Online refocusing algorithm for a satellite camera using stellar sources

Transcription

1 Online refocusing algorithm for a satellite camera using stellar sources Jeong-Bin Jo, Jai-Hyuk Hwang, * and Jae-Sung Bae School of Aerospace and Mechanical Engineering, Korea Aerospace University, Gyeonggi-do, South Korea * jhhwang@kau.ac.kr Abstract: In this study, an online refocusing algorithm is proposed for a satellite camera performing an Earth observation mission. Satellite cameras are vulnerable to misalignment in orbit because of their severe launching environments and the thermal vacuum environment in space. The proposed online refocusing algorithm is able to guarantee high quality images by aligning the satellite camera in real time. This alignment is achieved by precisely adjusting the movement mechanism of the secondary mirror (M2) and the focal plane. The target optical system used in this study was originally designed for the purposes of algorithm development. The system uses a Schmidt-Cassegrain-type satellite camera with a 200-mm diameter primary mirror (M1). The ground sampling distance (GSD) is 3.8 m from an altitude of 700 km. A fourth-order equation model is derived for the modulation transfer function (MTF) variation tendency for M2 de-spacing. Following this, the proposed online refocusing algorithm for the target optical system is developed. The algorithm is able to assess the de-space position from the MTF measurements using stellar sources. It is determined from the simulation that any misaligned satellite camera can be refocused within a ± 0.5μm M2 de-space error by applying the proposed refocusing algorithm in real time Optical Society of America OCIS codes: ( ) Image quality assessment; ( ) Modulation transfer function; ( ) Remote sensing and sensors. References and links 1. J. Yu, Satellite attitude scenario for image validation using stellar sources, Master s Thesis, Yonsei University (2011). 2. A. Meygret and D. Léger, In-flight refocusing of the SPOT1 HRV cameras, Aerospace/Defense Sensing and Controls, International Society for Optics and Photonics, (1966). 3. D. Léger, F. Viallefont-Robinet, E. Hillairet, and A. Meygret, In-flight refocusing and MTF assessment of SPOT5 HRG and HRS cameras, Progress in Biomedical Optics and Imaging (SPIE Proceedings Series) 4881, (2003). 4. F. Viallefont-Robinet, Edge method for on-orbit defocus assessment, Opt. Express 18(20), (2010). 5. J. Hwang, J. Yang, J. Park, J. Jo, M. Kang, and J. Bae, Design of the Active Optical Compensation Movements for Image Stabilization of Small Satellite, J. Kor. Soc. Aeronautical Space Sci. 43(5), (2015). 6. J. Jo, J. Hwang, and J. Bae, Optical Design for Satellite Camera with Online Optical Compensation Movements, Journal of the Korean Society for Aeronautical and Space Sciences 43(3), (2015). 7. D. Kim, D. Kim, N. Kim, J. Suk, H. Kim, G. Kim, and Y. Hyun, A Study on Basic Modeling Method for MTF Analysis of Observation Satellites, J. Kor. Soc. Aeronautical Space Sci. 36(5), (2008) Y. Cho, Characteristics of the Image Quality Parameter in Satellite Imaging Instrument, Aerospace Engineering and Technology 1(2), (2002). 10. E. Hecht, Optics, 4th ed. (Addison-Wesley, San Francisco, 2002) Chap D. Helder, T. Choi, and M. Rangaswamy, In-flight characterization of spatial quality using point spread functions, International Workshop on Radiometric and Geometric Calibration, (2004). (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5411

2 1. Introduction For a low orbit observation satellite performing an Earth observation mission, the satellite s optical system requires position precision within microns to ensure optical performance. Therefore, for successful mission implementation, an optical alignment process must be carried out before the satellite is launched. However, due to the severe launching environment and thermal vacuum environment in space, it is possible for the optical system s alignment to become distorted, which would in turn lead to image quality degradation. This is a serious problem; image quality degradation can result in the mission failure of the satellite. Therefore, in order to ensure high quality images, it is necessary to develop an algorithm that is able to carry out in-orbit, real-time autofocusing. Refocusing refers to the process in which a misalignment (de-space or defocus) is rectified. This involves returning the optical components to the normal position toward the direction of the optical axis. Off-line refocusing processes have been utilized for existing satellites to assess image quality after taking the images of a target object and then to determine the degree of defocus in the ground control center. Currently, the French SPOT and Pleiades series and the Korean KOMPSAT-3 series are using off-line refocusing performance mechanisms [1 4]. The refocusing processes for the aforementioned satellites rely on having the images transmitted to a ground control center. At the center, the degree of de-space is assessed, and the results are transmitted back to the satellite to enable refocusing to take place. The entire process requires significant financial resources and labor [2,3]. Therefore, the purpose of this study is to develop a refocusing algorithm that enables autofocusing in real time. In order to implement real-time refocusing, the algorithm requires compensation mechanisms to move the secondary mirror (M2) and focal plane toward the optic axis with precision. The compensation devices should be able to precisely control the relative distance in the optic axis from the primary mirror (M1) to M2, or from the focal plane to M2. The conceptual shape of the satellite camera mounted with a compensation device is shown in Fig. 1 [5]. The modulation transfer function (MTF) is used for the index to assess image quality in the refocusing algorithm. The MTF can provide analytical computations in the design stage of the optical system and provide measurements during the assembly and test stages. Also, after the satellite is launched, measurements can be acquired while in orbit by taking images of ground objects or stars. This index can evaluate optical systems from the design stage right to the very final stage in-orbit operation [5]. In this paper, an effective refocusing algorithm that precisely controls the position of the compensation devices is investigated. First, a target optical system, to which the refocusing algorithm will be applied, is designed. For this purpose, the specifications of an existing observation satellite camera are analyzed to determine the requirements and design an optical system that will satisfy those requirements. In the designed target optical system, by using the MTF variation tendency of the M2 and focal plane, a fourth-order equation model is derived and the online refocusing algorithm for the target optical system is developed. The proposed algorithm is able to assess the degree of de-space using MTF measurements of stellar sources. (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5412

3 Fig. 1. Conceptual design of the satellite camera equipped with compensation movements [5]. 2. Design of target optical system 2.1 Establishment of optical system design requirements In order to develop an algorithm that enables real-time refocusing alignments, an optical program (Code V) is used to model the target optical system in this study. The requirements of the target optical system are determined by referring to specifications for satellites that are currently in operation. Table 1 shows the specifications for SPOT-5/6/7, Pleiades, and KOMPSAT-3, all of which are using processes to carry out in-orbit refocusing. They are midsize to mid-large size satellites at low orbits of about 700 km with three-mirror anastigmat (TMA, Korsch) type optical systems whose M1 diameters are between 200 and 800 mm. In this study, considering the manufacturing of the optical system and the application of the algorithm, the diameter of M1 is determined to be 200 mm. At present, 200-mm M1s are the most commonly purchased M1s for use with commercial astronomical telescopes. It is easy to manufacture the optical system at the laboratory level [6]. 700 km is selected as the operation altitude; this is advantageous in that the visible radiation band can be observed. The performance of the ground sampling distance (GSD) for a satellite camera is determined as the distance covered when one pixel of the camera is projected on the ground. This relates to resolution power, and it is an important index for observation satellites. In Table 1, the GSD performance of the SPOT series with a 200-mm M1 is 3.5 m. This value is calculated based on the condition that several of the satellite s cameras take images of the same area on the ground several times for the post-processing of the images. The GSD performance is higher than that of the optical system itself. Therefore, in this study, the GSD requirement for the optical system with a 200-mm diameter M1 is determined within a 4-m GSD. The MTF performance of the optical camera is calculated as the multiplication of the MTF of the optical system by the MTF of the image sensor, as shown in the Eq. (1) [7]. MTFPayload = MTFOptics MTFDetector (1) When the MTF of the image sensor is assumed to be 40%, in order to ensure that the final MTF performance of electro-optical system(eos) in orbit is 10%, the MTF performance of the optical system should be above 25% (at a Nyquist frequency). In this study, the MTF requirement of the optical system is determined to be 30%. This value is determined by considering the performance degradation at the manufacturing and assembly stages. The contents are organized in Table 2. Table 1. Satellite camera data [8] SPOT-5/6/7 Pleiades KOMPSAT-3 Developers France France Korea Telescope type TMA (Korsch) TMA (Korsch) TMA (Korsch) Altitude 694 km 700 km 685 km GSD (PAN/MS) 1.5~3.5m / 6~10m 0.7m / 2.8m 0.7m / 2.8m Aperture diameter 200 mm 650 mm 800 mm Swath width 10~60 km 20 km 15 km (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5413

4 Table 2. Target optical system requirements Aperture diameter Altitude GSD MTF 200 mm 700 km 4m Nyquist freq. design, Nyquist freq. 2.2 Design of target optical system and its performance evaluation A Schmidt-Cassegrain-type target optical system is designed. The manufacturing costs for Schmidt-Cassegrain systems are much lower than the costs for other systems that have the same M1 size. It also offers advantages in manufacturing and assembly at the laboratory level because the number of optical components is small. The Schmidt-Cassegrain system is composed of a Schmidt plate, M1, and M2. The shape of the M1 is spherical, which enables the reduction in manufacturing costs. The Schmidt plate can correct the spherical aberration of the M1 and serve as a corrective lens in front of the image sensor, thus making it effective despite the reduction in the number of optical components. The 3-dimensional layout of the designed optical system is shown in Fig. 2. Equation (2) is used to determine the GSD [9]. In this equation, P x is the pixel size of the image sensor, EFL is the effective focal length of the optical system, H is the altitude, f is the f number of the optical system, and D is the diameter of the entrance pupil. Therefore, this design should be applied to an optical system with a 200-mm diameter M1 to have the appropriate f number and pixel size in order to obtain a GSD performance of less or equal to 4 m at the required 700 km altitude. Px Px GSD = H = H (2) EFL f D In this study, it is presumed that a 10-μm pixel size sensor is used. Accordingly, the f number of the optical system is designed to be Therefore, the GSD value computed by Eq. (2) is 3.8 m, which satisfies the design requirements. The MTF performance of the designed target optical system is shown in Fig. 3. The Nyquist frequency is the bandwidth of a sampled signal, and is equal to half the sampling frequency of that signal. At the optical system, the sampling frequency of a signal is determined by the pixel size of the detector. As it is presumed that the pixel size of the image 1 sensor is 10 μm, the Nyquist frequency becomes 50 cycle/mm ( ) [1]. At this frequency, 2Px it can be verified that the MTF performance in every field (0, 0.7, full field in the tangential(t)/sagittal(s) orientations) exceeds 30% of the design requirements [10]. (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5414

5 Fig. 2. 3D layout of the designed optical system. Fig. 3. MTF of the designed optical system. 3. De-space and MTF The MTF is an important index when evaluating the optical system in the overall development stages. Therefore, the MTF is also used as an index to assess the degree of despace in the algorithm proposed in this study. The methods to measure the MTF while in orbit vary by the type of target object. For this study, the impulse input method is used. When a point source is used as an input, an advantage is gained in that the MTF toward the x and y directions can be computed instantaneously. The handling method is also simple. Examples of target objects that correspond to the impulse input are a spotlight on the ground, the sunlight reflecting off of a convex mirror, and a stellar source in space [11]. In this study, the MTF is measured using a stellar source that can be observed easily from orbit. The handling method is relatively simple, and the algorithm is applied to carry out the autofocusing alignment. Information regarding the MTF measurement method utilizing the stellar source images will be provided in the next section. To develop an algorithm to carry out online autofocusing alignments, the variation tendency of the MTF to the de-space of the optical system is first analyzed. The tendency analysis is carried out for the designed target optical system. Figure 4 shows the variation of MTF values according to the de-space of the focal plane and M2. The MTF values are measured at the 0 field on the optical axis for tangential and sagittal orientations. Figure 4 shows optic axis, tangential and sagittal orientations, and de-space of M2 and focal plane. The autofocusing algorithm is developed so that the refocusing can also be done using the MTF value measurements in the center range of the image sensor. (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5415

6 y Sagittal Pattern Schmidt Plate Primary Mirror Tangential Pattern x y Optical Axis Secondary Mirror Focal Plane x dzsm dz fp Fig. 4. Schematic diagram of optical system. In the graph in Fig. 5(a), an MTF vertex according to the de-space of M2 occurs at about the 3.23 μm de-space position. In the case of the focal plane in Fig. 5(b), it is verified that an MTF peak point according to the de-space of the focal plane occurs nearby at about μm. This means that the sensitivity of M2 with respect to the MTF is about 21.7 times higher than that of the focal plane when the de-space of M2 occurs along the optic axis. In other words, a 1-μm de-space of M2 causes an approximate 21.7-μm de-space of the focal plane. As shown in Fig. 5, the MTF curve according to the de-space appears symmetrically near its vertex. Therefore, in Fig. 5(a), the MTF curve between 15 μm and + 10 μm is appropriate for curve fitting with a quartic function. Using this tendency, it is possible to estimate the degree of de-space when the optical components are defocused in orbit. The tendency of the MTF according to de-space is an important element in composing the autofocusing algorithm. 4. Refocusing algorithm 4.1 Existing refocusing methods Fig. 5. MTF vs de-space position. This section summarizes the refocusing methods used by the SPOT and Pleiades series. Both series have conducted off-line refocusing for satellites in orbit. The first SPOT satellite was launched in 1986, and SPOT-7 was put into operation in Refocusing for SPOT satellites (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5416

7 has been carried out by moving the de-space mechanism toward the +/ steps around the initial position and taking pictures of target objects on the ground. The images are downloaded to the ground control center, where the degree of de-space is assessed. In this method, it is possible for errors to be made when the vertex of the MTF curve is not included within the range of the +/ steps. Therefore, the process requires significant time and the use of more than eight images [2,3]. The Pleiades satellite was launched in 2011, and refocusing has been carried out using two methods. In the first method, the on-orbit optical transfer function is analyzed after taking checkerboard images on the ground. The degree of defocusing is estimated using the computation formula in reverse. In this method, a parametric optical transfer function model is used to find the parameters and the de-space simultaneously by fitting the optical transfer function model on the MTF measurement [4]. This method can only estimate the degree of de-space in the event of a large de-space, and its application is thus limited. In the second method, similar to the SPOT method, refocusing is carried out by moving the de-space mechanism toward the +/ steps, taking pictures of target objects, analyzing the taken images, and assessing the de-space [4]. As the examples above demonstrate, refocusing algorithms have thus far involved multiple iteration stages; images need to be taken to measure the MTF initially, and an analysis of the images then needs to be conducted to determine the degree of de-space. This iteration costs time and money, particularly when a ground control center is required to carry out the analysis. It also requires many satellite images to be taken in order to assess the despace. 4.2 Proposed refocusing algorithm An online refocusing alignment algorithm is developed to mitigate the limitations of existing refocusing methods. When a refocusing process is carried out online, as is done with existing refocusing methods, the technique of assigning +/ de-space steps is inefficient. The algorithm proposed in the present study is designed to carry out autofocusing in real time via two stages; the MTF is measured by taking images in Stage 1, and the degree of de-space is assessed by analyzing the measured MTF in Stage 2. In the algorithm, the MTF variation, which occurs due to the movement of compensation devices, is intentionally induced so as to efficiently assess the degree of de-space using the mathematical MTF tendency model. As discussed in References [2] and [4], mathematical models created from simulations are able to provide accurate results from MTFs obtained from images taken while in orbit. Therefore, if the MTF variation tendency according to the change of de-space is expressed with a mathematical equation, it becomes a very efficient method of finding the degree of de-space. When refocusing is carried out in real time, the alignment algorithm imports and exports the input and output files with the M2 and focal plane compensation mechanism, as shown in Fig. 6. The data file imported into the online refocusing algorithm is the information from the stellar source image taken by the satellite camera. The data file imported into the compensation mechanism is the command file to run the compensation mechanism. Between two and five iterations are carried out, after which the autofocusing alignment can be completed and excellent image quality can be obtained. For the simulation to verify the algorithm, an optical program (Code V) is used instead of a satellite camera, and the output file from the autofocusing algorithm is a command file, which moves the positions of optical components in the optical program. In the simulation, refocusing can be carried out within a ± 0.5-μm M2 de-space error when importing and exporting the input and output files in real time. (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5417

8 Fig. 6. Schematic diagram for the online optical alignment algorithm MTF measurements MTF values measured by using stellar source images are used for the index to assess the degree of de-space in the online refocusing algorithm. The characteristics of particular stars are well understood, and they are not affected by the atmosphere, seasons, or the climate on the ground. This is advantageous when measuring the MTF in orbit, and it allows the MTF to be computed directly into the algorithm [1,11]. Figure 7 shows a flow chart in which the MTF is computed by using stellar source images. The source point spread function can be obtained from the stellar source images. If necessary, the samples can be increased by using a mathematical function such as the Gaussian or Moffat functions [1]. The MTF can be obtained using the absolute values of the optical transfer function, which is the Fourier transform of the point spread function. In the refocusing algorithm, as the degree of de-space is computed by using the mathematical equation from the MTF tendency at a Nyquist frequency, the algorithm is designed to obtain MTF values at the Nyquist frequency using stellar source images in orbit. Fig. 7. Flow chart of the MTF measurement using a stellar source De-space assessment As discussed in the previous section, the MTF variation tendency according to the M2 despace is derived mathematically using Eq. (3), which expresses the curve fitting of the MTF values at a nearby peak point. A graph depicting the MTF value and curve fitting model is (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5418

9 shown in Fig. 8. In this equation, dz is the M2 de-space value, and the MTF is a function of dz. MTF dz = dz + dz dz dz + (3) ( ) field With this mathematical model, the degree of M2 de-space can be obtained immediately when the M2 de-space is between 15 μm and + 10 μm. The autofocusing algorithm is designed to assess the degree of de-space after computing the value of the MTF in the measured star images, and with this value, the image can be distinguished either in or out of the range. This enables the degree of de-space to be computed immediately if it is within the range. If it is out of the range, the position of M2 can be pushed into the range and the degree of de-space can be determined. As the MTF tendency model is a function obtained with an MTF curve computed through an optical component, it is necessary for the model to consider the MTF function of the image sensor when an actual optical system is manufactured field(x) Curve Fitting Model MTF M2 de-space(μm) Fig. 8. Curve fitting model for the MTF tendency Sequence diagram of the proposed algorithm In existing refocusing techniques, a restriction exists in that it is mandatory to include the peak point of the MTF curve in the +/ step movements. If this is explained in view of the target optical system, the M2 +/ step movements should be between 15 μm and + 10 μm. Therefore, if the peak point of the MTF is not included in the M2 +/ steps, the satellite will revert to the initial stage in which the images are taken. This will increase the amount of time needed for refocusing. However, the proposed algorithm carries out the entire process, including the image analysis, in real time so that it takes far less time. Figure 9 shows the entire sequence diagram of the algorithm. In the process, the measured MTF is assessed first at position 1(P1), and the M2 is subsequently moved to the range between 15 μm and + 10 μm where the degree of de-sapce can be computed immediately. In the M2 de-space within ± 35 μm, when M2 is moved to 25 μm(case 1) or + 20 μm(case 2) from the current position, the mirror is positioned within the computable range. When it is difficult to assess whether or not the mirror is within the computable range due to a misalignment(case 3) other than despace, the mirror is moved to the position of the highest MTF value among the measured values after the position movement of 25 μm de-space and the position movement of + 20 μm de-space, respectively. In order to move the position in the computable range via these procedures, the algorithm runs between one and three passes. Within the computable range, as the MTF curve is symmetrical near the vertex point, there are two solutions if the de-space is computed using a mathematical equation. In order to determine the solution, after the MTF assessment iteration is conducted at one + step position(p2) movement of the M2, an accurate de-space can be obtained. Therefore, the proposed algorithm can carry out the refocusing perfectly after two to five iterations of the image taking/analysis/running process. When the online autofocusing alignment is carried out in real time, the possible ranges for focus alignment are about ± 35 μm for the M2 or ± 735 μm for the focal plane. These are (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5419

10 determined by the driving range of the compensation mechanism. The compensation device used in this study can move the M2 by a maximum of ± 30 μm, and the compensation device of the focal plane can move the focal plane by a maximum of ± 500 μm. When driving the compensation device, the movement of the device to the range in which the degree of despace can be judged is carried out by only driving the M2. By design, the M2 should account for 30% and the focal plane should account 70% of the position movement when the degree of de-space is computed within the computable range. Therefore, because the possible range for autofocusing alignment uses two compensation devices, it is wider than the driving range of each device. The total de-space of the optical system can be expressed as the combination of the de-space of the M2 and the de-space of the focal plane, and it is expressed using Eq. (4). dz total is the value of the total de-space in view of the focal plane, and dz fp and dz sm are the de-space of the focal plane and the de-space of M2, respectively. dztotal = dz fp ( dzsm 21) (4) As the sensitivity of the M2 de-space upon the MTF is 21 times that of the focal plane, the total de-space of the optical system is induced via the above equation. This means that through the position movement of the focal plane, it is possible in terms of performance for the system to compensate for any MTF degradation caused by the position movement of the M2. Therefore, it is more useful to perceive the de-space in the optical system as the relative position difference between M2 and the focal plane rather than the absolute position difference of the designed optical components. In this study, the autofocusing algorithm carries out refocusing by controlling this relative position between M2 and the focal plane. 4.3 Performance analysis of the algorithm Fig. 9. Flow chart of the proposed algorithm. To analyze the performance of the online refocusing algorithm proposed in this study, an arbitrary de-space is intentionally created in the optical program to verify the performance of the autofocusing alignment algorithm. The performance analysis is carried out for several cases within the possible range of the proposed algorithm, and among these cases, the content for one case is included. As shown in Table 3, before the alignment, de-spaces of + 21 μm and μm are created for M2 and the focal plane, respectively, at the design position. This is equivalent to a distortion of about 650 μm in the focal plane only when it is converted to a total de-space value. At this time, the performance of the MTF is equivalent to the diagram in Fig. 10(a). At a Nyquist frequency of 50 cycle/mm, the value of the MTF is about 5% so that (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5420

11 an unusable image will be obtained because its resolution degrades significantly in actual satellite images. When the online autofocusing algorithm proposed in this study is carried out, the value of the de-space after the alignment improves within approximately 2 μm in view of the focal plane, or 0.01 μm in view of the M2. This verifies that the performance of the MTF, as depicted in the diagram in Fig. 10(b), is restored to the originally designed value. Because the technique proposed in this study uses stellar source images to measure the MTF, the location limitations of orbit are lowered while refocusing is carried out. Also, compared to the existing refocusing techniques, alignment is possible even when the de-space is relatively large. Between two and five images are needed for alignment to take place. The existing refocusing method of Pleiades requires at least three images, but the proposed algorithm in this study needs one pass and two images within computable range. The technique is time efficient; it takes fewer hours than conventional techniques. It also requires less labor. Table 3. Alignment results for the secondary mirror and focal plane Before alignment After alignment Secondary mirror(m2) 21 μm 5 μm Focal plane μm μm De-space of the FP ( dz total ) 652 μm 2 μm De-space of the M2 31 μm 0.01 μm MTF (at 50 cycle/mm) 5% 51% Fig. 10. MTF curves before and after alignment. 5. Conclusion In this study, an algorithm is developed to carry out online autofocusing alignment in real time. The satellite camera is an optical structure that requires position precision in microns. A precise optical alignment process must be conducted before the satellite is launched. However, due to the vibration environment during the launch and the severe environment of space, alignment distortions can occur so that the image quality of the camera degrades. Though existing refocusing processes can be carried out on the satellite, these processes require significant time and numerous images to assess the degree of de-space because they necessitate communication between the satellite and a ground control center. Therefore, in this paper, an autofocusing algorithm is developed to enable online refocusing in real time onboard the satellite camera while in orbit. (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5421

12 First, a target optical system is designed. The system corresponds to a satellite camera, and it is used to develop the autofocusing algorithm. In this case, a Schmidt-Cassegrain system with a 200-mm diameter M1 is used, and the performance parameters are assumed to include a GSD of 3.8 m from an operation altitude of 700 km. The MTF variation tendency due to the M2 de-space is derived mathematically to assess the degree of de-space for the optical system. Applying the obtained mathematical model, the refocusing algorithm is developed. The MTF measurement is carried out by using stellar source images. The autofocusing algorithm is connected with the M2 and focal plane compensation devices to enable refocusing in real time. It is determined from the simulations that about ± 35 μm de-focus for the M2( ± 735 μm de-focus for the focal plane) can be refocused within a ± 0.5 μm de-space error in terms of the M2 by applying the proposed refocusing algorithm in real time. Acknowledgment This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MEST) (No. 2013M1A3A3A ). (C) 2016 OSA 7 Mar 2016 Vol. 24, No. 5 DOI: /OE OPTICS EXPRESS 5422