EFFECT OF IMAGE MATCHING WINDOW SIZE ON SATELLITE JITTER FREQUENCY DETECTION

Satellite attitude jitter is a common and complex phenomenon for high-resolution satellites and it is detectable by multi-temporal image matching. This paper analyses the effect of image matching window size on jitter frequency detection. First, two simulation images with a given short time lag and line scanning frequency affected by a modelled jitter are generated based on the principle of dynamic imaging model. Then, the relative image distortions are obtained by dense image matching with different matching window size and the frequency is estimated through spectrum analysis of the obtained image distortions. The experimental results demonstrated the feasibility and reliability of high frequency jitter detection based on dense image matching, and the results indicated that the maximum detectable frequency is almost not affected by the change of image matching window size, which provided useful demonstration of image-based satellite jitter detection capacity.  Corresponding author


INTRODUCTION
With the development of earth observation technology, Highresolution satellite images (HRSIs) has been widely used in the fields of land resources exploration, city planning, the establishment and renewal of national basic geographic information database etc (Yang et al., 2017). However, attitude jitter, which refers to the periodic instability of attitude caused by the micro-vibration of satellite platform, affects geometric quality and imaging performance of high-resolution satellites. And it is an inevitable issue as the jitter can be induced by both the external space environment and internal mechanical operation, and the frequencies and amplitudes of jitters for different satellites are diverse (Wang et al., 2016). Many domestic and foreign satellites, such as IKONOS, High Resolution Imaging Science Experiment (HiRISE), ZiYuan-3, YaoGan-26 and GaoFen-9, have been affected by the attitude jitter (Grodecki, Dial, 2003;Mattson et al., 2009;Tong et al., 2014;Liu et al., 2016;Wang et al., 2016aWang et al., , 2016bPan et al.,2016;Zhang, Guan, 2018).
Nowadays, the published methods of jitter detection can be categorized into two categories. One is dependent on a highperformance attitude-measuring sensor since the attitude oscillation is the direct manifestation of jitter (Takaku, Tadono, 2009). The data of high-frequency angular displacement sensor were fused with satellite sensor attitude data for the jitter detection of YaoGan-26 satellite . After obtaining the high frequency angular displacement information through processing the measurements of the micro-vibration accelerometer installed on the GF-9 satellite, they corrected the attitude by taking the high-frequency attitude correction into account in the geometric model. And after the jitter correction, the jitter amplitude decreased from 0.9 pixels to 0.4 pixels (Zhang, Guan, 2018).
The other is based on imagery to obtain the distortion caused by satellite jitter, such as multispectral imagery, staggered CCD (Charge Coupled Device) images, and stereo image pairs, orthoimages, or linear objects in images. The attitude jitter detection method based on the parallax observation configuration were proposed, which estimated attitude jitter from the relative registration errors of adjacent bands of ASTER short-wave infrared sensor (Teshima, Iwasaki, 2008). Similar methods were used for detecting jitter of many satellites, such as HiRISE, Mapping Satellite-1, ZiYuan-1, ZiYuan-3 (Kirk et al., 2008;Sun et al.,2015;Jiang et al.,2014;Tong et al., 2015aTong et al., , 2015bTong et al., , 2015cTong et al., , 2015dTong et al., , 2017Tong et al., , 2019Wang et al.,2016;Pan et al., 2017;Zhu et al., 2018).
The choice of matching method directly affects the result of jitter detection. The phase correlation method can estimate the precise offset value between images, and it can be extended to high-precision dense matching between images by sliding window. At present, many scholars have made researches on the matching algorithm. The peak evaluation formula (PEF) method enables estimation of image displacements with 0.1-0.01pixels accuracy based on one-dimensional sinc function fitting (Nagashima et al.,2006). An improved phase correlation, which is based on 2-D plane fitting and the maximum kernel density estimator, combines the idea of Stone's method and robust estimator and increases the robustness when fitting the phase angle plane. (Tong et al., 2015c). And the SVD-RANSC subpixel phase correlation method integrates the advantages of Hoge's method and the RANSAC algorithm, avoiding the corresponding shortfalls of the original phase correlation method based only on SVD (Tong et al.,2015d).
It is generally believed that the matching window size affecting the jitter frequency detection. The maximum detectable frequency of ZiYuan-3 multispectral imagery can be up to 39.06 Hz ( 2 sr w , where the multispectral scanning rate sr is approximately 1250 Hz, the size of correlation window w is considered since the window is equivalent to low-pass filtering and is set as 16) (Ye et al., 2019). However, quantitative experimental analysis of matching window size on jitter detection has rarely been investigated.
Therefore, the purpose of this paper is to analyse the effect of image matching window size on jitter frequency detection based on simulation images, which are generated with a given short time lag and line scanning frequency affected by a modelled jitter. And the remaining parts of this paper are organized as follows. Following the introduction, the simulation images generation method and jitter detection method are presented in Section 2. And experiments and analysis of are described in Section 3. Finally, the conclusions are given in Section 4.

Generation of jitter affected images
For generating of simulation images，we have to establish the digital simulation model which refers to the mathematical relationship between the ideal sampled image F and the final distortion image I (Zhuang.,2011).
The degraded image caused by image shift and sampling is as follows: Where PSF is the point spread function caused by image shift, The ideal sample image of under the effect of image shift and sampling is as equation (4) and equation (5) (Zhuang, 2011, Tong., 2014:

the sampling interval, which is a pixel size
A =the amplitude of jitter f = the frequency of jitter  =the phase of jitter

Image based jitter detection
According to the principle of jitter estimation based on parallax images, relative image distortions of multi-temporal images with some imaging time interval can be denoted as formula (6) (Teshima, Iwasaki, 2008) Generally, the image distortions resulted from attitude jitter can be modelled as a sum of the sinusoidal functions with different amplitudes, phases, and frequencies. To simplify the derivation, it is assumed that the jitter contains only a single frequency, which can be denoted as follows: Where j A , j f , and j  denote the amplitude, frequency, and phase of the ith component of absolute jitter distortions, respectively. The relative residuals can also be expressed in a similar form as follows: Where A  , f  , and   denote the amplitude, frequency, and phase of the ith component of parallax difference value, respectively.
According to the synthesis and decomposition characteristics of the same frequency vibration (Liu et al., 2019), we can get the equation (9).
Therefore, the quantitative relationship between the jitter displacement and jitter parallax is determined.
It is worth noting that the frequency component of jitter should meet the equation (10), and it is difficult to detect the jitter information at the integral times of the blind spot frequency from the parallax image.
( 1, 2,3 ) Moreover, when the jitter frequency is close to the integral multiple of the characteristic frequency. We can get the follow equations (11)and (12) In the case of the same image quality and content richness, the imaging position difference at different frequencies is obtained by the image registration method. It can be approximately considered that the amplitude error of imaging position difference at different frequencies is the same, which is expressed by  . According to the error synthesis equations, the vibration amplitude error caused can be expressed as equation (13): Therefore, when the frequency is close to the blind spot frequency, the error of transfer coefficient between the jitter and the parallax is much greater than 1, which greatly enlarges the error of the imaging position difference, leading to the increase of jitter error.

The phase correlation algorithm based on PEF
The basic principle of phase correlation algorithm based on PEF (peak evaluation formula) is to use two-dimensional sin function fitting the peak matrix of cross power spectrum after inverse transformation, and getting accurate sub-pixel matching results according to the fitting results (Nagashima et al.,2006).
The result of inverse Fourier transform is a peak matrix. The coordinate position corresponding to the peak value is the whole pixel matching position of the matched image relative to the reference image. And the equation of the peak matrix can be approximately expressed as follows: Its peak value matrix ( , ) q i j can be approximately expressed by the above formula under ideal conditions. Therefore, by performing two-dimensional fitting on the peak matrix in two directions respectively, we can get accurate fitting peak vertices, and then get sub-pixel matching results.

The experimental workflow
High precision dense matching is the foundation of jitter detection. Therefore, we obtained lots of high-precision image points by point dense matching strategy of the PEF phase correlation. And the Figure 1 shows the workflow of the proposed method.  (2) Image pre-processing: Based on the generation method of 2.1, we generated two parallax images of different time. And to ensure the reliability of dense matching，we used Wallis filter for enhancing the image and improving image contrast (Barazzetti, Scaioni, 2010). The known disparity structure is used for initial image registration to eliminate the fixed offset between two images.
(3) Image dense matching：The correlation window size is set to oblong shape in order to estimate the relative registration error in the cross-track direction with a high time resolution, which makes analysis robust to scene features (Teshima, Iwasaki, 2008), and the sub-pixel offset is calculated by normalized cross correlation (NCC) (Barnea and Silverman, 1972) for coarse matching and PEF phase correlation method for accurate matching, so parallax images in two directions can be generated.
(4) Eliminating mismatch results: Before the follow-up analysis, it is necessary to eliminate mismatches and reduce the impact of noise. Firstly, setting threshold value to remove the abnormally large offset value in the disparity image. Secondly, the normalized correlation coefficient between matching windows is calculated to remove the low texture or low correlation area whose correlation coefficient is less than the threshold value of 0.7. The fitting residual value, returning by RANSAC robust estimation algorithm in the sub-pixel phase correlation calculation process, is used to eliminate the data of correlation quality difference. The image points on the same line are affected by the same jitter, so the outliers of each line are eliminated according to the principle of three times of mean square error.
(5) Obtaining jitter information: According to section 2.2, rebuilding of the jitter components, and converting to jitter information.

EXPERIMENTS AND ANALYSIS
To study the effect of image matching window size on satellite jitter frequency detection, simulation experiments are carried out. The reference image is from a panchromatic image of ZiYuan-3 satellite in Figure 2, which size is 1280*1280 pixels, and the simulation parameters are shown as

Generation of jitter affected images
For verifying the method of jitter detection, two simulation images with a given short time lag and line scanning frequency affected by a modelled jitter are generated based on the principle of dynamic imaging process, where the line scanning time is 0.0002s, the time lag is 135 lines. And based on line scanning time and Nyquist's law, the detectable frequency is no more than 2500 Hz, so we add jitter in the column direction with frequency varying from 50 Hz to 2500 Hz.
According to the principal of dynamic imaging in section 2.1, (1) we first load the reference image; (2) The simulation parameters are substituted into the dynamic imaging model;(3) we can calculated the coordinates of image points at each time and the position of the point may no longer be an integer, so we do bicubic convolution interpolation (Keys, 1981); (4) Traversing the whole image, we can obtain two multi-temporal images affected by a modelled jitter as shown in the Figure 3. (a) Simulation image 1 (b) Simulation image 2 Figure 3. Multi-temporal images which affected by a modelled jitter.

Analysis of effect of matching window size on jitter detection
To analyse the effect of the matching window size for jitter detection, we change the size of the matching window in the along-track direction, which is 128 pixels (cross)*16 pixels (along), 128 pixels (cross)*32 pixels (along), 128 pixels (cross)*64 pixels (along).
After carrying out multi-temporal image dense matching and removing mismatch, we obtained the relative disparity curves which are averaged the disparity maps at each line. The results of dense matching with three different match size are presented in Table 2, as shown in Table 2, jitter frequency as high as 2475 Hz can be effectively detected with all the three matching window size, but it failed for the detection of 2500 Hz, with the detected frequency as 2475Hz. The reasons may be that, the sampling frequency is closer to Nyquist sampling frequency, the aliasing effect will appear, that is, the frequency component higher than half of the sampling frequency is reconstructed into the signal lower than half of the sampling frequency.

CONCLUSIONS
The satellite jitter is one of the key problems affecting the ability of high-precision observation and mapping. This paper studied the effect of matching window size on jitter frequency detection. The experimental study demonstrated the feasibility and reliability of high frequency jitter detection based on multitemporal image matching. And, it indicated that the maximum detectable jitter frequency is almost not affected by the image matching window size. For simulated images with scanning rate of 5000 lines per second, the maximum detectable jitter frequency can be as high as 2475 Hz, close to the theoretical maximum detectable frequency of 2500 Hz, which provides useful demonstration for image-based satellite jitter detection capacity.