 -4 00

-2 -2

t=0.1T t=0.1T t=0.2T t=0.2T t=0.3T t=0.3T t=0.4T t=0.4T t=0.5T t=0.5T

0.50.5

1 1 1.51.5 Tim e (s) Tim e (s)

(a)(a)

22

t=0.9T t=0.9T t=0.8T t=0.8T t=0.7T t=0.7T t=0.6T t=0.6T t=0.5T t=0.5T

-3 -3

2.52.5

-4 -4 00

0.50.5

1 1 1.51.5 Tim e (s) Tim e (s)

22

2.52.5

(b) (b)

Figure3. 3.The Thesimulated simulatedregistration registrationerror errorcurves curvescaused causedbybysatellite satellite jitter(SJ) (SJ)atatdifferent differentimaging imaging Figure Figure 3. The simulated registration error curves caused by satellite jitterjitter (SJ) at different imaging time  t  0.1 T ,0.2 T ...0.5 T  t  0.5 T ,0.6 T ...0.9 T time intervals: (a) ; (b) . t  0.2T...0.5T; 0.1T ,0.2T ...0.5 T ;= t 0.6T...0.9T.  0.5T ,0.6T ...0.9T . time intervals: (b)0.5T, intervals: (a) ∆t(a) = 0.1T, (b) ∆t

3.2.Real RealImage ImageExperiments Experiments 3.2. Jitterdisplacements displacementscaused causedbybySJSJare areininboth bothcross-track cross-trackand andalong-track along-trackdirections. directions.Because Because Jitter jitterdisplacements displacementsininalong-track along-trackdirection directionare arecovered coveredupupbybyerrors errorscaused causedbybythe theterrain terrainfactor, factor, jitter jitter displacements in cross-track direction are usually more striking than those in the along-track jitter displacements in cross-track direction are usually more striking than those in the along-track

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Jitter displacements caused by SJ are in both cross-track and along-track directions. Because jitter displacements in along-track direction arein covered up bydirection errors caused by the factor, direction [16]. Therefore, jitter displacements cross-track are used to terrain evaluate the jitter displacements cross-track direction are usually more those in the along-track performance of the in jitter displacement estimation based on striking the builtthan model. The presented data direction [16]. Therefore, jitterofdisplacements in cross-track direction are used to (HR) evaluate the include multispectral images ZY-3 and images from the High Resolution camera performance of the jitter displacement estimation based on the built model. The presented data of ZY1-02C. include multispectral images of ZY-3 and images from the High Resolution (HR) camera of ZY1-02C. 3.2.1. Experiment Using Multispectral Images of ZY-3 Satellite 3.2.1. Experiment Using Multispectral Images of ZY-3 Satellite The presented multispectral image of ZY-3 is a level 1A product which is generated by The presented multispectral image of ZY-3 isAs a level 1A product which is generated radiometric radiometric correction and sensor correction. shown in Figure 4, the image wasbycaptured on correction and sensor correction. As shown in Figure 4, the image was captured on 3 February 2012, 3 February 2012, covering Dengfeng, Henan province, China. The image size is 8813 by 9307 pixels covering Dengfeng, Henandistance province,(GSD) China.ofThe sizecorresponding is 8813 by 9307 pixels with a ground with a ground sampling 5.8 image m. The high accuracy ground sampling distance (GSD) of 5.8 m.aThe corresponding accuracy ground dataaccuracy include reference data include DOM with plane accuracy of 1high m and DEM data withreference an elevation DOM of 2 m.with a plane accuracy of 1 m and DEM data with an elevation accuracy of 2 m.

Figure 4. Multispectral satellite. Figure 4. Multispectral image image of of ZY-3 ZY-3 satellite.

The multispectral camera of ZY-3 satellite has four bands including blue (B1), green (B2), red The multispectral camera of ZY-3 satellite has four bands including blue (B1), green (B2), red (B3), and near-infrared (B4), and the corresponding CCD arrays are parallel to each other [18]. The (B3), and near-infrared (B4), and the corresponding CCD arrays are parallel to each other [18]. integration time of each scanning line in a multispectral image is about 0.8 ms. The physical The integration time of each scanning line in a multispectral image is about 0.8 ms. The physical distance between B1 and B2 is 152 lines, and the physical distances between B2 and B3 and B3 and distance between B1 and B2 is 152 lines, and the physical distances between B2 and B3 and B3 and B4 B4 are both 128 lines [22]. This means the imaging time interval between different adjacent bands is are both 128 lines [22]. This means the imaging time interval between different adjacent bands is either either 121.6 ms or 120.4 ms. Considering the difference of visible band and near-infrared band, B1, 121.6 ms or 120.4 ms. Considering the difference of visible band and near-infrared band, B1, B2 and B3 B2 and B3 were chosen to conduct the experiment. Then multispectral bands were divided into were chosen to conduct the experiment. Then multispectral bands were divided into three kinds of three kinds of combinations (B1-B2, B2-B3, B1-B3) to compute registration error at different imaging combinations (B1-B2, B2-B3, B1-B3) to compute registration error at different imaging time intervals time intervals since the relative registration error caused by SJ varied with the imaging time interval. since the relative registration error caused by SJ varied with the imaging time interval. The detected and fitted relative registration error curves of different band combinations are The detected and fitted relative registration error curves of different band combinations are shown shown in Figure 5. In Figure 5, the red and blue curves represent detected and fitted registration in Figure 5. In Figure 5, the red and blue curves represent detected and fitted registration error curves error curves respectively. Table 3 shows the detailed fitted parameters of relative registration error respectively. Table 3 shows the detailed fitted parameters of relative registration error curves and the curves and the corresponding fitting residuals are shown in Table 4. The jitter displacement can be corresponding fitting residuals are shown in Table 4. The jitter displacement can be obtained based on obtained based on the jitter displacement estimation model using this information. The detailed the jitter displacement estimation model using this information. The detailed estimated parameters of estimated parameters of jitter displacement are shown in Table 5. In Table 5, the estimated jitter displacement are shown in Table 5. In Table 5, the estimated parameters of jitter displacement parameters of jitter displacement by different band combinations are very similar and the average amplitudes and frequencies of the estimated values of jitter displacement are around 0.9463 pixels and 0.6568 Hz, respectively.

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Figure 5. The relative registration error curves of different band combinations (a) B1-B2; (b) B2-B3;

Figure 5. The relative registration error curves of different band combinations (a) B1-B2; (b) B2-B3; (c) B1-B3. (c) B1-B3. Table 3. The parameters of registration error curves of different band combinations.

Table 3. The parameters of registration error curves of different band combinations.

Band Combination Amplitude/Pixel Frequency/Hz B1-B2 0.4941 0.6567 Band Combination Amplitude/Pixel Frequency/Hz B2-B3 0.3830 0.6566 0.8045 0.6570 B1-B2 B1-B3 0.4941 0.6567

Phase/Rad Time Interval/ms 1.6854 121.6 Phase/Rad Time 2.1558 102.4 Interval/ms 1.8928 224.0 121.6 1.6854

B2-B3 0.3830 0.6566 2.1558 102.4 B1-B3 0.6570 1.8928band combinations. 224.0 Table 4. Fitting residuals0.8045 of relative registration error curves of different

Band Max/Pixel RMSE Table 4. Fitting residuals of relative curves×of band combinations. −4 B1-B2 registration 0.0758 error4.1023 10different B2-B3 0.0557 2.0165 × 10−4 BandB1-B3 Max/Pixel RMSE 0.1003 0.0011 B1-B2 0.0758 4.1023 × 10−4 −4 Reference data (DOM/DEM accuracy) are2.0165 further utilized to obtain the absolute B2-B3 with high0.0557 × 10 0.1003 of jitter displacement 0.0011 registration error curve asB1-B3 the reference values in order to validate the accuracy of estimated values of jitter displacement. The detected and fitted results of absolute registration error curves obtained based on high accuracy ground reference data are shown in Reference data (DOM/DEM with high accuracy) are further utilized to obtain the absolute Figure 6. The detected result is shown as a red curve in Figure 6. The blue curve in Figure 6 is the registration error curve as the reference values of jitter displacement in order to validate the accuracy fitted result based on the detected result and the corresponding fitting residuals of absolute of estimated values of jitter displacement. detected fitted results absolute registration error curves are shown inThe Table 6. The and parameters of theoffitted curveregistration of absoluteerror curves obtained based on high accuracy ground reference data are shown in Figure 6. The detected registration error (blue curve in Figure 6) are considered as the reference values of jitter resultdisplacement is shown as and a red curve Figure The 5,blue in Figure 6 jitter is thedisplacement fitted resultcomputed based on the shown inin Table 5. In6.Table the curve reference values of by different bands also very similar andresiduals the amplitudes and frequencies of error the reference detected result and the are corresponding fitting of absolute registration curves values are shown of jitter displacement are around 0.9191 pixels and 0.6561 Hz, respectively. The evaluation of the6) are in Table 6. The parameters of the fitted curve of absolute registration error (blue curve in Figure accuracy of jitter displacement estimation was also conducted, and is shown in Table 5. Obviously, considered as the reference values of jitter displacement and shown in Table 5. In Table 5, the reference values of jitter displacement are very close to the reference values. The estimated valuesthe ofestimated jitter displacement computed by different bands are also very similar and the amplitudes frequencies of jitter displacement by different band combinations are almost identical to the and frequencies of the reference values of jitter displacement are around 0.9191 pixels and 0.6561 Hz, reference values. The average relative error is 0.11% and the maximum relative error is 0.23%. The respectively. The evaluation of the accuracy of jitter displacement estimation was also conducted, estimated amplitudes of jitter displacement by different band combinations are also similar to the and isreference shown in Tableand 5. Obviously, estimated values error of jitter displacement are very close to the values vary slightly.the The average relative is 2.96% and the maximum relative reference values. The estimated frequencies of jitter displacement by different band combinations error is 5.37%. The band combination of B2-B3 achieves the highest estimation accuracy. are almost identical to thejitter reference values. The average relative error is 0.11% and the Therefore, the built displacement estimation model is suitable to estimate themaximum jitter displacement for multispectral images of ZY-3 and B2-B3 is the optimal combination for jitter relative error is 0.23%. The estimated amplitudes of jitter displacement byband different band combinations displacement are also similar to estimation. the reference values and vary slightly. The average relative error is 2.96% and the

maximum relative error is 5.37%. The band combination of B2-B3 achieves the highest estimation accuracy. Therefore, the built jitter displacement estimation model is suitable to estimate the jitter displacement for multispectral images of ZY-3 and B2-B3 is the optimal band combination for jitter displacement estimation.

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Figure 6. The absolute registration error curves of different bands (a) B1; (b) B2; (c) B3. Table 5. The comparison of parameters of jitter displacement for multispectral images of ZY-3.

Table 5. The comparison of parameters of jitter displacement for multispectral images of ZY-3. Estimated Values

Band Combination

Band B1-B2 Combination B2-B3

B1-B3 B1-B2 average B2-B3 B1-B3 average

Amplitude Frequency Estimated Values /Pixel /Hz

Amplitude 0.9911 /Pixel 0.9096

Frequency 0.6567 /Hz 0.6566

Reference Values

Band

Band B1 B2

Amplitude Reference Frequency Values /Pixel /Hz

Amplitude 0.9406 /Pixel 0.9145

Frequency 0.6552 /Hz 0.6573

Relative Error (Amplitude) Relative Error 5.37% (Amplitude) 0.54% 4.00% 5.37% 2.96% 0.54%

0.9383 0.6570 B3 0.9022 0.6558 0.9911 0.6567 B1 0.9406 0.6552 0.9463 0.6568 average 0.9191 0.6561 0.9096 0.6566 B2 0.9145 0.6573 0.9383 0.6570 B3 0.9022 0.6558 4.00% 0.9463 0.6568 of absolute average registration 0.9191error curves 0.6561of different 2.96% Table 6. Fitting residuals bands.

Relative Error (Frequency) Relative Error 0.23% (Frequency) 0.11% 0.18% 0.23% 0.11% 0.11%

0.18% 0.11%

Band Max/Pixel RMSE Table 6. Fitting residuals of absolute registration error curves of different bands. B1 0.3265 0.0234 B2 0.3251 0.0248 Band Max/Pixel RMSE B3 0.2232 0.0122 B1 0.3265 0.0234 B2 of ZY1-02C0.3251 0.0248 3.2.2. Experiment Using HR Images B3 0.2232 0.0122 The presented HR image of ZY1-02C is a level 0 product including rational function coefficients. As shown in Figure 7, the image was captured on 13 April 2012, covering Dengfeng 3.2.2. area, Experiment Using HR Images ZY1-02C Henan province, China. The of image size is 4096 by 27,575 pixels with a GSD of 2.36 m. The HR camera consists of three non-linear CCD arrays (CCD1, CCD2, CCD3), which are staggered on the The presented HR image of ZY1-02C is a level 0 product including rational function coefficients. focal plane along the track. The interval of along-track is about 2600 pixels and overlap of As shown in Figure 7, the image was captured on 13 April 2012, covering Dengfeng area, Henan cross-track is about 30 pixels in the adjacent CCD arrays. The exposure time of each scanning line is province, China. The image size is 4096 by 27,575 pixels with a GSD of 2.36 m. The HR camera consists approximately 0.35 ms, that is to say, the imaging time interval between adjacent CCD arrays is of three non-linear about 0.91 s. CCD arrays (CCD1, CCD2, CCD3), which are staggered on the focal plane along the track. The The process interval isof similar along-track about 2600and pixels and overlap cross-track isimages. about 30 pixels to theis estimation validation usingofmultispectral The in thedetected adjacentand CCD arrays. The exposure of each is approximately ms,in that is fitted relative registrationtime error curvescanning between line CCD1 and CCD2 are 0.35 shown 8a and the fittingadjacent residualCCD of relative error to say,Figure the imaging timecorresponding interval between arraysregistration is about 0.91 s. curve is shown in Table 7. In Figure 8, theto red and blue curves represent detected fitted registration error curves The process is similar the estimation and validation usingand multispectral images. The detected respectively. The detailed estimated parameters of jitter displacement obtained based on the jitter and fitted relative registration error curve between CCD1 and CCD2 are shown in Figure 8a and the displacement estimation model are given in Table 8. The amplitudes and frequencies of the corresponding fitting residual of relative registration error curve is shown in Table 7. In Figure 8, estimated values of jitter displacement are around 3.5643 pixels and 0.3112 Hz, respectively. the red and blue curves represent detected and fitted registration error curves respectively. The detailed Reference data (DOM/DEM with high accuracy) are also used to get the absolute registration error estimated parameters of jitter displacement obtained based on the jitter displacement estimation model curve as reference values of jitter displacement to verify the accuracy of estimated values of jitter are given in Table 8.Figure The amplitudes and frequencies of the estimated values of jitter displacement displacement. 8b shows the detected and fitted absolute registration error curve of CCD1 are around pixels and 0.3112 Hz, respectively. with high accuracy) and3.5643 the corresponding fitting residual of absolute Reference registrationdata error(DOM/DEM curve is also shown in Table 7. are also to getof the registration errorascurve as reference jitter displacement Theused parameters theabsolute fitted curve are considered the reference valuesvalues of jitterof displacement and also the shown in Tableof8.estimated The amplitudes andoffrequencies of the reference values of jitter the displacement to verify accuracy values jitter displacement. Figure 8b shows detected and are around 3.2863 pixels and 0.3123 Hz, respectively. The evaluation about the accuracy fitted absolute registration error curve of CCD1 and the corresponding fitting residual of of jitter absolute

registration error curve is also shown in Table 7. The parameters of the fitted curve are considered as the reference values of jitter displacement and also shown in Table 8. The amplitudes and frequencies of the reference values of jitter displacement are around 3.2863 pixels and 0.3123 Hz, respectively.

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The evaluation estimation about the accuracy jitter displacement is also in Table 8. Obviously, displacement is also ofshown in Table 8.estimation Obviously, the shown estimated values of jitter the estimated values of jitter displacement are very close to the reference values. The relative of displacement very close to the reference values. The relative error of the estimated amplitude is Sensors 2017,are 17, 83 10 oferror 13 the estimated amplitude 8.46%ofand theestimated relative error of the estimated frequency is 0.35%. Therefore, 8.46% and the relative iserror the frequency is 0.35%. Therefore, the built jitter displacement estimation is also shown in Table 8.suitable Obviously, the estimated values of images jitter for the built jitter estimation displacement estimation model istoalso estimate the jitter displacement displacement model is also suitable estimate thetojitter displacement for HR of displacement are veryOf close to the 8.46% reference values. The relative error oflarge. the estimated amplitude is may HR images of ZY1-02C. course, as a max error is relatively The relative error ZY1-02C. Of course, 8.46% as a max error is relatively large. The relative error may be caused by 8.46% by and the relative error and of the estimated is 0.35%. Therefore, built jitter may be caused matching fitting errors.frequency Furthermore, satellite jitter is the complex matching accuracy andaccuracy fitting errors. Furthermore, satellite jitter is complex and mayand contain displacement estimation model is also suitable to estimate the jitter displacement for HR images of contain multiple frequencies. Since the obtained relative registration error curves show obvious single multiple frequencies. registration curves showbeobvious single ZY1-02C. Of course,Since 8.46%the as aobtained max errorrelative is relatively large. Theerror relative error may caused by frequency characteristics, they are are processed in thisinwork bywork the single frequency jitter displacement frequency characteristics, they processed this the single frequency matching accuracy and fitting errors. Furthermore, satellite jitter by is complex and may contain jitter estimation model. In this procedure, error also may be introduced. displacement estimation model. In this procedure, also may be introduced. multiple frequencies. Since the obtained relativeerror registration error curves show obvious single frequency characteristics, they are processed in this work by the single frequency jitter displacement estimation model. In this procedure, error also may be introduced.

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(a)registration error curve between CCD1 and (b)CCD2; (b) The absolute Figure 8. (a) The relative registration error curve of CCD1. registration error of CCD1. Figure 8. (a) Thecurve relative registration error curve between CCD1 and CCD2; (b) The absolute Table 7. Fitting residuals of registration error curves. registration error curve of CCD1. Table 7. Fitting residuals of registration error curves. Band Max/Pixel RMSE Table 7. Fitting residuals of registration error curves. CCD1-CCD2 1.1391 0.1827 Band Max/Pixel RMSE CCD1 1.0861 0.3128 Band Max/Pixel RMSE CCD1-CCD2 1.1391 0.1827 CCD1-CCD2 1.1391 0.1827 CCD1 1.0861 0.3128 CCD1

1.0861

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Table 8. The comparison of parameters of jitter displacement for HR images of ZY1-02C. Estimated Values

Reference Values

CCD Combination

Amplitude /Pixel

Frequency /Hz

CCD

Amplitude /Pixel

Frequency /Hz

Relative Error (Amplitude)

Relative Error (Frequency)

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0.35%

4. Conclusions In this paper, the feasibility of jitter displacement estimation only using parallax images was analyzed in theory. Then the jitter displacement estimation model was built based on the quantitative relationship between the jitter displacement and the relative registration error obtained from parallax images. Finally, simulation experiments analyzed the change rule of jitter displacement estimated by registration error at different imaging time intervals and verified the feasibility of using the built jitter displacement model to estimate the jitter displacement. Real image experiments further evaluated the performance of the jitter displacement estimation based on the built model. In real image experiments, two typical images including multispectral images of ZY-3 satellite and HR images of ZY1-02C satellite were chosen as experimental data and high accuracy ground reference data were also used to evaluate the accuracy of the estimation. Experimental results indicated that the built jitter displacement estimation model can estimate the jitter displacement accurately and may be used to do the jitter detection for other sensors with parallax images. This paper summarizes the work performed to improve upon our previous works [18,19], and includes three key developments: (1)

(2)

(3)

The deduced quantitative relationship between the jitter displacement and relative registration error obtained from parallax images in the presented paper is more accurate than our previous work. In our previous works, the relationship has only been approximated. However, the present work deduced an accurate quantitative relationship between the jitter displacement and relative registration error obtained from parallax images without any approximation. The deduced quantitative relationship between the jitter displacement and relative registration error obtained from parallax images in the presented paper is more robust than our previous work. The deduced quantitative relationship in our previous work is not an accurate expression, as its accuracy decreased when the imaging time interval increased. Obviously, such a relationship is not suitable to parallax images with large imaging time intervals such as those from the High Resolution camera of ZY1-02C. In the presented paper, the deduced quantitative relationship is accurate without any approximation. It is suitable to parallax images with any imaging time interval. Two typical images, the multispectral images of ZY-3 satellite (with a short imaging time interval between different adjacent bands of either 121.6 ms or 120.4 ms) and the HR images of ZY1-02C satellite (with a long imaging time interval between adjacent CCD arrays of about 0.91 s), were chosen as experimental data and validated the deduced relationship. High accuracy ground reference data were used to evaluate the accuracy of the jitter displacement estimation in real image experiments in the presented paper. It is an objective method. In our previous work, the accuracy of the jitter displacement estimation in real image experiments wasn’t evaluated. Other references also didn’t evaluate the accuracy of the jitter displacement detection. Generally, they evaluated the images after jitter correction [3].

Future work must also determine whether the integration stage has a certain impact on the accuracy of the jitter displacement estimation. Whether the proposed jitter estimation approach is appropriate to estimate these high frequency satellite jitters also needs to be confirmed experimentally. More accurate fitting methods may also improve the performance of the jitter displacement estimation. These will be addressed in our future work.

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Acknowledgments: This work was supported by the National Natural Science Foundation of China (No. 91438112), the National Basic Research Program of China (973 Program, No. 2014CB744201 and No. 2012CB719901) and a Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD, No. 201249). Author Contributions: Jun Pan conceived the study and designed the experiments; Chengbang Che performed the experiments and analyzed the data; Ying Zhu wrote the program for data analysis; Mi Wang gave valuable suggestions and constructive discussions, and contributed to manuscript preparation. Conflicts of Interest: The authors declare no conflict of interest.

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