Yalnız genlik bilgisinden yaralanarak mikrodalgalarla görüntüleme

Abstract

Bilindiği üzere difraksiyon tomografisi yöntemiyle görüntü oluşturma problemlerinde alınan dalganın hem genliğini hem de fazını bilmek gereklidir. Bu durum difraksiyon tomografisi uygulamalarını kısıtlayıcı bir neden olarak karşımıza çıkar. Bu tez çalışmasında faz ve genlik bilgilerinin birbirinden tamamen bağımsız olmadığından hareket edilerek sadece alınan dalgaya ait genlik bilgisini data olarak kullanmak suretiyle cismin görüntüsünün oluşturulabilmesi problemi ele alınmıştır. Bunun için de difraksiyon tomografisinin genel formülleri ve Rytov yaklaşımı kullanılarak genlik ve faz arasındaki ilişki gösterilmiştir. Kolaylık sağlamak amacıyla da skaler dalgalar ele alınmış ve klasik ölçüm yöntemlerinin kullanıldığı varsayılmıştır. Difraksiyon tomografisindeki görüntü oluşturma yöntemlerinden de filtrelenmiş geri propagasyon algoritması kullanılarak cisim fonksiyonlarının görüntüleri elde edilmiştir. Bulunan sonuçlar genlik ve faz bilgisinin birlikte kullanılmasıyla elde edilenlerle karşılaştırmıştır.

In diffraction tomography, the reconstruction algorithms are based on the use of both intensity and phase of the transmitted wavefields which are obtained from the experimental measurements. These data are the input of the diffraction tomography reconstruction algorithm. This requirement has limited the use of these reconstruction procedures and has adversely affected the use of diffraction tomography in practical applications. A weakly inhomogeneous scattering object which is illuminated by a known monochromatic incident wavefield characterized by a complex index of refraction n( r ) = 1+5n( r ) and the transmitted wavefield is measured over a recording surface with the goal of deducing the complex index perturbation 8n( r ). We consider the classical measurement configuration ( shown in Figure 1. 1 ) so that the scattering object is imbedded in a homogeneous nonabsorbing, isotropic medium having a constant index of refraction which will be taken unity with no loss in generality. Following the usual procedure in diffraction tomography we express the wavefield U = exp( ikW ) resulting from the interaction of the incident plane wave U, = exp( iks0r ) with the object in terms of its wavenumber normalized complex phase W. Here k = 2nlX is the wavenumber in the background medium in which the object is imbedded, s0 is the unit propagation vector of the incident plane wave, and r is the vector location of a general field point. The complex phase W can be decomposed into the sum of the phase of the incident wave and a perturbation 8W that is introduced by the presence of the scattering object. W = s0r + 8W ( 1 ) Clearly, all information about the scattering object is contained in the complex phase perturbation 5W. The complex phase which was given in ( 1 ) satisfies a non - linear equation of the Ricatti type. This equation can be linearized if the phase perturbation 8W is small relative to the incident wave phase s0r. The linearizing procedure is known as the Rytov approximation and forms the basic concept of reconstructive diffraction tomography. Within this approximation, the following linear mapping from the complex index perturbation of the object to the complex phase perturbation 8W can be obtained. SWr (r ; s0 ) = JdP 8n( r' ) r( f - r' ; s0 ) ( 2 ) Here 5Wr is the Rytov approximation to the complex phase perturbation and quantity r is, T( R ; s0 ) = 2ik exp( -iks0R ) G( R ) ( 3 ) where r = r - 1" and G( R ) is the Green function. G( R ) = - ( 1/471 ).( exp(ik |R| ) / |R| ) ( 4 ) The integral transform defined in ( 2 ) is the diffraction tomography counterpart to the Radon transform of X - ray computed tomography. The reconstruction problem in diffraction tomography reduces to estimating the index perturbation, 8n from a given set of generalized projections SWr obtained in a suite of experiments employing a set of incident wavevectors s0 e Q. The backpropagation of the complex phase from the plane r\ = l0 into the half - space r\ < l0 that contains the scattering object by means of the formula BP{8WR}(f;s0) = Jdf.'8WR(r';s0)Y<(f- r';s0) (5) In this formula, S denotes the plane i\ = l0 and f = fJ + s0 l0 = (&', [, ) is the vector location of a general point on 2 with 7.' being the projection of this vector onto this plane. The integral in ( 5 ) is taken over E and the quantity y< is backpropagation kernel and given in terms of the outgoing wave Green function Gq that vanishes on E as follows : Y{ ( r - r'; s0) = exp[-iks0(f-f')]^-G^(r-r') (6) where the partial derivative is taken with respect to the primed coordinate and where superscript asterisk * stands for the complex conjugate. The Green function Gd is expressible in terms of the outgoing wave Green function G via the formula VI GD(r- f')=G(r- r')-G( f - P) (7) where f = (r., 2I0 - rj) is the mirror image of the point r = (I,r\) about the plane £. We can see from ( 5 ) and ( 6 ) that the real and imaginary parts of the complex phase are not independent. Let us consider the forward propagation of the complex phase from its boundary value on S into the half - space r| > l0 ; 5WR(r ;s0) = Jdr' 8% (r'; s0 ) y > (r- r'.s0) (8) where kernel y> is defined as follows : Y > ( r - r'. s0) = -exp[-1ks0(f - r')\^-GD{f -f) ( 9 ) Now, we consider the backpropagation of the complex conjugate of phase perturbation 8Wr on the plane S. From ( 5 ), BP{8W£}(F;s0) = Jjdf.'8WR(r';s0)Y;(r- r'.s0) I (10) is obtained. Also from ( 6 ), ( 7 ) and ( 9 ) it is obvious that for ti < l0 ; y*(r- r'.s0) = -Y)(r-f';s0) (11) Upon substituing ( 11 ) into ( 10 ) we find that BP{8W;}(r;s0) = -8W;(r;s0) (12) where 8Wr( f ; s0 ) is the forward propagation of the boundary value of the phase perturbation SWr to the conjugate image point f. This is the important result for connection with the reconstruction problem using intensity data. If we remember that the total wavefield is U = exp( ikW ) and the compleks phase W is as defined in equation ( 1 ), we can write vn (1 / ik ) In I U |2= 2i lm( 8WR) (13) Now if we use the above result, we can rewrite ( 13 ) as follows : (1 / ik ) In İUİ2= 5WR - 8Wr (14) We can put this value in place of the complex phase perturbation 8Wr in the filtered backpropagation algorithm. The filtered backpropagation algorithm is one of the diffraction tomography reconstruction methods and is analogous of the filtered backprojection algorithm in X - ray computational tomography. This algorithm reduces to the filtered backprojection algorithm in the limit where the wavelength tends to zero. Object function, 0( x, y ), can be find as given below in this algorithm : 271 0( x. y ) = ( 1 I 27t ) fd^n,^ xsin<|) - ycos(|), xcoa|) + ysin(|) ) ( 15 ) 0 where ru( Ç, r\ ) is the filtering function and defined as, oo n,( Ç, r\ ) = ( 1 / 2% ) Jr4>(o))H(ra)Gtl(cû)exp( ia>Ç ) do ( 16 ) -00 In ( 16 ), G^co) is the transfer function of depth - dependent filter (17) (18) (19) Vffl Also the coordinate systems ( a>, <|>) and ( Ç, r\ ) are shown in Figure 3. 1 and 3.2 The computational procedure for reconstructing an image on the basis of equations (15 ) and ( 16 ) may be presented in form of the following steps : Step 1. In accordance with equation ( 16 ), filter each projection with a separate filter for each depth in the image frame. Step 2. To each pixel ( x, y ) in the image frame, in accordance with equation ( 15 ), allocate a value of the filtered projection that corresponds to the nearest depth line. Step 3. Repeat the preceding two steps for all projections. As a new projection is taken up, add its contribution to the current sum at the pixel located at (x.y). If we apply this algorithm to our problem, we can represent this algorithm mathematically in the following form 8n(r) = £BP{5\AU(r;s0) (20) s0eQ where 8n is the reconstruction and where the overbar indicates a two - dimensional convolutional filtering operation applied to the boundary value of the phase perturbation on the plane £. The backpropagation operation in the above equation is performed into the half - space ti < l0 for each value of the propagation vector s0 ; i.e., for each view of the object. Chapter 1 is an introduction to the problem of image reconstruction from intensity data. In this chapter the basic aim of this thesis is represented. In chapter 2, signal reconstruction from Fourier transform magnitude is investigated and a method studied and developped by Hayes is introduced. Chapter 3 the diffraction tomography and Rytov approximation are briefly reviewed. In section 3. 3, the relation between the phase and the intensity are described. Chapter 4 contains the presentation of the filtered backpropagation algorithm. EX Finally, in the last chapter the obtained results are compared with the optimal reconstruction generated from both the magnitude and real phase of the transmitted wavefields.