Mikrodalga difraksiyon tomografisi için adaptif bir görüntü oluşturma algoritması

Abstract

Mikrodalga sistem ve teknolojilerindeki gelişmeler mikrodalga düzenlerinin değişik uygulamalarda kulanılmasım mümkün kılmaktadır. Bir cisim yada ortamın fiziksel paremetrelerine dokunmaksızın Özelliklerini belirlemek, pratikte birçok uygulama alanı bulabilen bir ölçme yöntemidir. Halen değişik kaynaklar ( X-ışını, elektromagnetik dalgalar, manyetik rezonans, akustik dalgalar ) kullanılarak cisimlerin fiziksel paremetre dağılımları bulunmaya çalışılmaktadır. Elektromagnetik dalgalar yardımı ile mikrodalga difraksiyon tomografisinde Fourier difraksiyon teoremine dayanarak zayıf kontrastlı yani Born yaklaşımının geçerli olduğu cisimlerin iki boyutlu kesit parametrelerinin bulunması mümkün olmaktadır. Ancak yüksek kontrastlı dağılımların belirlenmesinde bu teorem yetersiz kalmaktadır. Araştırmacılar bu gibi kuvvetli saçıcı cisimlerin dağılımlarını belirlemek için hızlı ve güvenilir ters saçılma algoritmaları aramaktadırlar. Bu çalışmada halen geliştirilmeye çalışılan algoritmalardan farklı olarak çözüme adaptif hata azaltma yöntemi ile ulaşılması araştırılmıştır. Tezin birinci bölümünde silin dirik bir cisimden saçılan alan değerlerinin ifadesi sunulmuştur. İkinci bölümde ters saçılma bağıntısının moment methodu ile oluşturulması ve saçıcı sistemin modelinin adaptif sisteme uygunluğu gösterilerek, ters saçılma bağıntısının adaptif bir çözümü verilmiştir, üçüncü bölümde algoritmanın mikrodalga difraksiyon tomografisine uygulanması değişik örnekler ile gösterilip performansı incelenmiştir. Bu tezde halen kullanılan mikrodalga difraksiyon tomografisinde görüntü oluşturma algoritmaları ile sağlanamıyacak resolüsyonda görüntülerinin adaptif algoritmalar yardımı ile basit ve hızlı bir şekilde oluşturulabileceği gösterilerek, adaptif algoritmaların deterministik algoritmalara iyi bir rakip ve tercih edilebilir bir alternatif oluşturduğu ispatlanmıştır.

In the past few years, there has been a growing interest in microwave imaging. However, concerned about the complexity of the mathematical models involved in microwave imaging processes, researchers are currently working on the development of simple, yet efficient and reliable, imaging systems. In the past, several algorithms based on the well-known Fourier diffraction projection theorem were developed for electromagnetic diffraction tomography. Since such algorithms are generally based on Born's or Rytov's approximations, they have been successfully utilized in case of weakly scattering objects. However, they usually fail when applied to strong scatters. The aim of this thesis is to show that general microwave diffraction imaging formulation illuminates alternative, computationally efficient solution methods for several classes of signal processing problems. Diffraction imaging problems arise in physics, geophysics and acoustics; and in one class of formulations they require a procedure to determine parameters of medium from wave measurements. There exists a close relationship between the physical microwave imaging problems and some issues in signal processing such as the design of digital filters, the development of linear prediction algorithms and their adaptive filter implementations. For many of these problems several efficient algorithms already exist in the literature, but the connection between the different solutions are not always clear. In this thesis alternative procedures are shown which correspond to conceptually extremely simple, basic ways of solving microwave imaging problems: the so called adaptive prediction methods. Examples include LMS method for determining the optimal medium parameters. - IV Consider the two dimensional scattering problem shown in Fig. 1.; where the formation is characterized by an index of refraction n(?) which is equal to a real constant ( no loss of generality ) except over some finite region S that is contained between the source and receiver arrays. No assumption will be made concerning the magnitude of index n(r") so that the developed theory will be valid for both strongly and weakly scattering formations. However, the formation is two-dimensional so that n(?) is a function depending only on x and y as illustrated. The plane wavefield is generated by a suitable linearly phased line sources. The wavefield satisfies the reduced wave equation which can be written in the form [v2+fc2]tf(r)«0(?)£/(?) (1) where O(r) is called as object function 0(?)=fc2[l-n(T)] (2) where r denotes the two-dimensional position vector, V is the two-dimensional Laplace operator and k=2rc/A is the wave number in the region exterior to S; in which the index of refraction n(f)=l. ntr>»l Figure 1. Geometry of the scattering system. The solution to equation (1) are given as - V - U(?)-U,n (?)+ j 0(T-)U(r-)G(r,?*)<&' (3) where Ujn(r) represents incident wave into the object from source and GC?,?) is the Green function for a homogeneous medium ca JO- -U)fio\k\7 -f'\) (4) with Ho the Hankel function of the second kind represntingad outgoing wave for assumed exp(jwt) time dependence. The scattered wavefield is defined to be the difference between the total and incident wavefields. US(?)~U(7)-U ln (?) (5) U^r^^k'jo^U^H^Ck^-r'ncfr' (6) Let us divide the cross section of the object into L cells sufficiently small so that the index n(?)=ni and total wavefield intensity U(î)=Uı are essentially constant over each cell. The scattered wavefield outside the object can then be written [1] 2 I-l where a is the radius of the circular region and Ji is the Bessel function of the first kind. The scattered wavefield at any point r outside the object is given by tf.(?)- Y. "><»<(?) (8) 1-1 w.-^^p^ft.-Otf.a.y.ffca,) (9) «,(?)- tf$2)(kl?-r,l) (10) - VI - The scattering system of Fig.l models a general linear system with input ( vı(î) ) and output ( UB(f) ) relation. Where the coefficients wi are simply the spatial mapping of the system's weighting. In the scattering terminology, the wi provide the local parametrization of the propagation medium. For each scattered wave, one equation can be written relating these unknowns to the measured data. Doing this for each scattered wave anables one to obtain a system of K equations in L unknowns. The problem of estimating wi has now been reduced to solving the set of linear equation in (8). However, several problems remain. First, the data Us(r) is not known exactly in any practical case as a result of various sources of noise in the system and approximations in the discrete model. Generally this will cause (8) to be inconsistent. Second, the number of independent equations, as determined by the scanning geometry, is usually insufficient to determine wi uniquely. Finally, in many cases the number of equations is too large for direct matrix inversion or pseudo-inversion tecniques. For example, if we desire a resolution corresponding to 100x100 cells, then there will be 10000 unknowns and an equatin coefficient matrix of perhaps 10000x10000. Although the equation coefficient matrix is sparse, standart solition methods are impractical. Basicly, an iteratif method treats one equation at a time. In the case of system ( model parameters ) identification, the model takes the form of linear combination equations (8) where the significant terms, as well as coefficients, are to be estimated. The difference between Uo(rm) and its estimated value at any time m and location rm is called estimation error. e(rJ = tfB(?m)-l/HF(rm) OD l/r-[iö, w2....»J (12) ^py-MrJ v2[rm)....Vl{?m)} (13) where the superscript T denotes transposition and H denotes Hermitian transposition. The minimum mean-square error criterion is used to optimize the coefficients. Specifically, the weights are chosen so as to minimize an index of performance, J(W), defined as the mean-square value of the estimation error or mean-square error ( mse ): J(V)-E[e{rm)e'[?m)} (14) - VII where the asterisk denotes complex conjugation and E denotes the expectation operator. Substituting equations (11) and its complex conjugation in (14), one can write the mse as J(V) = al~ pHW-WHp+WHRW (15a) ol-E[U,(ri)U't{7Z)] (15b) p = E[v[l?m)Ul[r*m)] (15c) R~E[v{Tm)VH{rm)] (15d) By minimizing J(W), one can obtain the best or optimum coefficient in the minimum mean-square sense. The method of steepest descent uses gradients of the mean-square J(W) in seeking its minimum. The gradient at any point r may be obtained by differentiating the mse function, equation (15a), with respect to the coefficients vector W. The gradient vector is VJ = -2p + 2Kl/ (16) Setting the gradient to zero to find the optimal coefficients W one obtains, RV op =p (17a) Vop-K-'p (17b) which is the Wiener-Hopf equation in matrix form. If it were possible to make exact measurements of the gradient vector at any r point, the coefficients computed by using the steepest descent method would indeed converge to optimum Wiener solution. However, exact measurements of the gradient vector are not possible, and gradient vector must be estimated from the available data. In other word coefficients are updated in accordance with an algorithm that adapts to incoming scattering waves at any point. One such algorithm is the least-mean-square ( LMS ) algorithm. The simplest choice of estimators for R and p is the instantaneous estimate which is based on sample values of the scattering wave and V(r) function, as defined by, R-V{Tm)VH{Tm) (18a) - VIII - P-V[Tm)U\[Tm) (18b) v--2v[rwl)u;(rm)+2v[rm)va[rm)v (18c) Substituting the estimate of gradient vector (18c) in the steepest-descent algorithm [2], one can get an updating relatim for the coefficients e{?m)-us{rm)-w"mv{rm) (19) i/mH -vm+pv[Tmy[?m) (20) Figure 2. Flow graph of adaptive system Fig.2 describes the flow graph of the adaptive system defined by (19) and (20). The iterative procedure is started with initial guess W ( one may set Wo=0 ). p is a positive real- valued constant and controls the size of the incremental correction. The convergence speed of the LMS algorithm depends on the choice of the step size p. The larger the step size is, the faster the convergence becomes. However, large step sizes lead to instability. The step size must lie in the flowing range to insure stability, 0 < fi < total input power (21) IX - where the total input power refers to the sum of the mean-square values of the system inputs U(rm). If the step size parameter p could not be chosen suitable, one can be use the normalize LMS algorithm, equation (21) turns as V'«.i ~Vm+ Ü 2 V{?mW{rm) (22) il nr«)ii where

.)i is the norm operation and 11 have to be chosen as 0<(j,<2 (23) To demonstrate the performance of the LMS adaptive algorithm, three different type object functions are used in section fifth. These objects shown in Figüre 3.1 and Fig. 3.2 which does not have any symmetry properties. The adaptive system used in the simulation is depicted in Fig.2. Reconstructed object are given in Fig. 3.4, Fig. 3. 6 " and Fig. 3. 8» Numerical performance of the algorithm presented in for different kinds of SNR. The adaptive system identification algorithm presented does not require a priory knowledge of, or assumptions about. LMS algorithm can be find the optimal solution slowly. However, in practice, the use of LMS algorithm is wide-spread due to its computational simplicity. Alternatively, the convergence behavior of the another recursive least square ( RLS ) algorithm is independent of the spectral characteristics of the scattering wavefield. For the LMS algorithm, such a dependence is very strong. The LMS algorithm lead to a computational complexity of 0(L), but the RLS solution can be recursively computed using the well-known conventional algorithm with a complexity of 0(L2) computations. Otherwise the usual way of solving linear equations ( deterministic inverse scattering problem ) requires an 0(L3) computation.