Modeling and processing of SAR images Nikita Andriyanov Danila Andriyanov JSC "RPC "Istok" named after Shokin Telecommunication department Fryazino, Moscow Region, Russia; Ulyanovsk State Technical University Telecommunication department Ulyanovsk, Russia Ulyanovsk State Technical University boss.renome@mail.ru Ulyanovsk, Russia nikita-and-nov@mail.ru Abstract—The text is devoted to the method of simulating detection characteristics of small point objects is also radar images based on harmonic analysis. The possibilities of investigated. impelenting small and distributed objects in the coordinates of inclined and track ranges are considered. Moreover, for a II. SYNTHETIC APERTURE RADARS number of reference objects, a detection algorithm based on the Neyman–Pearson criterion was implemented and The quality of radar images is characterized by their investigated, and an algorithm for recognizing reference resolution. To determine the linear resolution in azimuth, it targets was also proposed. is possible to use the expression Keywords—SAR images; radar images; object detection; x   А r  r / d A , (1) image processing where  А is a horizontal beamwidth of the antenna, r is a I. INTRODUCTION slant range,  is wavelength, d А is horizontal aperture size The processing of Earth remote sensing data is of of the antenna. particular interest these days. Moreover, such data is usually Spatial resolution on slope  r and horizontal range represented by multidimensional arrays of information. Today there are a lot widespread methods for processing resolution  y determined respectively by the duration of satellite images based on mathematical models of random the probe pulse  и and viewing angle (observation angle) fields (RF) [1-4] and machine learning methods [5-7].  Indeed, in the course of such processing a number of applied c и problems are solved. In particular, the work [1] is devoted to r  ,  y   r sec  , (2) methods of filtering and reconstructing satellite images 2 based on doubly stochastic models. The work [2] describes a where c is speed of light. method of image segmentation, which in combination with other segmentation algorithms can improve the quality of An analysis of expressions (1) and (2) shows that it is segmentation. A wide study of methods for processing possible to improve the resolution by increasing the size of satellite images using doubly stochastic models is presented the aperture of the antenna and reducing the duration of the in [3]. In addition to doubly stochastic models, models based probe pulses. However, in the second case, the energy of the on autoregressions with multiple roots of characteristic probe signal decreases, and with it the observation range. equations were proposed for describing and processing The horizontal size of the antenna is limited by the size of images providing lower computational costs [4]. the aircraft from which the sounding is performed. Therefore, the parameter d А is artificially enlarged by However, mainly in the literature on image processing, authors are talking about processing two-dimensional synthesizing the image along the route of the aircraft. And brightness fields in grayscale or two-dimensional brightness such systems are called SAR stations. fields in several color channels. This refers to optical III. MATHEMATICAL MODEL OF THE PROCESS OF FORMING images. However, despite the wide distribution, such images SAR IMAGES have some drawbacks. One of them is the dependence of the work of registrars on weather conditions. Clouds and other For adequate modeling of the radar image formation disturbing natural factors can often be found in images. In process, it is necessary to simulate a number of processes order to register the Earth’s surface regardless of weather and factors important for the formation of such an image. In conditions, synthetic aperture radar (SAR) can be used, the [9] it was proposed to use the following models: result of which are radar images (SAR images) [8-11]. By 1) Model of reflection (scattering) of an and large, such images also represent a two-dimensional electromagnetic wave (EMW) by the Earth (sea) surface and array of brightness elements, however, the method of their objects. Let such a model be described by the operator L рр . formation is of interest. It is close to processing of multichannel data [12]. Radar images and it's processing are 2) Model for generating a radar signal, including 2 considered in [13-16]. The main attention of such data procedures: conversion of the electromagnetic field (EMF) processing is devoted to tasks of modeling and objects scattered by the observed surface, described by the remote recognition. New modeling methods was suggested [16], sensing operator L з , and a procedure for scanning an EMF however in the number of task it is expedient to use classical having two-dimensional coordinates into a one-dimensional models of SAR. In this paper, an example of modeling radar images and objects on its background are considered. The Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0) Image Processing and Earth Remote Sensing radar signal. The last procedure is described by the spatio- The method is called “harmonic analysis”, since the temporal scan operator L пвс . multiplication of signals (4) by the reference function (5) corresponds to their linear-frequency (LFM) demodulation 3) Model of noise and interfering factors, for the and for their isolation from the mixture with noise mi (t ) description of which the distortion operator L и is used. harmonic analysis is used. Thus, the totality of responses to 4) Model for processing the complex envelope of the accurate reflectors can be obtained for the m -th strip by signal and the formation of radar data, which, as well as the model for generating the radar signal, is represented by two taking the Fourier transform of the products (4) and (5). procedures such as the data processing procedure itself and Returning to the discrete time, to represent the line of the the procedure for converting the radar signal into an image. radar signal, it is necessary to take the module of the discrete Fourier transform (DFT) The first is described by the processing operator L обр , the N с 1 1 second is described by the operator L пвс , the inverse of the eˆ ( m , k )    ( m , n ) h ( m , n ) exp(  j 2  nk / N ) , (6) 0 с n0 spatial-temporal scan operator. Thus, according to [9], to describe the mathematical where m  0 ,1,..., N r  1; k  0 ,1,..., N с  1 , Nr and model of radar data, 6 operators and the radar terrain function (RTF) are used. RTF can be written as follows N с are sizes of SAR image. e ( r , x )  e ( r , x ) exp[ j  ( r , x )], (3) Fig. 2a and Fig. 2b show examples of point small-sized object (a) and an extended distributed object (b) formation where e ( r , x ) and  ( r , x ) are amplitude and phase on the radar image using the method of harmonic analysis characteristics of scattered EMF. with following parameters 2 7 Fig. 1 shows the radar image of a point object.   10 m , V п  200 m / s ,  и  10 s and signal to noise (signal / background) ratio q  10 . Fig. 1. SAR image with point object. It should be noted that not only the point object itself is clearly visible in the image, but also the area in the point object neighborhood. IV. RADAR IMAGES MODELING BY HARMONIC ANALYSIS Let the path signal s ( m , n ) is signal reflected along the m -th strip of range. Then it represents the sum of signals from elementary reflectors s i ( m , n ) . Passing to continuous time, it is possible to write the following equation a b Fig. 2. Imitation of a point (a) and an extended (b) object on a radar image. s m i ( t )  U i G ( t  t i ) exp{  j [  ( t  t i ) 2   0 i ]}  , (4)  U i G ( t  t i ) exp(  j  t 2 ) exp[ j (  дi t   i )] An analysis of Fig. 2 shows that even with the proposed modeling method, a point object on a radar image is where U i G ( t  t i ) is signal envelope determined by the represented by several points. This is due to the model of shape of the antenna pattern in the azimuthal plane, formation of such an image, one of the stages of the implementation of which is the use of DFT. 2 V п 2   , r m  r0  r m   r ,  r0 V. DETECTION AND RECOGNITION OF OBJECTS ON SAR 4 V п ti 4 V п xi 2  дi   is circular Doppler frequency IMAGES  r0  r0 A mixture of signals from the object (if it is present, in the center of the observed from i -th point signal object otherwise its component can be taken equal to zero) and the (t  0 ) , xi  V пt , background with noise is a total signal at the detector input  i   0 i  2  V п t i /  r0  const 2 2 V п is ground speed.  ( m , k )  s ф ( m , k )  s о ( m , k )  n ( m , k ) . (8) The reference function of the signal (4) is written as To detect useful signal in this case, it is also possible to h m 0 ( t )  H ( t ) exp( j  t ) , 2 (5) use the classic comparison of the likelihood ratio with the threshold h , found from the Neyman-Pearson test, which where H ( t ) is envelope of the reference function on the provides a given probability of false alarm synthesis interval. VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 90 Image Processing and Earth Remote Sensing p (  )  h  H 1 l [  ( r , x )]  1 . (9) p 0 (  )  h  H 0 Here H 1 is hypothesis of the presence of a signal from an object, H 0 is hypothesis of the absence of a signal from the object. Expression (9) can be used in the case of complete a priori certainty. In the case of using uncertain parameters, the likelihood ratio is determined to within finite Fig. 5. Detection efficiency depending on threshold. aggregates of undefined parameters  1 and  0 . Then An analysis of the curves in the graphs of Fig. 3–5 shows equation (9) can be written in the form that the best detection characteristics are achieved under conditions of accumulation of several information frames. It p 1 (  /  1 )  h  H 1 is also should be noted that at N = 5, if q> 1, good correct l [  /  0 ,  1 ]  , (10) detection probabilities are achieved. It is also seen that for p (  /  )  h  H 0 0 0 small q it is recommended to carefully choose the threshold, since with its increase the detection efficiency drops sharply. where  0 and  1 are sets of informational and non- Finally, based on detection results objects can be further informational unknown distribution parameters p 0 (  ) and recognized. For this, the correlation with the known pattern is considered in the detected area and the object for which it p 1 (  ) under hypotheses H 0 and H 1 . made the highest value is selected. Estimation for calculating the correlation is as follows The likelihood functions in (10) can be found with the known parameters of observation noise N r 1 N с 1  1 N r 1 N с 1 2    R i     ( m , k )  s ф s оi . (12) p 1 [  ( m , k /  1 )]  k n exp      ( m , k /  1 )  s о  s ф ( r , x )  , (11) m 0 k 0  N 0 m 0 k 0   1 N 1 N r 1 с 2  Thus, the characteristics of the detection of small objects p 0 [  ( m , k /  0 )]  k n exp      ( m , k /  0 )  s ф ( r , x )  , on the radar are considered and an algorithm for recognizing  N 0 m 0 k 0  such objects is proposed. where k n is normalization factor, N 0 is spectral density of VI. CONCLUSION observation noise. The article discusses the main features of the formation Fig. 3-5 show the detection characteristics. Here Pd is a of radar data and the principle of operation of a radar of probability of the correct detection. Fig. 3 presents the SAR type. It is shown how to generate radar images using dependence of the correct detection on the ratio of the signal the harmonic analysis method. In addition, characteristics of level to the background level for different numbers of the detection efficiency of small objects are obtained. It was accumulated frames, Fig. 4 shows the dependence of the revealed that with the probability of false alarm P f  10  3 it probability of correct detection on the number of is possible to get good detection results ( Pd  0 . 9 ) if at least accumulated frames, Fig. 5 shows the dependence of the probability of correct detection on the threshold value. In all 3 frames of SAR images is accumulated and the ratio of the modeling and detection processes the probability of false signal level to the background level q  10 . alarm is P f  10  3 . ACKNOWLEDGMENT This work was funded by the RFBR Grant, Project No. 19-29-09048. REFERENCES [1] K.K. Vasiliev, V.E. Dementiev and N.A. Andriyanov, “Filtration and restoration of satellite images using doubly stochastic random fields,” Fig. 3. Detection efficiency based on signal to noise ratio. 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