This thesis is concerned with problems related to Synthetic Aperture Radar (SAR). The thesis is structured as follows: The first chapter explains what SAR is, and the physical and mathematical background is illuminated. The following chapter points out a problem with a divergent integral in a common approach and proposes an improvement. Numerical comparisons are shown that indicate that the improvements allow for a superior image quality. Thereafter the problem of limited data is analyzed. In a realistic SAR-measurement the data gathered from the electromagnetic waves reflected from the surface can only be collected from a limited area. However the reconstruction formula requires data from an infinite distance. The chapter gives an analysis of the artifacts which can obscure the reconstructed images due to this problem. Additionally, some numerical examples are shown that point to the severity of the problem. In chapter 4 the fact that data is available only from a limited area is used to propose a new inversion formula. This inversion formula has the potential to make it easier to suppress artifacts due to limited data and, depending on the application, can be refined to a fast reconstruction formula. In the penultimate chapter a solution to the problem of left-right ambiguity is presented. This problem exists since the invention of SAR and is caused by the geometry of the measurements. This leads to the fact that only symmetric images can be obtained. With the solution from this chapter it is possible to reconstruct not only the even part of the reflectivity function, but also the odd part, thus making it possible to reconstruct asymmetric images. Numerical simulations are shown to demonstrate that this solution is not affected by stability problems as other approaches have been. The final chapter develops some continuative ideas that could be pursued in the future.
Abstract
This thesis is concerned with problems related to Synthetic Aperture Radar (SAR), a technique of making images of the surfaces of planets using electro- magnetic waves. Reconstructing images of the surfaces from the gathered data is an inverse problem as are other young and thriving imaging techniques used for example in optical tomography and transient elastography. In optical tomography one tries to create images of the human body employing light, whereas in tran- sient elastography ultrasound is used to measure the propagation of shear waves and thus reconstruct the stiffness of human tissue. But the field of inverse prob- lems covers older and established topics as well like computerized tomography that creates images of the human body by means of x-rays and magnetic reso- nance imaging using electromagnetic fields to measure the distribution of atoms. All these very different applications have in common that the gathered data is difficult to interpret. Therefore mathematical processing is necessary in order to create an intelligible image of the measured object. Some of the problems related to this mathematical processing necessary in creating SAR-images are analyzed in this thesis.
The thesis is structured as follows: The first chapter explains what SAR is, and the physical and mathematical background is illuminated.
The following chapter points out a problem with a divergent integral in a common approach and proposes an improvement. Some numerical comparisons are shown that indicate that the improvements allow for a superior image quality.
Thereafter an important problem is analyzed – the problem of limited data. In a realistic SAR-measurement the data gathered from the electromagnetic waves reflected from the surface can only be collected from a limited area. However the reconstruction formula requires data from an infinite distance. The chapter gives a comprehensive analysis of the artifacts which can obscure the reconstructed images due to this problem. Additionally, some numerical examples are shown that point to the severity of the problem.
In chapter 4 the fact that data is available only from a limited area is used to propose a new inversion formula. This inversion formula has the potential to make it easier to suppress artifacts due to limited data and, depending on the application, can be refined to a fast reconstruction formula.
In the penultimate chapter a solution to the problem of left-right ambiguity is presented. This problem exists since the invention of SAR and is caused by the geometry of the measurements. This leads to the fact that only symmetric images can be obtained. With the solution from this chapter it is possible to recon- struct not only the even part of the reflectivity function, but also the odd part, thus making it possible to reconstruct asymmetric images. Numerical simulations are shown to demonstrate that this solution is not affected by stability problems as other approaches have been.
The final chapter lists some conclusions drawn from the preceding chapters and develops some continuative ideas that could be pursued in the future.
Acknowledgements
First, I would like to thank Prof. Dr. Dr. h. c. F. Natterer for stimulating the work on this thesis and for offering helpful advice.
My thank also goes to Dr. F. Wübbeling who contributed to this thesis in many fruitful discussions and to all members of the institute for the pleasant atmosphere. Finally I would like to especially thank my parents and my significant other, Birgit, for their ceaseless support that made this thesis possible.
Subjects: Classical Analysis and ODEs (math.CA); Numerical Analysis (math.NA)
MSC classes: 65-XX
ACM classes: G.1
Cite as: arXiv:1010.4859 [math.CA]