Study of full-link on-orbit geometric calibration using multi-attitude imaging with linear agile optical satellite

Yingdong Pi; Bo Yang; Xin Li; Mi Wang

doi:10.1364/OE.27.000980

1. Introduction

Agile optical satellite (AOS) [1,2] usually refers to a satellite platform equipped with optical sensors, and capable of rapid orbital and attitude maneuver. The three-axis active control system utilizes the entire satellite to control the attitude, and can theoretically realize any maneuver in three directions. With its high level of maneuverability, the AOS can obtain ground target images rapidly and accurately, thereby providing first-hand information that can be applied to aid disaster relief, military strikes and planning decisions. This renders AOS an extremely important component for future earth observation systems. Compared with the conventional optical satellite, AOS boasts the following advantages:

1) The increased side-swing angle can broaden the observation bandwidth, further extending the observation strip width and observation range. Therefore, an equivalent quantity of information can be obtained in a shorter time when an AOS is employed.
2) AOS can achieve sustained staring imaging of a specific target on the earth, near earth space, or in the universe, which is beneficial for natural disaster monitoring. Additionally, AOS can provide real-time tracking images according to its rapid attitude control properties.
3) Via its attitude maneuver capability, AOS can obtain stereo images with different baseline lengths, along shared or different tracks, to further single-linear array stereo mapping.

Regardless of the precise nature of the application a superior degree of geometric accuracy is required. As such, on-orbit geometric calibration is routinely employed during the commissioning phase to improve the geometric quality of satellite imagery. The space resection technique is usually applied in the conventional method [3,4] to achieve a high-precision calibration using ground control points (GCPs) identified from the reference data of the calibration sites. However, the reference data covering the calibration site are usually obtained through aerial photogrammetry, associated with inferior timeliness and high production costs. Additionally, the difficulty in obtaining cloud-free images over calibration sites, and the limited matching precision caused by the textural inconsistency between the calibrated and reference images, further impedes the applicability of this method for AOS.

With the aim of circumventing the limitations associated with the conventional method, an independent geometric calibration method is proposed to calibrate systematic parameters without using GCPs. This method uses the self-constraint of a set of images that meet specific attitude requirements to achieve high-precision calibration; such images can be collected by an AOS given their agility.

As Fig. 1 illustrates, a reasonable combination of multi-attitude images was seen following systematic analysis of the validity of various self-constraint conditions. In Fig. 1, E1, E2 and E3 are the push-broom images that were collected by an AOS with an attitude of 0° in the yaw direction, at moments t1, t2 and t3. E2 was obtained from standard nadir imaging, while E1 and E3 were obtained from both sides of E2 via different orbits, or by using AOS mobility in the same orbit. The overlap between E1 and E2, and that between E3 and E2, should exceed 50%. E4 is the push-broom image collected from an attitude angle of 180° in the yaw direction at the moment t4, with the same covering area as E2. From just the self-constraint of these four images and a corresponding digital surface model (DSM), the high-precision geometric calibration of an AOS can be realized. A stepwise calculation is performed to estimate calibration parameters, within which the external parameters are calculated; internal parameters are subsequently calculated in a generalized camera frame determined by the calculated external parameters. Regarding the external parameters, only the constraint between E1 and E4 is used to perform calibration, whereas a block adjustment is applied to all four images for estimating the internal parameters.

Fig. 1 The combination of multi-attitude images.

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2. Literature review

Geometric calibration is a key technology for guaranteeing the geometric quality of high-resolution optical satellite imagery [5,6]. Rigorous calibration of an optical satellite sensor is routinely undertaken in the laboratory in advance of the sensor’s launch. However, the geometric parameters are usually subject to a certain degree of alteration as a result of a complex set of factors, such as changes to the environmental temperature and stress release. As such, geometric calibration must be repeated on orbit to ensure the geometric quality of the satellite images. Several methods have been developed for the on-orbit geometric calibration of different satellites. SPOT-5 satellite was calibrated using the GCPs identified from the French MANOSQUE calibration site, achieving an internal accuracy of around 0.1 pixels [7]. Breton et al. also performed a geometric calibration for the SPOT-5 satellite, calibrating the sensor alignment, the high-resolution geometrical (HRG) instrument steering mirror of the nadir model, and the HRG/HRS detection lines [8]. Mulawa et al. calibrated the OrbView-3 satellite using GCPs determined from the reference data of the Lubbock calibration site, improving the internal accuracy from 30 to 0.4 pixels [9]. Li et al. used the northeast digital calibration site to calibrate the focal length, position of the main point, camera intersection angle, and included angle between the star camera and the ground observation camera of the TianHui-1 satellite, achieving a positioning accuracy of around 10.3 m [10]. For ZiYuan-3 satellite, a comprehensive calibration was also performed during its commissioning phase, using the reference imagery covering the Songshan calibration site to calibrate the triple-linear array cameras and multispectral camera [11–13]. Based on the calibration results, the internal distortion of the image product generated by the sensor calibration is controlled in the sub-pixel, and the geometric registration accuracy of each of band of the multispectral image is superior to 0.25 pixels [14]. Based on the similar method, Wang et al. used star as control condition to calibrate the navigation camera [15].

The aforementioned methods mostly used a dense network of GCPs identified from the reference data of calibration sites as a constraint. As such, these methods are highly dependent on the calibration sites, rendering them high in cost and low in efficiency. As an alternative to these conventional methods, an independent geometric calibration method without the use of calibration sites was successfully used by the PLEIADES-1A/1B satellite. In this method, an image pair with an included angle of almost 180° in the yaw direction—collected from the same orbit in opposite directions—was used to calibrate the pitch and roll angles of the installation; block adjustment was applied to regional images with certain overlaps to estimate the yaw angle and focal length of the installation, and the cross imagery pair, with an included angle of almost 90° in the yaw direction, was used to calibrate the focal plane. The specific calibration model and method, however, have not yet been published [16,17]. However, this method requires the satellite to obtain the swing scanning image during motion, exacerbating the difficulty of satellite attitude control, and may lead to error due to an unstable attitude during imaging; this can further reduce the accuracy of the calibration in practical application.

Compared with previous studies, this investigation is innovative in several aspects. First, a complete, full-link, independent geometric calibration designed to compensate for both external and internal systematic errors is proposed, and the image combination that can be used for this full-link calibration is elucidated. Second, regarding the constraint of multi-attitude images, the calibration parameters that may be estimated are defined as equivalent to twice the compensation, and compensate for all systematic errors. Third, the images used in the method are obtained via push-broom imaging with a stabilized satellite attitude, avoiding errors caused by unstable attitude during swing scanning imaging in motion.

3. Analysis of systematic errors and establishment of the calibration model

Considering that geometric calibration, by its nature, corrects systematic errors in the imaging model parameters, the systematic errors that affect the geometric quality of the AOS images are analyzed first; subsequently, a calibration model based on equivalent compensation is constructed with reference to analysis of the systematic errors.

3.1 Analysis of systematic errors

The systematic errors can be divided into two types: the internal camera errors and the external installation error. The former is mainly composed of CCD physical error (CCD translation and rotation), lens distortion and focal length error. Assuming the measured coordinates of an imagery point are $(x_{s}, y_{s}, z_{s})$ in a camera coordinate system, the translation of the CCD is $(d x, d y)$ , the rotation angle of the CCD is $θ$ , the size of the CCD detector is $(λ_{x}, λ_{y})$ , the optical lens distortion is $(Δ x, Δ y)$ , the designed value of the focal length is $f$ , and the focal length error is $Δ f$ . Therefore, we may obtain the final imagery coordinate $(x, y, z)$ by correcting these internal geometric errors, as Eq. (1).

{\begin{cases} x = (f + Δ f) \cdot (x_{s} + d x + s \cdot λ_{x} \cdot \sin θ + Δ x) / f \\ y = (f + Δ f) \cdot (y_{s} + d y + s \cdot λ_{y} \cdot \cos θ + Δ y) / f \\ z = - (f + Δ f) \end{cases}

where,

s

is the number of CCD detectors.

Another systematic error is the external installation error. The attitude of the camera system is indirectly determined by the attitude determination system (ADS) of the satellite. An installation matrix $R_{body}^{cam}$ (determined by the installation angles $p i t c h, r o l l, y a w$ ) is used to represent the transform relationship. Limited by their assembly technology, and subject to possible displacement during launch and orbit, the true camera installation angles tend to deviate from the preset angles determined on the ground, so that a camera installation error is present in the initial image model, thus requiring on-orbit calibration.

3.2 Rigorous imaging model of linear AOS

Based on the error analysis, a collinear AOS imaging model may be established, as Eq. (2).

(\begin{array}{l} x \\ y \\ z \end{array}) = λ R_{body}^{cam} (p i t c h, r o l l, y a w) (R_{J2000}^{body} R_{wgs}^{J2000} {[\begin{array}{l} X_{g} - X_{gps} \\ Y_{g} - Y_{gps} \\ Z_{g} - Z_{gps} \end{array}]}_{wgs} - {[\begin{array}{l} B_{X} \\ B_{Y} \\ B_{Z} \end{array}]}_{body})

In which,

(x, y, z)

represent the real coordinates of the CCD detector in the camera coordinate system;

λ

is the scale factor;

(X_{g}, Y_{g}, Z_{g})

and

(X_{gps}, Y_{gps}, Z_{gps})

represent the corresponding ground coordinates and the center coordinate of the GPS antenna in the WGS84 coordinate system, respectively;

R_{wgs}^{J2000}

,

R_{J2000}^{body}

, and

R_{body}^{cam}

are the rotation matrixes from the WGS84 to the J2000 coordinate system, from the J2000 to the body-fixed coordinate system of the satellite and from the body-fixed to the camera coordinate system, respectively; and

{(B_{X}, B_{Y}, B_{Z})}_{body}

represents the translation vector from the projection center to the GPS antenna center in the satellite’s body-fixed coordinate system.

3.3 Calibration model based on equivalent compensation

The systematic errors affecting the geometric quality of AOS images comprise CCD errors, focal length errors, lens distortion and camera installation errors. These errors should be estimated using the appropriate constraints. Compared with the strong absolute constraint used in the conventional calibration method, the relative constraints of multi-attitude images are weak. When used as the only condition for calibrating parameters, a more complex calibration method and model, with more unknown parameters, will be the result. Based on the correlation between the system error parameters [4], the equivalent compensation is used to simplify the model in accordance with the stepwise estimation model. The equivalent compensations manifest primarily in two aspects: the yaw angle and the internal parameters.

3.3.1 Yaw angle

For a single CCD, the installation angle yaw and the rotation angle $θ$ of CCD in the focal plane are completely equivalent. The relationship is illustrated in Fig. 2, in which $O - X Y Z$ is the body-fixed coordinate system of the satellite, and $o - x y$ is the focal plane coordinate system of the camera. It may be observed that the installation error in the yaw direction is completely equivalent to the rotation error of the CCD, and may be compensated equivalently with CCD rotation, therefore no longer requiring estimation in the practical calibration. The results of the quantitative analysis of the correlation between these two parameters are given in [4].

Fig. 2 Correlation between CCD rotation and yaw.

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It is important to point out that the pitch and roll angles of the installation are also completely related to the translation of the CCD. However, the positioning errors caused by pitch and roll errors far exceed those caused by similar yaw errors, and as such their effects are far more evident. Additionally, external systematic errors are inconsistent and usually vary within a particular range over time. However, the geometric error caused by yaw variation is usually negligible, but very conspicuous for pitch and roll. Thus, pitch and roll usually need to be calibrated sequentially for practical utility, and so they are inappropriate for substitution with internal parameters. As a result, the internal parameters are only used to equivalently compensate yaw, and pitch and roll are regarded as unknown parameters requiring calibration under the constraints of the multi-images obtained using this independent method.

3.3.2 Internal parameters

The rigorous internal model of the camera is unsuitable for on-orbit calibration due to underfitting and the correlation between the parameters. To address these problems, an equivalent viewing angle model is adopted to replace the rigorous internal model of the camera system. As illustrated in Eq. (3), the line of sight (LOS) of each CCD detector in the camera coordinate system may be determined by the viewing angle $(φ_{x}, φ_{y})$ :

{(V_{i m a g e})}_{c a m} = {(x, y, z)}^{T} = λ {(\tan (φ_{x}), \tan (φ_{y}), - 1)}^{T}

However, it is overly ambitious, and unnecessary, to attempt to calculate all viewing angles of the CCD detectors. A continuous internal camera model may be established by fitting the viewing angles of all CCD detectors with two polynomials. Considering the long focal length of the geometric characteristics, and the narrow field view of the AOS, the contribution of high-order errors to the overall internal distortion of the sensor is limited. Therefore, two three-order polynomials are usually appropriate for description of the internal parameters [3,4], as Eq. (4).

{\begin{matrix} φ_{x} (s) = a_{0} + a_{1} s + a_{2} s^{2} + a_{3} s^{3} \\ φ_{y} (s) = b_{0} + b_{1} s + b_{2} s^{2} + b_{3} s^{3} \end{matrix}

where

s

is the sequential number of the CCD detector.

By introducing the equivalent viewing angle model Eq. (4) into the rigorous imaging model Eq. (2), the most appropriate model for independent calibration of the AOS may be derived as Eq. (5), in which the unknown parameters are the coefficients of the viewing angle model $(a_{ω}, b_{ω})$ $(ω = 0, 1, 2, 3)$ , the camera installation angles $(p i t c h, r o l l)$ , and the ground 3D coordinates $(X_{g}, Y_{g}, Z_{g})$ .

(\begin{array}{l} \tan (φ_{x}) \\ \tan (φ_{y}) \\ - 1 \end{array}) = λ R_{body}^{cam} (p i t c h, r o l l) (R_{J2000}^{body} R_{wgs}^{J2000} {[\begin{array}{l} X_{g} - X_{gps} \\ Y_{g} - Y_{gps} \\ Z_{g} - Z_{gps} \end{array}]}_{wgs} - {[\begin{array}{l} B_{X} \\ B_{Y} \\ B_{Z} \end{array}]}_{body})

4. Approach and feasibility of the method

4.1 Effect of elevation error on calibration accuracy

The effect of elevation errors on calibration should be afforded due consideration, owing to their strong correlation in cases where GCPs are not used. The orientation error $Δ θ$ resulting from the elevation error $Δ H$ may be calculated by Eq. (6) [18]:

Δ θ = θ - θ' = arc \tan (B / H) - arc \tan (B / (H + Δ H))

in which,

B

is the baseline length,

H

is the orbit height of the satellite.

If we use the orientation error to describe the calibration accuracy, the quantitative relationship between the calibration accuracy and elevation accuracy may be determined with Eq. (6). Toward a more intuitive demonstration of this relationship, we drew the tendency curve of the calibration accuracy under different baseline lengths, in which the orbit height was set as 500 km, by referring to the real orbit heights of most earth observation satellites. Consequently, the relationship between calibration accuracy and elevation error could be ascertained, as shown in Fig. 3.

Fig. 3 Relationship between calibration accuracy and elevation error.

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It may be observed that the relationship between elevation error and calibration accuracy is basically linear for a fixed baseline, indicating a high degree of correlation between them. If both kinds of parameters are regarded as unknowns in a solution, this correlation will ensure that they cannot be separated; as a result, optimal estimation of calibration parameters cannot be achieved. Therefore, an elevation constraint is essential for achieving a satisfactorily stable geometric calibration.

4.2 Adjustment model for geometric calibration

Assuming that:

[\begin{matrix} \bar{X} \\ \bar{Y} \\ \bar{Z} \end{matrix}] = R_{body}^{cam} (R_{J2000}^{body} R_{wgs}^{J2000} {[\begin{array}{l} X_{g} - X_{gps} \\ Y_{g} - Y_{gps} \\ Z_{g} - Z_{gps} \end{array}]}_{wgs} - {[\begin{array}{l} B_{X} \\ B_{Y} \\ B_{Z} \end{array}]}_{body})

and replacing right side of equal sign in Eq. (5) with Eq. (7):

Therefore, the adjustment model used for calibration may be established as Eq. (8):

{\begin{matrix} F = \bar{X} + \tan (φ_{x} (s)) \cdot \bar{Z} \\ G = \bar{Y} + \tan (φ_{y} (s)) \cdot \bar{Z} \end{matrix}

where the internal and external parameters to be calculated are the respective coefficients of the viewing-angle model

X_{I} = (a_{ω}, b_{ω})

(ω = 0, 1, 2, 3)

and the camera installation angles

X_{E} = (p i t c h, r o l l)

.

4.3 External calibration

4.3.1 Constraint of multi-attitude images

Only images E2 and E4, obtained with 0° and 180° attitude, respectively, in yaw direction are used to estimate the external parameters of pitch and roll.

As Fig. 4 illustrates, the AOS achieves attitude maneuver by turning the whole satellite with the aid of a momentum flywheel, thus the installation errors of pitch and roll for 0° and 180° yaw attitude are equal in magnitude and opposite in direction. The ground positioning biases resulting from these inverse installation errors are also equal in magnitude and opposite in direction. By using the special geometric relationship from E2 and E4, the external parameters of the pitch and roll angles can be compensated, and high-accuracy estimation of the external parameters can be achieved without recourse to GCPs.

Fig. 4 Installation errors for 0° and 180° yaw attitude.

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4.3.2 Estimation of external parameters

To ascertain the external calibration parameters, it is assumed that that the initial internal calibration parameters are “true”. The internal calibration parameters are initialized with the on-ground calibration result $X_{I}^{0}$ . The least squares method is adopted to estimate the calibration parameters. The error equation Eq. (9) in the kth iteration solution, for the ith corresponding point pair (CPP), identified from the jth image, may be obtained by linearizing Eq. (8).

V_{i, j, k}^{E} = A_{i, j, k} x_{k}^{E} + B_{i, j, k} t_{i, k}^{E} - L_{i, j, k}^{E} \begin{matrix} P_{i, j, k}^{E} \end{matrix}

in which

A_{i, j, k} = [\begin{matrix} \frac{\partial F_{i, j, k}}{\partial p i t c h} & \frac{\partial F_{i, j, k}}{\partial r o l l} \\ \frac{\partial G_{i, j, k}}{\partial p i t c h} & \frac{\partial G_{i, j, k}}{\partial r o l l} \end{matrix}], B_{i, j, k} = [\begin{matrix} \frac{\partial F_{i, j, k}}{\partial L a t_{i, k}} & \frac{\partial F_{i, j, k}}{\partial L o n_{i, k}} \\ \frac{\partial G_{i, j, k}}{\partial L a t_{i, k}} & \frac{\partial G_{i, j, k}}{\partial L a t_{i, k}} \end{matrix}], x_{k}^{E} = {[\begin{matrix} Δ p i t c h \\ Δ r o l l \end{matrix}]}_{k},

where

x_{k}^{E}

is the correction of the external parameters,

L_{i, j, k}^{E}

is the residual vector calculated by the current external and internal parameters

(X_{E}^{k}, X_{I}^{0})

, and the ground coordinate determined by the current

(X_{E}^{k}, X_{I}^{0})

,

P_{i, j, k}^{E}

is the weight matrix.

It is important to note that the ground geodetic coordinates $(L a t, L o n, H e i)$ of each CCP are also unknown parameters, and must be estimated in the calculation. However, the strong correlation between the calibration accuracy and elevation will result in an ill-conditioned adjustment model if the elevations are also calculated. In response to this problem, an elevation constraint is introduced into the adjustment model to ensure a stable solution. For the external calibration, the reference elevation may be a surface height with constant elevation, or a DSM. The precision of the reference elevation is determined by the convergence geometry of the images used. If a DSM is applied, the calibration model may be improved by interpolating the CCP elevation from the reference DSM, and considering it to be the true elevation in each iterative calculation.

Therefore, the error equation for ith CCP, in the kth iteration solution can be derived as Eq. (10), considering that only two images are used for estimating the external parameters.

V_{i, k}^{E} = A_{i, k} x_{k}^{E} + B_{i, k} t_{i, k}^{E} - L_{i, k}^{E} \begin{matrix} P_{i, k}^{E} \end{matrix}

in which

A_{i, k} = [\begin{matrix} A_{i, 1, k} \\ A_{i, 2, k} \end{matrix}], B_{i, k} = [\begin{matrix} B_{i, 1, k} \\ B_{i, 2, k} \end{matrix}], L_{i, k}^{E} = [\begin{matrix} L_{i, 1, k}^{E} \\ L_{i, 2, k}^{E} \end{matrix}], P_{i, k}^{E} = [\begin{matrix} P_{i, 1, k}^{E} \\ P_{i, 2, k}^{E} \end{matrix}] .

then,

x_{k}^{E}

may be estimated according to the least squares theory as Eq. (11):

M_{k}^{E} x_{k}^{E} = W_{k}^{E}

where,

M_{k}^{E} = \sum_{i = 1}^{n} (A_{i, k}^{T} P_{i, k}^{E} A_{i, k} - A_{i, k}^{T} P_{i, k}^{E} B_{i, k} {(B_{i, k}^{T} P_{i, k}^{E} B_{i, k})}^{- 1} B_{i, k}^{T} P_{i, k}^{E} A_{i, k})

W_{k}^{E} = \sum_{i = 1}^{n} (A_{i, k}^{T} P_{i, k}^{E} L_{i, k}^{E} - A_{i, k}^{T} P_{i, k}^{E} B_{i, k} {(B_{i, k}^{T} P_{i, k}^{E} B_{i, k})}^{- 1} B_{i, k}^{T} P_{i, k}^{E} L_{i, k}^{E})

and

n

is the number of CCP.

Finally, $X_{k}^{E}$ can be updated by Eq. (12).

X_{k + 1}^{E} = X_{k}^{E} + x_{k}^{E}

The adjustment calculation is an iterative process, with the iterative calculation not undertaken until the difference between the two successive results is less than the predefined tolerance limits.

4.4 Internal calibration

4.4.1 Constraint of multi-attitude images

To estimate the internal parameters, the relative constraints from images E2 and E4 alone will not suffice. It can be seen in Figs. 5 (a) and 5(b) that, when the CCP identified from E2 and E4 alone are used, the internal distortion may only constitute an odd function distortion, so that the constraint condition is insufficient for elimination of the internal distortion. Based on the odd function distortion, elimination of the internal distortion may be achieved by adding the two constraint images, E1 and E3, on both sides of E2 and E4. With these two constraints, the odd function distortion may be further compensated, thereby eliminating the internal distortion (Fig. 5(c)). In practical calculation, the block adjustment may be applied to the CCPs identified from these four images, and thus the optimal estimation of internal parameters achieved directly.

Fig. 5 Internal distortions for different constraint from multi-attitude images.

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4.4.2 Estimation of internal parameters

In estimating the internal parameters, the external calibration parameters are initialized with the estimated parameters, $X_{E}$ , in external calibration, and regarded as true values. Estimation of the internal parameters is also achieved by the least squares method. The differences between the estimations of the internal and external parameters are the parameter type, and that a CCP may include more than two points here, considering that not only two images are used. The error equation Eq. (13) in the kth iteration for the ith CPP identified from the jth image can be obtained by linearizing Eq. (8).

V_{i, j, k}^{I} = C_{i, j, k} x_{k}^{I} + D_{i, j, k} t_{i, k}^{I} - L_{i, j, k}^{I} \begin{matrix} P_{i, j, k}^{I} \end{matrix}

in which

C_{i, j, k} = [\begin{matrix} \frac{\partial F_{i, j, k}}{\partial a_{ω}} & \frac{\partial F_{i, j, k}}{\partial b_{ω}} \\ \frac{\partial G_{i, j, k}}{\partial a_{ω}} & \frac{\partial G_{i, j, k}}{\partial b_{ω}} \end{matrix}], D_{i, j, k} = [\begin{matrix} \frac{\partial F_{i, j, k}}{\partial L a t_{i, k}} & \frac{\partial F_{i, j, k}}{\partial L o n_{i, k}} \\ \frac{\partial G_{i, j, k}}{\partial L a t_{i, k}} & \frac{\partial G_{i, j, k}}{\partial L a t_{i, k}} \end{matrix}], x_{k}^{I} = {[\begin{matrix} Δ a_{ω} \\ Δ b_{ω} \end{matrix}]}_{k}, (ω = 0, 1, 2, 3)

where

x_{k}^{I}

is the correction of the internal parameters,

L_{i, j, k}^{I}

is the residual vector calculated by the current external and internal parameters

(X_{E}^{}, X_{I}^{k})

and the ground coordinate determined by current

(X_{E}^{}, X_{I}^{k})

, and

P_{i, j, k}^{I}

is the weight matrix. The elevation of each CCP used here should be interpolated from the reference DSM to ensure calibration accuracy, rather than to calculate it.

Similar to Eq. (10), the error equation for the ith CCP, in the kth iteration solution may be derived as Eq. (14):

V_{i, k}^{I} = C_{i, k} x_{k}^{I} + D_{i, k} t_{i, k}^{I} - L_{i, k}^{I} \begin{matrix} P_{i, k}^{I} \end{matrix}

in which

C_{i, k} = [\begin{matrix} C_{i, 1, k} \\ ⋮ \\ C_{i, m, k} \end{matrix}], D_{i, k} = [\begin{matrix} D_{i, 1, k} \\ ⋮ \\ D_{i, m, k} \end{matrix}], L_{i, k}^{I} = [\begin{matrix} L_{i, 1, k}^{I} \\ ⋮ \\ L_{i, m, k}^{I} \end{matrix}], P_{i, k}^{I} = [\begin{matrix} P_{i, 1, k}^{I} \\ ⋱ \\ P_{i, m, k}^{I} \end{matrix}],

_$m$ is the number of corresponding points in the ith CCP.

Subsequently, $x_{k}^{I}$ may also be estimated according to the least squares theory as Eq. (15):

M_{k}^{I} x_{k}^{I} = W_{k}^{I}

where

M_{k}^{I} = \sum_{i = 1}^{n} (A_{i, k}^{T} P_{i, k}^{E} A_{i, k} - A_{i, k}^{T} P_{i, k}^{E} B_{i, k} {(B_{i, k}^{T} P_{i, k}^{E} B_{i, k})}^{- 1} B_{i, k}^{T} P_{i, k}^{E} A_{i, k})

W_{k}^{I} = \sum_{i = 1}^{n} (A_{i, k}^{T} P_{i, k}^{E} L_{i, k}^{E} - A_{i, k}^{T} P_{i, k}^{E} B_{i, k} {(B_{i, k}^{T} P_{i, k}^{E} B_{i, k})}^{- 1} B_{i, k}^{T} P_{i, k}^{E} L_{i, k}^{E})

and

n

is the number of CCP.

Finally, $X_{k}^{I}$ can be updated by Eq. (16). The estimation is also an iterative process, and will not be undertaken until the estimated internal parameters tend to be constant.

X_{k + 1}^{I} = X_{k}^{I} + x_{k}^{I}

5. Experimental results and discussion

5.1 Approach and feasibility of experiment

At present, satellites equipped with imaging capabilities, such as those described above, are in the minority. Only the PLEIADES-1A/1B satellite is known to have these capabilities [16], but is unsuitable for our purposes, as geometric calibration requires confidential satellite parameters and initial imagery. To verify the proposed method satisfactorily, we designed an original and rigorous simulation experiment (Fig. 6).

Fig. 6 Flow of simulated experiment.

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First, multi-attitude images consistent with the imaging conditions in this paper were simulated to emulate the rigorous imaging process of a satellite; the imaging model used was an ideal model devoid of deformities. Second, three internal camera systems with different systematic distortions were designed to obtain three sets of internal camera parameters. Third, the camera parameters (internal and external) with the designed errors, served as the initial values for the geometric calibration in accordance with the proposed method. Fourth, the geometric accuracy was verified by comparing the LOSs determined by the calibrated parameters and the camera parameters designed to be devoid of irregularities.

This experiment may be regarded as a procedure that is inverse to the more usual mode of geometric calibration. Although the procedure differs from the normal geometric calibration process, in that the true camera used for imaging is deformed (whereas the initial parameters in the calibration are ideal), both processes verify that the proposed method is effective for calibrating the systematic errors inherent in the imaging model, considering that the essential purpose of geometric calibration is to determine systematic bias relative to the initial camera parameters. Additionally, this inverse experiment boasts an incomparable merit: the images only need to be simulated once using the designed camera without systematic distortion; it is unnecessary to simulate different images for different internal camera systems having different distortions, and so the experiment is more cost-effective.

5.2 Simulation of experimental data set

5.2.1 Simulation of multi-attitude images

The camera was designed with a central FOV, in which the CCDs were completely parallel to the X axis of the focal plane coordinate system, and the centroid of the CCDs coincided completely with the coordinate origin. The physical parameters used for the simulation of the multi-images are presented in Table 1.

Table 1. Basic parameters of designed satellite

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On the basis of the physical parameters, we estimated the internal viewing-angle coefficients in accordance with the rigorous camera model. These parameters can determine the LOS of each CCD detector precisely, and can serve as true values for accuracy assessment. The precise DOM and DSM of the Chinese Songshan calibration site were then used as the basic data for simulating the multi-attitude images. The resolutions of the DOM and DSM were as high as 0.2 m and 1.0 m to restrain potential errors from interpolating the DSM. To ensure the truth and rationality of the orbit and attitude of the designed satellite, these parameters were obtained by proportionately interpolating the actual auxiliary data from the ZiYuan-3 satellite. In accordance with the imaging conditions, the multi-attitude images were simulated based on the rigorous imaging model.

The simulated multi-attitude images are illustrated in Fig. 7, in which Figs. 7(a) and 7(d) represent the push-broom image simulated under 0° and 180° yaw attitude angles, respectively, with standard nadir imagery, and the attitudes of images Figs. 7(b) and 7(c) can subsequently be determined according to the 50% overlap with image Fig. 7(a). Both these images can furthermore be simulated by applying the same method to Figs. 7(a) and 7(d). The average baseline length of the simulated images is around 5.3 km.

Fig. 7 The simulated multi-attitude images.

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5.2.2 Simulation of sensor internal distortion

To verify the feasibility of the proposed method, the internal sensor parameters had to be determined with distortions. For this experiment, we designed three sets of internal distortions, which can be described as Sinx, Sqrtx and X2 [19]. As illustrated in Fig. 8, the designed distortion in direction X is up to 2 pixels at either end of the CCD, achieved by changing the size of the CCD detector (the effect of the detector size error is equivalent to the effect of the focal length error), and the maximum error in direction Y is up to 10 pixels. Additionally, the internal sensor distortions were designed holistically, as the designed distortions are of a high-order and can incorporate the CCD error, focal length error and optical error. As such, it was not necessary to design the errors separately.

Fig. 8 The designed internal distortions with different orders.

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According to the designed distortions, the internal parameters, i.e., the coefficients of the viewing-angle model, may be determined with high accuracy through mathematical fitting. These parameters represent the designed internal distortion precisely, and will be used as inputs for the proposed calibration method.

5.3 Assessment of calibration accuracy

The LOS accuracy can be used to estimate the accuracy of on-orbit calibration. The statistical residual between the calculated and true LOS of each detector was used as an index of LOS accuracy. The main statistical index can be calculated by Eq. (17):

R M S_{x} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {(φ_{x} - φ'_{x})}_{i}^{2}}, R M S_{y} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {(φ_{y} - φ'_{y})}_{i}^{2}}

where

m

is the sum of the CCD detectors, and

(φ'_{x}, φ'_{y})

and

(φ_{x}, φ_{y})

are the estimated and true LOS in both directions, respectively.

5.3.1 External calibration accuracy

First, the feasibility of the proposed external calibration method was verified. As described in this paper, only the pitch and roll of the installation angles were estimated in the external calibration. In verifying the accuracy of the external calibration, the effect of the internal distortion should be disregarded. As such, the true designed values of internal parameters were used to avoid incorporating the internal distortion into the estimated external parameters. Additionally, the effect of the elevation errors on the external calibration accuracy was also assessed, in which the planes of −1,000 m, 0 m, 500 m, and 9,000 m, and a DSM, were used as reference elevations. The adjustment solutions based on these elevation references all achieved optimal estimation within three iterations. The estimated external calibration results based on these elevations are listed in Table 2.

Table 2. Estimated external parameters based on different elevations

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Since the true designed values of pitch and roll are both zero, a relatively high calibration accuracy was achieved when we performed the external geometric calibration based on the DSM. When the DSM was replaced with a surface height with constant elevation, the calibration accuracy deteriorated. Considering that the average elevation of the test area is around 430m, and the topographic relief is around 100 m, we may conclude that the external accuracy deteriorates gradually with the increase in elevation error. This decline is extremely limited, however, even when the reference surface height reaches the limit of the earth surface’s elevation. As such, a satisfactory calibration accuracy can be achieved, as long as a rough estimate is achieved for the reference elevation of the test area prior to completing the external calibration. For practical utility, the question of whether it is appropriate to use DSM, and the estimated elevation of the fixed surface height, if used, should be determined according to the intersection conditions of the images. If the intersection angle (baseline) of the images is limited, a rough reference elevation usually suffices for external calibration without using calibration sites.

5.3.2 Internal calibration accuracy

The feasibility of the proposed internal calibration method was subsequently verified. Just as in the assessment of the external calibration, the effects of the external distortions should also be disregarded. As such, the designed installation angles (0, 0, 0) were used to avoid incorporating the external errors into the estimated internal parameters. Additionally, the effect of the elevation errors on the accuracy of the internal calibration was also evaluated by adding systematic errors of varying sizes to the elevation interpolated from the reference DSM. Based on the initial internal parameters Sinx, all optimal solutions finished within 5 iterations, and the estimated parameters for the various elevation errors are listed in Table 3.

Table 3. Estimated internal parameters based on DSM with different errors

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Based on both the estimated and true designed internal parameters, the residual of the LOS may be calculated for all CCD detectors. Therefore, the root mean square (RMS), mean value (Mean) and maximum value (Max) of these residuals can be determined. Since the camera installation angles used in the internal calibration are the same as the design values, the statistical residuals in the camera coordinate system may be used to evaluate the calibration accuracy. The statistical results are listed in Table 4.

Table 4. Accuracy assessment for the internal calibration based on DSM with different errors

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As may be seen from Table 4, for the internal calibration based on DSM (with an added elevation error of zero), the Mean of all LOS residuals was better than 0.001 arc seconds in both the X and Y directions, while the maximum residual was better than 0.06 arc seconds in both directions. The RMSs in the X and Y directions were both better than 0.003 arc seconds, indicating that a high internal calibration accuracy and good geometric consistency of the designed sensor were achieved. Furthermore, the proposed internal calibration method can achieve optimal estimates of internal parameters without using calibration sites. With an increase in the elevation error, the calibration accuracy deteriorates gradually according to the overall RMS (RMS of XY). To demonstrate this tendency toward variation more clearly, the variation in overall calibration accuracy with the increase in the elevation error is illustrated in Fig. 9.

Fig. 9 The variation of internal calibration accuracy with elevation error.

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It is clear that the relationship between the internal calibration accuracy and the elevation error is almost linear, which is consistent with the analysis presented in section 4.1. Furthermore, the importance of the reference elevation constraint in the internal calibration is further elucidated. In accordance with the designed physical parameters, 0.03 arc seconds is equivalent to around 0.4 pixels. As such, the proposed method can still achieve a sub-pixel calibration accuracy, despite an introduced elevation error as high as 50 m for the simulated images. In practical application, the accuracy of the adopted DSM depends on the intersection condition of the multi-images. For images with large baseline lengths, a flat area should be chosen to minimize the effects of the elevation error from the reference DSM. It is fortunate, however, that the internal accuracy is not sensitive to the elevation error, thereby reducing the requirement for calibration accuracy to achieve DSM quality; such DSM is far easier and more cost efficient with respect to acquiring calibration site data.

5.3.3 Evaluation of overall accuracy

In both experiments outlined above, the feasibilities of the external and internal calibrations were verified separately. However, the overall accuracy and effectiveness has not yet been verified, and neither has the compensatory effect of the internal parameters on the yaw of the installation angle. In this section, an experiment aiming to evaluate both of these points is proposed. First, the initial internal parameters corresponding to three designed distortions were used as true values, and subsequently the external parameters were estimated. Since the DSM was still required for the internal calibration, we also used DSM as the elevation reference for the external calibration. The estimated external parameters (pitch and roll) for the different internal parameters and yaw (not estimated) are listed in Table 5.

Table 5. Estimated external parameters based on different internal distortions

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The internal parameters were then estimated in a generalized camera frame (satellite body coordinate system) determined by the above estimated external parameters. The corresponding estimated internal parameters are listed in Table 6.

Table 6. Estimated internal parameters for different internal distortions

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Finally, the overall accuracy of the calibration was verified based on the estimated and designed camera parameters. However, owing to the mutual compensation between the internal and external parameters, the low-order error inherent in the internal distortion was incorporated into the estimated external parameters during the external calibration process, and the remaining high-order errors were then compensated by the estimated internal parameters during the internal calibration process. Consequently, the proposed method is generalized; neither the estimated internal nor external parameters are accurate when used separately, but when used in tandem can achieve very high calibration accuracy. Thus, compensation of the external parameters should be considered when on increasing the calibration accuracy. Furthermore, the statistical residuals of the LOS in the body coordinate system of the satellite should be used to index accuracy, rather than that in the coordinate system of the camera. The statistical results of the LOS residuals, in the coordinate system of the camera and the body coordinate system of the satellite, are compared in Table 7.

Table 7. Overall calibration accuracy for different internal distortions

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As shown in Table 7, the statistical accuracy of the coordinate system of the camera is as low as 0.1 arc seconds, because the estimated internal parameters absorb the errors of the external parameters, particularly the yaw angle error (which extends to 0.001 rad). The statistical accuracy is improved to around 0.002 arc seconds; however, the accuracy is determined by the body coordinate system of the satellite, due to the compensation of the external parameters. Thus, we may conclude that the internal parameters are effective in compensating for the error in the yaw angle; furthermore, the calibration parameters used in this method are reasonable. Since a pixel is roughly equivalent to 0.075 arc seconds, according to the designed physical parameters, the overall calibration accuracy is about 0.04 pixels in reference to the RMS, with a maximum of around 0.1 pixels. This result is consistent across the three designed sets of internal distortions, indicating the efficacy and soundness of the proposed method. Based on the calibrated parameters, estimated geometric residual curves of Sinx were drawn in accordance with both the coordinate system of the camera and the body coordinate system of the satellite to more clearly demonstrate satisfactory realization of the calibration. The residual curves are illustrated in Fig. 10.

Fig. 10 The estimated geometric residual curves of Sinx.

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It is evident from Fig. 10(a) that a rotation error is present in the internal parameters. This was caused by the installation error in the yaw direction’s absorption into the internal parameters during the internal calibration process, based on the external parameters with the error considered. However, this error is compensated perfectly when the calibration result is assessed in accordance with the body coordinate system of the satellite (Fig. 10(b)). The mutual compensation of the internal and external parameters results in a satisfactory overall geometric accuracy of better than 0.1 pixels in both directions. Additionally, the good geometric consistency in the errors of all CCD detectors indicates the soundness of the proposed viewing-angle model and the adopted calibration parameters. Consequently, the proposed independent geometric calibration method without recourse to GCPs may be deemed effective, and the image combination is appropriate for high-accuracy geometric calibration, offering a new technique for high-precision geometric processing of satellites with potential as a guiding principle for future design of satellite control systems.

6. Conclusion

We proposed a new on-orbit geometric calibration method for linear AOS using the self-constraint of multi-attitude images. Based on the analysis of systematic errors present in the rigorous imaging model, the calibration parameters applied to the discovered images combination were defined using equivalent compensations. In formulating the calibration model, the self-constraint of the corresponding bundles was used in place of GCPs to estimate the optimal calibration parameters. A reverse experiment was conducted on the simulated multi-attitude images to verify the feasibility and accuracy of the proposed method. Through theoretical analysis and experimental validation, we reached the following conclusions:

1) The satisfactory convergence and accuracy of the external calibration indicate that the imagery combination used in the external calibration was reasonable, and that the elevation reference is essential to ensure the stable estimation of the external calibration. However, the limited variation in calibration accuracy among the different reference elevations also indicates that a rough elevation is usually sufficient to ensure accuracy in the external calibration.
2) The internal calibration results also indicate that the imagery combination used for internal calibration was reasonable, but a DSM with specific accuracy is essential for internal calibration, considering the correlation between elevation and internal calibration accuracy.
3) An overall theoretical accuracy of around 0.002 arc seconds in both directions was achieved, indicating the feasibility of the method for calibrating the systematic errors present in the rigorous imaging model of the AOS.

Consequently, this investigation indicates that the satellite can be used independently to achieve geometric accuracy, without recourse to absolute references, to perform on-orbit calibration. This result is not only significant for the practical application of independent on-orbit calibration, but also for follow-on satellite design. Future efforts will focus on verifying the application of this method to real AOS.

Funding

National Natural Science Foundation of China (NSFC) (41601492, 91638201); National Key Research and Development Program of China (2016YFB0501402).

Acknowledgments

The authors would like to thank the editors and anonymous reviewers for their valuable comments, and checking the English in this paper, which helped improve this manuscript.

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Item	Value
Focal length	5.5 (m)
Number of CCD detector	6000
Size of CCD detector	2.0(µm)
Orbit height	500 (km)
Ground sample distance	0.2 (m)
Camera installation angle	(0,0,0)

Reference elevation	pitch/rad	roll/rad
True value	0.0	0.0
Initial value	0.1	0.1
H = DSM	6.56149E-09	6.92743E-12
H = 500m	1.96664E-07	3.60682E-11
H = 0m	1.13732E-06	1.91999E-10
H = −1,000m	3.79347E-06	5.09642E-10
H = 9,000m	2.32543E-05	2.71499E-09

Error/m	a₀	a₁	a₂	a₃	b₀	b₁	b₂	b₃
0	−1.090909E-03	3.63636E-07	1.36E-17	−2.29E-21	−2.74363E-08	2.1467E-11	−5.34E-15	5.39E-19
10	−1.090909E-03	3.63636E-07	4.24E-17	−5.93E-21	2.09647E-08	3.6750E-11	−2.35E-14	3.35E-18
20	−1.090909E-03	3.63636E-07	8.22E-17	−1.06E-20	−3.94829E-08	1.5421E-10	−6.84E-14	8.42E-18
30	−1.090909E-03	3.63636E-07	1.01E-16	−1.38E-20	−3.89020E-08	2.2979E-10	−9.93E-14	1.15E-17
40	−1.090909E-03	3.63636E-07	7.09E-17	−8.87E-21	1.13912E-07	1.4609E-10	−8.04E-14	9.60E-18
50	−1.090909E-03	3.63636E-07	7.98E-17	−1.02E-20	1.32217E-07	1.8299E-10	−9.59E-14	1.11E-17

Error/m	RMS/arc seconds			Mean/arc seconds		Max/arc seconds
Error/m	X	Y	XY	X	Y	X	Y
0	0.0000116	0.0025415	0.0025415	−0.0000040	−0.0003997	0.0000337	0.0056591
10	0.0000132	0.0079225	0.0079225	0.0000089	−0.0062286	0.0000464	0.0246160
20	0.0000136	0.0153590	0.0153590	0.0000097	−0.0116270	0.0000434	0.0495260
30	0.0000135	0.0195570	0.0195570	0.0000061	−0.0166350	0.0000380	0.0518620
40	0.0000137	0.0251600	0.0251600	0.0000112	−0.0217870	0.0000413	0.0392270
50	0.0000130	0.0306540	0.0306540	0.0000091	−0.0265780	0.0000424	0.0480270

Internal parameter	a₀	a₁	a₂	a₃	b₀	b₁	b₂	b₃
Sinx	−1.09091E-03	3.63636E-07	1.116E-17	−2.11E-21	−1.0449E-06	3.85825E-10	−4.820E-15	4.37E-19
Sqrtx	−1.09091E-03	3.63636E-07	9.340E-18	−2.55E-21	−1.1053E-06	3.83256E-10	−5.561E-15	6.63E-19
X2	−1.09091E-03	3.63636E-07	8.791E-17	−8.69E-21	6.2262E-07	3.99339E-10	−9.017E-15	7.03E-19

Study of full-link on-orbit geometric calibration using multi-attitude imaging with linear agile optical satellite

Abstract

1. Introduction

2. Literature review

3. Analysis of systematic errors and establishment of the calibration model

3.1 Analysis of systematic errors

3.2 Rigorous imaging model of linear AOS

3.3 Calibration model based on equivalent compensation

3.3.1 Yaw angle

3.3.2 Internal parameters

4. Approach and feasibility of the method

4.1 Effect of elevation error on calibration accuracy

4.2 Adjustment model for geometric calibration

4.3 External calibration

4.3.1 Constraint of multi-attitude images

4.3.2 Estimation of external parameters

4.4 Internal calibration

4.4.1 Constraint of multi-attitude images

4.4.2 Estimation of internal parameters

5. Experimental results and discussion

5.1 Approach and feasibility of experiment

5.2 Simulation of experimental data set

5.2.1 Simulation of multi-attitude images

5.2.2 Simulation of sensor internal distortion

5.3 Assessment of calibration accuracy

5.3.1 External calibration accuracy

5.3.2 Internal calibration accuracy

5.3.3 Evaluation of overall accuracy

6. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (10)

Tables (7)

Equations (25)

Optics Express

Parameters	pitch/rad	roll/rad	yaw/rad
True values	0.0	0.0	0.0
Initial values	0.001	0.001	0.001
Sinx	8.248394E-08	6.376867E-12	0.001
Sqrtx	1.273133E-08	7.992140E-11	0.001
X2	1.751164E-06	1.835072E-09	0.001

Internal parameters		RMS/arc seconds			Mean/arc seconds		Max/arc seconds
Internal parameters		X	Y	XY	X	Y	X	Y
Sinx	Cam	0.000066	0.133600	0.133600	0.000011	−0.016150	0.000103	0.245600
Sinx	Body	0.000014	0.002854	0.002854	0.000007	−0.000862	0.000037	0.007525
Sqrtx	Cam	0.000067	0.132700	0.132700	0.000018	−0.002793	0.000116	0.234600
Sqrtx	Body	0.000013	0.002831	0.002831	0.000001	0.000167	0.000050	0.006924
X2	Cam	0.000738	0.384200	0.384200	0.000736	−0.361000	0.000799	0.587000
X2	Body	0.000018	0.002244	0.002244	0.000004	−0.000165	0.000056	0.007763