Difraksiyon tomografisinde hankel dönüşümü algoritması ile görüntü oluşturma

Abstract

Homojen bir ortam içinde bulunan ve genellikle homojen olmayan dielefctrifc bir cismin, düzlem elektroman yetik dalgalarla aydınlatılıp daha sonra saçılan alanlarının toplanarak kesit üzerindeki cisim parametrelerinin belirlenmesi ve cismin kesit görüntüsünün oluşturulması kısaca Mikrodalga Difraksiyon Tomografisi olarak adlandırılır. Mikrodalga difraksiyon tomografisi, teoremin temel ifadesi olan integral denklemin çözümün de yapılan Born ve Rytoy yaklaşıklıklarının gerektirdiği bazı ön koşulları sağlayan cisimler için doğru sonuçlar verir. Bu tez çalışmasında, görüntülemek üzere dielektrik silindirik bir cisim gözönüne alınmıştır. Silindirin z- ekseni boyunca sonsuz uzun ve cisim parametrelerinin, yani dielektrik permi ti vites inin ve permeabilitesinin de yalnızca ( x - y ) koordinat düzleminde değiştiği düşünülmüştür. Kullanılan düzlem dalga ise z- doğrultusunda polarize olmuş bir dalga olarak seçilmiştir. Böylece yapılan hesaplar kolaylaşmış ve vektörel denklemlerin skaler forma indirgenmesi sağlanmıştır. Silindiri aydınlatan elektromanyetik dalga kaynağı, silindirin çevresinde periyodik açı değerleri boyunca değiştirilmiş ve her bir geliş açısı için de yine periyodik aralıktaki açılarda saçılan alanlar ölçülmüştür. Ölçümler uzak alanda (20X) yapılmıştır.

Microwave diffraction tomography is based on defi nition of dielectric constants on a scattering dielec tric object's cross-section using the samples of the electromagnetic waves scattered forward and backward from the object. The cross -sectional image reconstruc tion is obtained by applying transformation algorithms. The main difference between the microwave diffrac tion tomography and the X- ray tomography which is based on the projection-slice theorem is that X- rays propa gates along straight lines and gives informations about only a slice of the object since it is nondiffracting and interacts with only a slice of the object. But the electromagnetic waves being diffracting, interacts with whole the object and after scattering, the scattered fields can define the dielectric properties and the shape of the dielectric object. The electromagnetic waves scattered by the dielec tric object is characterized by the Helmholtz Differen tial equation, [ V2 + k2 ] Ec = -g(r) (1) where g(r) = k2 Oir)E(î) <2) 0(r) and E(r> are object function and total fields in the medium respectively. (vi) The differential equation (1) is reduced to a in tegral equation by using the Green's function, E (r) = J G(r,r') g"(r') d2r' E (r) = k2 j G(r,?'> O(r') E d2r' (3) This integral equation can be solved under some certain approximations only. This approximations made to solve Eqn. <3) are named as Born and Rytov approximations. In this thesis only the Born approximation has been applied to solve the integral equation. For the solution, the total electromagnetic field in the medium is written as E = E. + E (4) ı s and substitution of this equality to Eqn. (3) by applying the Born approximation which is given as yields E = k2 fG(?,?') (Mr"') E.(î') d2r' (5) s J ' x so that it can be said that according to the Born ap proximation there should be a weak scattering and the object should not be complex. In Born approximation the more complex the object, the worse the quality of the image. For the two dimensional case the Green's function is given as (vii) (2 > Gtr,r') = - (j/4) H Ck|r-r'|) <2> where H (. ) represents -the Hankel function of the o second kind and zero order. In this thesis the scattered electromagnetic fields is measured in the far field (2 OX) and the object has a cylindirical geometry. under these considerations, the reconstruction algorithm is developed. After organizing Eqn. 5 by doing appropriate analy sis, the equation -jkr « #-*» *. -,a/2 jrc/4 r,-./-*»» -jk(s +s)r* 2, E (r) * -j k eJ [ O(r') e J o dr* S. _, 1/2.> (8rrr) (6) is got. As seen from Egn. (6) the integrand is the value of O(r) at k(s +s) the frequency domain and the integral is limited by the bounds of the object. Since this thesis involves with the polar coordi nate based Hankel transforms, the tans mi t ter and the receiver antennas are thought as a circular array; so that "r" in Eqn. (6) becomes is a constant. In the circular antenna array case, the samples of the scattered electromagnetic waves are located on the points defined by r = k V 2 fl + cos(*> - $> )1, 0 £ r £ 2k sin ?>. + sin 0^.A* **" * t ma reev ] COS i> + COS = l/2n f r E Cr> J (rp> dr n J n n o and as the last step the object is obtained as n=-co The operation procedures can be summed as the fol lowing, (1). A 1-D PFT is applied to the sampled values E(r,Ö) and E (r) are obtained, n (2). A Hankel transform is applied to E (r) and O (p> are obtained, A 1-D FFT is function is obtained. (3). A 1-D FFT is applied to O _(p) and 0(p,*>) object Because of the fact that there is not any interpo lation operation in the Hankel transformation recon struction Algorithm, it reconstructs more sensitive and' correct results. Because, the interpolation methods themselves cause errors in the results. But the one thing that makes the Hankel transform reconstruction algorithm hard to work with is that it is a relatively slow algorithm since calculating the Bess el functions takes time. Passing through the chapters: Chapter 1 is an introductory chapter to microwave diffraction tomography and the reconstruction problem. In this chapter the main equations of electromagnetic (ix) diffraction tomography are described. The Hankel trans forms are introduced and previous studies about Hankel Transforms in electromagnetic diffraction tomography are briefly reviewed. In Chapter 2, the Hankel transform reconstruction algorithm is presented and the relation between the Hankel transforms and Fourier transforms is defined. Chapter 3 includes the image reconstructions of three different sample objects. These three samples are one-layered, two-layered and three-layered dielec tric cylinders which are infinitely long along z- axis. In this chapter, the surface topologies of the recon structions and Signal to Noise Ratios (SNR) are given as well. Additionally, the comparison between the Hankel transform reconstruction algorithm and Forier transform based algorithm is made by comparing the signal to noise ratios <£NR). The flowchart of the computer programs used to re construct the image is also given in this chapter. Finally, in the last section, the conclusions and suggestions are presented. At the end of the thesis, there are five appendices Appendix. 1 explains finding the Hankel Transforms from the main mathematical integral transform relation choosing the kernel function. Appendix. 2 gives the properties of the Hankel transforms. Appendix. 3 represents the Hankel transforms of some functions as a table. Appendix. 4 gives the properties of the Bessel functions. Appendix. 5 explains the signal to noise ratio (SNR).