Please use this identifier to cite or link to this item: http://hdl.handle.net/11527/16207
Title: Mikrodalga Difraksiyon Tomografisi İçin Yeni Bir Görüntüleme Algoritması Ve Diğer Yöntemlerle Karşılaştırılması
Other Titles:  a New Algorithm For Microwave Diffraction Tomography And Its Comparison With The Other Reconstruction Methods
Authors: Yazgan, Bingül
Kent, Sedef
14189
Elektronik Mühendisliği
Electronics Engineering
Keywords: Görüntüleme yöntemleri
Kırınım tomografisi
Imaging methods
Diffraction tomography
Issue Date: 1990
Publisher: Fen Bilimleri Enstitüsü
Institute of Science and Technology
Abstract: Mikrodalga difraksiyon tomografisi, dielektrik bir cisimden ileriye veya geriye saçılan elektromagnetik dalgaların örneklenmiş değerleri ile o cismin herhangi bir kesitindeki dielektrik sabitlerinin belirlenmesi ve dönüşüm algoritmaları kullanılarak cismin kesitsel görüntüsünün oluşturulması esasına dayanır. Birinci mertebe difraksiyon tomografisinde, saçılan alanlarla ilgili integral denklemin çözümü için yapılan Born ve Rytov yaklaşımları görüntüleme yöntemini belirli bazı özellikleri sağlayan cisimlerle sınırlar. Bu tez çalışmasında, genelde homogen olmayan dielektrik bir cismin görüntülenmesi amacı ile uzak saçılan alanları kullanan yeni bir difraksiyon tomografisi algoritmasının geliştirilmesi ve görüntü leme için uygulanması amaçlanmıştır. Bunun için saçılan alanları analitik olarak bilinen homogen ve tabakalı sonsuz uzun silindirler simülasyon amacı ile kulla nılmıştır. Geliştirilen algoritma ile saçılan alanlar uzak alan bölgesinde ve cismi çevreleyen bir daire üzerinde örneklenerek cisim fonksiyonlarının görüntüleri çıkarılmıştır. Bu yöntem, frekans uzayında daha geniş bir bölgenin doldurulmasına ve dolayısıyla görüntü kalitesinin iyileşmesine olanak sağlamaktadır. Ayrıca saçılan alanların uzak alanda örneklenmesi sonucu cisim fonksiyonu ile saçılan alanları birbirine bağlayan ifadede ortaya çıkan basitleşme yöntemin bilgisayar işlem zamanını azaltmıştır. Limit halde dairesel algoritmaya eşdeğer olan ve saçılan uzak alanların, cismin iki tarafında birbirine paralel sonlu uzunluklu iki çizgi boyunca sıralanmış, hem alıcı, hem de verici olarak çalışan anten dizileri ile örneklenmesi durumunda görüntüleme algoritmasının yapısı araştırılmıştır. Uzak alan algoritması, uzak saçılan alanların cisim etrafında kaynakla birlikte döndürülen düz bir çizgi üzerinde Örneklenmesi hali için uygulanmış ve görüntü çıkarıl mıştır. Kaynak sabit tutulup alıcı çizgisi ve cisim döndürüldüğünde algoritmanın göstereceği değişim ince lenmiştir. Dairesel görüntüleme algoritması ile elde edilen görüntüler, saçılan alanların cisimden belli bir uzaklıktaki düz bir alıcı çizgisi üzerinde yakın alanda örneklenmesine dayanan ve difraksiyon tomografisinde iyi bilinen Fourier difraksiyon izdüşüm algoritmasından bulunan görüntülerle, görüntü kalitesinin bir ölçüsü olan işaret/Gürültü oranı ve Ortalama Karesel Hata yönünden karşılaş t ırılm ıştır, örnekleme sayısının bu yöntem üzerindeki etkisi Nokta Dağılım Fonksiyonu (PSF) yardımıyla da incelenmiştir. Bütün yöntemler görüntü kalitesi, bilgisayar işlem zamanı ve işgal ettikleri bellek hacmi yönünden karşılaştır ılm ışlardır.
The field of computed tomography involves the nondestructive reconstruction of a slice of an object from measurements made external to the object. There are a variety of ways in which the object is irradiated and in which the measurements are made. Electromagnetic waves are diffracted and refracted when they encounter scattering objects. For this reason the type of tomography is named diffraction tomography. Reconstruc tions can only be accurately obtained when the object under examination is a weakly scattering object. As object complexity increases, reconstruction quality. decreases. Electromagnetic diffraction tomography is the inversion of scattered waves from an object, based on the Helmholtz wave equation. For the case of weak scattering, the first order Born approximation gives the relation between the object function and the scattered fields as £*(*,;) = A:2 Jg(F-?)0(FM*^V (1) In the two dimensional case the Green's function is written as G(;-;') = -(//4)//a|2,(A: |r-r1 ) (2) where Ho(.) represents the Hankel function of the second kind and order zero. Here we used both the far field scattered data and the cylindrical geometry to develop a reconstruction algorithm for microwave imaging. (vi) Let us consider an infinite dielectric cylinder whose axis is taken to be parallel with the z-axis of a rectangular coordinate system. The incident plane wave, propagating in the direction of the unit vector so, which is normal to the a-axis is given by B'ikjy -Jta,.r (3) where k is the wavenumber of the surrounding medium, r is the position vector and exp(+jwt) is implied. The scattered fields from the object can be measurejd by a circular antenna array at a constant distance \~r\ from the center of the object as shown in Fig: 2. 2. If we assume that the observation point is in the far field, the distance from the scatter to the receiver point can be simplified as A-|r-?|- {r2+r'2-2r?f. (-?)) U2»r + ?'.s (4) where U is the unit vector in the opposite direction of r. By using the incident plane wave as in Eqn. (3) and the Hankel function representation of the Green's function in Eqn. (1) the scattered field is obtained as E'{k,r)-[jk2/4)fHT CkR)0[r)& ""^ d: (5) Since we assumed that the receiver point is in the far field region, the asymptotic expansion of the Hankel function can be used to obtain, £'{*>?)"-jJ -ğ^T e ""4 e ~Jkr J0{?)e -**«&' dzr' (6) The integral term in Eqn. (6) represents the two dimensional^ 2-D) Fourier transformation of the object function 0("r). In this case, the relation between the far field scattered wave and the 2-D Fourier transform of the object function can be obtained as £s(fc,;) = -/fc3/2 )"/*/¦ -/fcr V 8/CJ 3(*(s + 3.)) (7) Since the distance r is constant in the proposed circular antenna array, the multiplicative terms in Eqn. (7) yield a constant factor. To obtain the 0(x,y) distribution of the object function in Cartesian coordinate system, by inverse transformation, first the location of the O(k(s+so)) in the 0(u,v) frequency domain must be found. (vii) In the circular antenna array every antenna element in its turn, is considered to be the transmitter, while the others are receivers. Let the angle between the _^ direction of propagation of the plane wave (given by so) and the horizontal axis be Xo. Similarly, let the angle between the antenna position vector Is and the horizontal axis_J>e^X. The known values of the Fourier transform Q(k(s+so)) of the object function are on a circle as shown in Fig: 2. 3. This circle has its center at kiTo, and its radius is k. When the angle X changes with the receiver antenna element, the circle also changes position, but its radius k remains constant. When the circular antenna array has regular j^onstan t spacing, the points on the circle given by k(et+so) are also arranged regularly, with the incremental angle between subsequent points being constant._<. A point on the circle can be shown with the vector K given by /?=fcs0 + fc? = |£|e'* (8) The equations of the circle on the (u,v) -plane can be written as {u-k.cQsX0)2+{v-k.sinXa)z~k2 (9) Thus the center of the circle is at (k.cosXo, k.sinXo). As the angle Xo changes, so does the center of the circle. Since \K |.cosa = k.cosX0 + k,CQsX \K\.sina = k.sinXa + k.sinX (10) the amplitude |KI and the angle <* between K and the horizontal axis can be written as |£|-fcV 2{l+cas{X0-X)) (11) a» arctanf - - } (12) cos X a + cos X The denominator of Eqn. (12) is aero when cos Xo =-cos X or cos Xo =0 and cos X =0. The first case is obtained when x.~x+a, thus cos(Xo-X)=-l. Then lit 1=0 according to Eqn. (11). The second case is obtained when x.-nx/2. x-ma/2 with m,n (viii) being odd positive integers. If m=n then a-na/2. When m + n.x-xa-in, and thus |K|=0. Then, there is no need to compute « for both cases. In summary, each set of data obtained when one of the antenna elements transmit while others receive results in a circle in the frequency plane of diameter 2k. If there are N antenna elements, the received field values are measured at all the elements for each element used as the transmitter. This means the field values Es(so,s) can be considered to be the result of one transmitter-receiver configuration. These values are converted into the frequency domain, using Eqn. (11)-(12). The wave vector matrix elements can be written as K(i,j) and «(£,/). *U.j) is the phase matrix representing the angles of K. The i,j denote the numbers of the transmitter and the receiver, respective ly. In Fig: 2. 3, a 4-element antenna array and the location of the circles in the frequency domain is^ shown. There are some pairs of points with same |K| values but different angles at each circle. To separate these points from each other, the circle çan be divided into two halves as shown in Fig: 2. 3. The |K| values are assumed to be positive at upper half and negative at lower half of the circle. Then K(i,j) matrix is rearranged with the negative elements. The phase matrix a[i.j) corresponding the new K(i,j) is also rearranged between 0-2 tr as an NxN matrix. To determine the object function Q(K.a) which represents the locus of points of Es(i,j) in frequency domain the following steps should be taken: i- An alfa(i) vector is set up as alfa(i) = (i-l) tt/N for i=l, 2N ii- To obtain the sampled K values, a K(j) vector is formed. Each element of these vector must be equal to the projection of the K(i,j) points on the u-axis. Then for a point (i,j) in frequency domain, one can write, f'(fc.i). if K(ktl)-KU) and a(k,l)- alfa(i) (13) 0, otherwise Q(i,;)- The resulting Qi*c.a) will be a 2Nx2N matrix. To obtain the object function on a rectangular grid as required by the 2-D inverse FFT algorithm, the nearest-neighbor (ix) interpolation technique is used. By taking the 2-D inverse FFT of the resulting 0(u,v) matrix, the object function 0(x,y) is obtained. Chapter 1 is an introduction to microwave diffraction tomography reconstruction problem. In this chapter, electromagnetic diffraction tomography is defined and the previous studies are briefly reviewed. Basic contents of the thesis are also outlined. In Section 1.4, object function and the integral equation, showing the relation between the scattered fields and the object function are described. First order Born and Rytov's approximations to solve the integral equation are also given. Then, the Fourier Diffraction Theorem which is the fundamental concept of the diffraction tomography is introduced and the reconstruction procedure is examined. In Chapter 2 a new reconstruction algorithm is presented for the circular data sampling system geometry and electromagnetic far field scattered data from a dielectric object. Chapter 3 covers the reconstructed images of the objects that obtained by using the near and far scattered fields for the algorithm based on the Fourier Diffraction Theorem and the new far field circular geometry algorithm respectively. The obtained results are compared using image quality parameters such as Mean Squared Error (MSE) and Signal to Noise Ratio (SNR). The performance of the tomographic imaging technique based on the circular measurement geometry is examined by a computer-aided reconstruction of the point spread function (PSF), taking into account the effect of the number of the frequency values of the exploring radiation. The reconstruction procedure by using Fourier Diffraction Theorem and interpolation techniques, the numerical calculations of the scattered fields for the homogeneous and layered dielectric cylinders, the iterative techniques that more accurately model the scattered fields, the flow charts of the computer programs, the image quality criteria and the effect of the distance to the scattered fields are given as appendices. Finally, in the last section the obtained results are compared with each other. The MSE for the reconstructed object function of the two layered infinitely long circular cylinder obtained via conven tional near field planar geometry is equal to %3.686 (Fig: 3. 3) and for the image quality of the same cylinder one obtains MSE=%3.468 (Fig: 3. 17) by using far field scattered data and cylindrical geometry. Signal to Noise Ratios of these images are 10.75 dB for the first case (x) and 12.40 dB for the circular geometry. For planar sampling geometry the required computer processing time is longer than that for the circular case. For the reconstructed image (Fig: 3. 9) obtained by using the forward far field scattered data at a line perpendicular to the direction of the incident wave as given in Section 3.3, the MSE is equal to 7.360 and SNR is 10.681. Finally the value of MSE for the three layered cylinder (as shown in Fig: 3. 18) is %5.40 and SNR is equal to 11.762 dB for the circular geometry far field algorithm.
Description: Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1990
Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1990
URI: http://hdl.handle.net/11527/16207
Appears in Collections:Elektronik Mühendisliği Lisansüstü Programı - Doktora

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