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Implementation of a fast method for reconstruction of ISAR images

LiTH-ISY-EX-3437-2003

2 Implementation of a fast method for reconstruction of ISAR images av

LiTH-ISY-EX-3437-2003

Handledare: Christer Larsson, AerotechTelub AB

Examinator: Per-Erik Danielsson

Avdelning, Institution

Division, Department

581 83 LINKÖPING

Datum Date

2003-12-15

Språk

Language

Rapporttyp

Report category

ISBN

Svenska/Swedish

X Engelska/English Licentiatavhandling

X Examensarbete

ISRN LITH-ISY-EX-3437-2003

C-uppsats

D-uppsats

Serietitel och serienummer

Title of series, numbering

ISSN

Övrig rapport

____ Titel Title Implementation of a fast method for reconstruction of ISAR images

Author

Sammanfattning

Abstract

By analyzing ISAR images, the characteristics of

military platforms with respect to radar visibility can be evaluated. The method, which is based on the Discrete-Time Fourier Transform (DTFT), that is currently used to calculate the ISAR images requires large computations efforts. This thesis investigates the possibility to replace the DTFT with the Fast Fourier Transform (FFT). Such a replacement is not trivial since the DTFT is able to compute a contribution anywhere along the spatial axis while the FFT delivers output data at fixed sampling, which requires subsequent interpolation. The interpolation leads to a difference in the ISAR image compared to the ISAR image obtained by DTFT. On the other hand, the FFT is much faster. In this quality-and-time trade-off, the objective is to minimize the error while keeping high computational efficiency. The FFT-approach is evaluated by studying execution time and image error when generating ISAR images for an aircraft model in a controlled environment. The FFT method shows good results. The execution speed is increased significantly without any visible differences in the ISAR images. The speed-up- factor depends on different parameters: image size, degree of zero-padding when calculating the FFT and the number of frequencies in the input data.

Nyckelord

Keyword

radar, RCS, radar cross section, ISAR, image reconstruction, algorithm, FFT 3

Abstract

By analyzing ISAR images, the characteristics of military platforms with respect to radar visibility can be evaluated. The method, which is based on the Discrete-Time Fourier Transform (DTFT), that is currently used to calculate the ISAR images requires large computations efforts. This thesis investigates the possibility to replace the DTFT with the Fast Fourier Transform (FFT). Such a replacement is not trivial since the DTFT is able to compute a contribution anywhere along the spatial axis while the FFT delivers output data at fixed sampling, which requires subsequent interpolation. The interpolation leads to a difference in the ISAR image compared to the ISAR image obtained by DTFT. On the other hand, the FFT is much faster. In this quality-and-time trade-off, the objective is to minimize the error while keeping high computational efficiency. The FFT-approach is evaluated by studying execution time and image error when generating ISAR images for an aircraft model in a controlled environment. The FFT method shows good results. The execution speed is increased significantly without any visible differences in the ISAR images. The speed-up-factor depends on different parameters: image size, degree of zero-padding when calculating the FFT and the number of frequencies in the input data. 4

Acknowledgements

I want to thank the following persons:

Christer Larsson for sharing his knowledge about ISAR and for his valuable discussions.

Per-Erik Danielsson for his support.

Andreas Granstedt for his valuable comments on this thesis. The people at AerotechTelub for a pleasant atmosphere. 5

Table of contents

1 INTRODUCTION....................................................................................................... 8

1.1 BACKGROUND........................................................................................................... 8

1.2 PURPOSE.................................................................................................................... 9

1.3 METHOD.................................................................................................................... 9

1.4 CONSTRAINTS AND PRE-REQUISITES........................................................................ 10

1.5 METHODOLOGICAL CONSIDERATIONS..................................................................... 10

1.6 TERMINOLOGY AND ABBREVIATIONS...................................................................... 10

2 THEORY.................................................................................................................... 11

2.1 RADAR - AN INTRODUCTION.................................................................................... 11

2.2 ISAR....................................................................................................................... 19

2.3 MODEL AND GEOMETRY.......................................................................................... 20

2.4 FOURIER TRANSFORMS............................................................................................ 22

2.5 IMAGE RECONSTRUCTION........................................................................................ 25

2.6 IMAGE QUALITY....................................................................................................... 30

3 INTERPOLATION................................................................................................... 32

3.1 CUBIC SPLINE INTERPOLATION................................................................................ 33

4 THE METHOD.......................................................................................................... 37

4.1 IMAGE RECONSTRUCTION........................................................................................ 37

4.2 TIME-COMPLEXITY.................................................................................................. 40

4.3 COMPUTATIONAL CONSIDERATIONS........................................................................ 42

5 TEST CASES............................................................................................................. 45

5.1 AIRCRAFT MODEL.................................................................................................... 45

5.2 EVALUATED QUANTITIES......................................................................................... 46

6 IMPLEMENTATION............................................................................................... 47

6.1 TOOLS AND LANGUAGES.......................................................................................... 47

6.2 PROGRAM DISCUSSION AND NOTES.......................................................................... 48

7 RESULTS................................................................................................................... 53

7.1 ISAR IMAGES.......................................................................................................... 54

7.2 PARAMETER STUDY................................................................................................. 57

8 DISCUSSION AND CONCLUSIONS..................................................................... 61

8.1 DISCUSSION............................................................................................................. 61

8.2 CONCLUSIONS.......................................................................................................... 62

8.3 FUTURE WORK AND IMPROVEMENTS....................................................................... 62

REFERENCES 64 6

Table of figures

Figure 1: Reflection in backdirection............................................................12

Figure 2: Direct imaging................................................................................13

Figure 3: Frequency table with frequency bands...........................................14

Figure 4: Explanation of RCS. ......................................................................16

Figure 5: RCS plot of an aircraft model........................................................18 Figure 6: Difference between SAR and ISAR. (a) illustrates a SAR system while (b) illustrates an ISAR system.....................................................20 Figure 7: Geometry model for ISAR system.................................................21 Figure 8: Difference between DTFT and FFT...............................................23 Figure 9: Difference between FFT without zero-padding and FFT with zero-

padding. .................................................................................................24

Figure 10: Linear system...............................................................................26

Figure 11: ISAR as a linear system...............................................................26 Figure 12: Range profile of an aircraft model...............................................27 Figure 13: Illustration of the image reconstruction problem.........................28 Figure 14: ISAR image of an aircraft model.................................................30 Figure 15: Cubic spline function consisting of piece-wise polynomials.......34 Figure 16: Illustration of the image reconstruction problem using the approximative method...........................................................................38 Figure 17: Block scheme of the approximative method................................40

Figure 18: Picture of RAK.............................................................................45

Figure 19: RAK on tripod..............................................................................46

Figure 20: Scheme of Columbus2000...........................................................47 Figure 21: ISAR images for test case where N=200 and F p =1024................54 Figure 22: ISAR images for test case where N=200 and F p =2048................55 Figure 23: ISAR images for test case where N=500 and F p =2048................56 Figure 24: Relation between image error and length of zero-padded signal.57 Figure 25: Relation between decrease in execution time and size of image Figure 26: Relation between decrease in execution time and length of zero-

padded signal.........................................................................................59

Figure 27: Relation between decrease in execution time and number of frequencies for N=500...........................................................................60 7

Table of tables

Table 1: Frequency bands for radar applications...........................................15

Table 2: RCS of different objects..................................................................18

Table 3: Overview of properties of different interpolation techniques.........33 8

Chapter 1

1 INTRODUCTION

This chapter gives the background and the reason for writing this thesis. It also contains the method used, constraints and pre-requisites, the methodological considerations and disposition.

1.1 Background

In modern warfare it has always been important to avoid that military objects are discovered by hostile forces. Since radar systems are frequently used for detection, it is important to make the own forces less visible on radar. So called stealth [Schleher, 1999] characteristics are therefore highly desirable. The object should ideally be virtually impossible to detect. One of the services that the company AerotechTelub offers is to analyze radar reflectivity of military objects, such as fighter aircrafts. This is done with inverse synthetic aperture radar (ISAR). In ISAR, the radar response of the object is processed to an image that shows the reflectivity of the different parts of the object. The results from this process can be used to improve certain parts of the object with respect to radar reflectivity. The procedure of retrieving this image accurately involves a kernel of Discrete-Time Fourier Transform (DTFT) calculations. This DTFT calculation is very time- consuming, which is a major problem. For example, a 3D ISAR image may take a couple of weeks to compute. This problem is the reason for studying methods to increase the performance of the calculations, which is done in this master's thesis. The idea is to replace the DTFT kernel with a Fast Fourier Transform (FFT) kernel. The FFT is much faster than the DTFT. However, the FFT calculation outputs a discrete set of values. That is, the FFT values are only calculated for a discrete set of points. That leads us to the problem of finding the output values for input values that resides between those input values that outputs correct result. A standard procedure for doing this is to apply interpolation, in this case to the FFT function. An FFT together with interpolation at a certain input value should approximate the DTFT value at that input value. In this thesis the FFT approach is called the approximative method since its objective is to estimate the DTFT method. It should be kept in mind that the DTFT method is not an exact reconstruction of the target object. There are tough requirements for the results of the approximative calculations since the results describe the characteristics of military crafts. The quality of the ISAR image has to be good in comparison with the ideal image that is obtained using the DTFT method. At the same time the approximative method has to show a significant

9change when it comes to computation speed for the method to be useful. In other

words there is a quality-and-time trade-off. The FFT approach is called the approximative method since its objective is to estimate the result of the DTFT. The approximative method is not only applicable to ISAR systems. There are other applications in imaging systems where the method can be used to increase the computation speed, such as imaging based on microwaves. See [Kim et al, 2003] for an application example. Comuterized tomography [Smith et al, 1999] is another are where the results of this thesis may be used.

1.2 Purpose

The purpose of this thesis is to investigate whether cubic spline interpolation together with FFT calculations can be used to obtain sufficiently good results when calculating ISAR images. The main question is: Can the method increase the computation speed with a factor of ten or more and at the same time produce images of sufficiently high quality?

1.3 Method

Existing sources of information have been studied to support the decision of choosing an interpolation method. ISAR literature is also included in this thesis since it is required to fully understand the process of calculating an ISAR image. The mathematics behind Fourier transforms will also be presented since that is an important part of understanding the main problem. The interpolation method that is implemented is the cubic spline interpolation technique. The reason for this is its good approximation properties as well as its implementation simplicity. Another reason for choosing the cubic spline interpolation method is that existing empirical tests have shown promising results. Further information about this choice of technique is presented in this thesis. The algorithms included in this thesis are written in MatLab[MathWorks MatLab, 2001]- and C-code[Kernighan et al, 1988]. The ISAR images will be constructed using the framework of the Columbus2000 software[Berlin et al, 2000]. Columbus2000 uses the DTFT to calculate the images, which will be replaced by the new approximative method. It is important to remember that the result of the calculations depends on several parameters, for example the size of the ISAR image. A parameter study is therefore included in this thesis since it will make it easier to understand how the parameters affect the results. The result consists of two variables; Execution time and image quality. Execution time is the total time that is consumed for the ISAR image to be calculated while image quality is measured by comparing the images constructed by the two different methods. The quality measurement is described later. 10

1.4 Constraints and pre-requisites

It is assumed that the DTFT kernel existing today is correctly implemented. Otherwise it would not be possible to measure the quality of the approximative method.

1.5 Methodological considerations

An obvious problem using the method described above is that the results obtained are heavily influenced by the implementation. The code has to be correct in order to draw any conclusions. A minor bug in the code may strongly affect the result.

Therefore it is important to validate the code.

The compiler that is used to compile the program containing the algorithms may affect their relative execution time, which would make it harder to interpret the results.

1.6 Terminology and abbreviations

Down-range The direction of the radar's line of sight Cross-range The direction orthogonal to the radar's line of sight dBsm decibel square meters Scatterer A point that reflects radar waves

RCS Radar Cross Section

SAR Synthetic Aperture Radar

ISAR Inverse Synthetic Aperture Radar

DTFT Discrete-Time Fourier Transform

FFT Fast Fourier Transform

RRMSE Relative Root-Mean-Square Error

MARE Mean-Absolute Relative Error

11

Chapter 2

2 THEORY

This chapter presents theory about radar and ISAR, mathematics behind Fourier transforms and quality metrics. The sections covering the radar information are fundamental. More detailed information can be found in [Skolnik, 1980]. The chapter also contains a section about the model and geometry used in this thesis. Further on it is described how an ISAR image is calculated based on that model, using the DTFT method.

2.1 Radar - an introduction

General

Radar is a system that uses the characteristics of electromagnetic waves to detect and locate objects. Of course, this can also be performed by the human eye, but the radar has several advantages. Since electromagnetic waves can "see through" for example clouds, darkness and rain the radar is of high use in many applications. Radar stands for RAdio Detection and Ranging. Not surprisingly, the radar can also be used to measure the range to an object. The radar is able so detect object far away which is also a significant advantage compared to the human eye. [Skolnik, 1980] The radar was developed in order to be able to detect hostile aircrafts in the beginning of the 20 th century [Knott et al, 1993]. After years of development, new application areas have been found. Today the radar is used in, for example, air traffic control, ship safety, environmental observations and spacecrafts. [Skolnik, 1980] A typical radar system consists of two parts; a transmitting antenna and a receiving antenna. The transmitter transmits electromagnetic radiation using an oscillator. The electromagnetic waves are then reflected when they reach an object that is able to reflect them. Not all of the electromagnetic waves sent by the transmitter will be returned to the receiver. The target that is hit by the waves will reradiate them in many directions and not only straight back to the radar. It is only the waves reflected in the back direction that are of interest, see Figure 1. The waves are collected at the receiver and they can now be used to calculate the distance to the target as well as its velocity in relation to the radar. [ibid] 12

Reflection in

backdirectionTransmitted signal

Figure 1: Reflection in backdirection.

Radar systems where the transmitting antenna is located at the receiving antenna are called monostatic. Radar systems where the transmitter and receiver are separated are called bistatic. [Analytical Graphics]

Imaging radar systems

To determine the scattering properties of an object that is illuminated by electromagnetic energy emitted from a radar, the features of the object that leads to the scattering has to be visualized. This visualization is called a radar image. Mensa (1991) defines a radar image as the spatial distribution of reflectivity corresponding to an object. In other words a radar image is an array with values partitioning the object space. [Mensa, 1991] According to Mensa (1991), the most common areas of use for radar images are:

1. Identification and characterization of radar reflectivity parts of complex

object.

2. Simulation of radar signatures in order to determine responses of radar

sensors.

3. Recognition systems that uses radar images as an identifier unique to

certain objects. Complex objects may require images taken from several angles in order to output correct information about the object's characteristics. The reason for this is that the scattered fields from the target object are determined by an integrated effect since radars can "see through" objects. [ibid] It is possible to create a radar image of a three-dimensional object by scanning it piece by piece with a range-gated, short-pulse radar as shown in Figure 2 and collecting the values of reflectivity for each piece. This method requires no additional calculations in order to retrieve the image, but this method, also called the direct imaging method, has several disadvantages. For example, a high spatial resolution leads to impractical configurations and cross-range resolution gets worse at long distances. Synthetic imaging methods deal with this complex of problems. Synthetic imaging uses the result of several observations of the object from different angles and frequencies. The observations are processed with signal processing resulting in an image with higher resolution. [ibid] 13

Figure 2: Direct imaging.

Frequencies

A radar may operate at any frequency, since it is basically a system that emits electromagnetic energy and retrieves the part of that energy that is reflected from an object. In practice however, there are limits for the radar frequencies. The reasons for this are for example availability of components and resolution and range requirements. In general, radars operate within the frequency bands between 3 Mhz and 300 Ghz. See Figure 3 for an overview of the frequency spectra and the intended usage of the different bands. [Knott et al, 1993] 14 1 km 100 m
10 m 1 m 1 cm 1 m 10 cm 1 mm

0.1 mm

0.01 mm

0.001 mm

1000 Å

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