Bu çalışmada yanına erişilemeyen mükemmel elektrik ileten cisimlerin şekillerinin elektromagnetik dalgalar kullanılarak uzaktan belirlenebilmesi için yeni bir yöntem geliştirilmiştir. Ele alınan ters saçılma probleminde yanına erişilemeyen cisim elektromagnetik dalgalarla aydınlatılır ve cismin bu gelen dalgalarla etkileşimi sonucu ortaya çıkan saçılan dalga uzak alan bölgesinde ölçülür. Çalışmadaki esas amaç bu gürültülü ölçüm bilgisini kullanarak cismin şeklinin belirlenmesidir. Bu problem teorik açıdan baştan kararlı bir çözümün varlığının ve tekliğinin öngörülemeyeceği doğrusal olmayan kötü kurulmuş bir problemdir. Bu çalışmada sunulan yöntem ele alınan problemin kötü kurulmuş ve doğrusal olmayan kısımlarının ayrı ayrı ele alındığı bir analitik devam yöntemidir. Öncelikli olarak uzak alan verisi cismi çevreleyen minimum yarıçaplı bir daire üzerinde tanımlanan bir tek-katman potansiyel yoğunluğu yardımıyla ifade edilir. Ardından elde edilen integral denklem kesilmiş ayrık değer ayrıştırma yöntemi kullanılarak, regülerize edilmiş bir biçimde çözülüp, minimum daire üzerindeki bilinmeyen tek-katman potansiyel yoğunluğu belirlenir. Bu potansiyel yoğunluğu yardımıyla minimum daireden biraz daha geniş bir daire üzerinde hesaplanan saçılan alanın Taylor serisi açılımıyla, saçılan alan cismin yüzeyine kadar analitik olarak devam ettirilir. Cismin üzerinde toplam elektrik alanın sıfıra gitmesi biçiminde tanımlanan sınır koşulu yardımıyla, cismin şeklinin belirlenmesi problemi doğrusal olmayan bir denklemin köklerinin bulunmasına indirgenir. Bu polinom biçimindeki denklem Gauss-Newton algoritması kullanılarak yinelemeli bir biçimde çözülür. Sayısal sonuçlarla gösterilmiştir ki yöntem hem konveks hem de konkav tarafları olan yıldızbiçimli yüzeyler için başarılı sonuçlar vermektedir.

In this study, a new method for shape reconstruction of a perfect electric conducting target through the use of electromagnetic waves is presented. The aim in this inverse scattering problem is to retrieve the shape of an unknown target remotely, provided that the scattering wave which is the result of electromagnetic interaction between the target and the incident wave is known in the far field region. Theoretically the problem considered is a nonlinear illposed problem in which the existence and the uniqueness of a stable solution cannot be anticipated initially. The method presented in this study can be classified as an analytical continuation method in which the ill-posedness and the nonlinearity of the underlying problem are handled separately. The shape reconstruction problem is studied in twodimensional case. More precisely, the inaccessible target which is located in an infinite, lossless, nonmagnetic and homogeneous space is assumed to be infinitely long and unchanging at one direction. The target is illuminated with a set of time harmonic plane waves of different incidence angles at a fixed frequency whose electric field is always parallel to the direction of the target. The scattered field which satisfies the Helmholtz equation together with appropriate boundary conditions is sampled on a full aperture in the far field region. Since ill-posed problems have a strong sensitivity to small perturbations on the input data, the scattered field is corrupted synthetically in order to model the inevitable measurement noise. The initial step of the method is concerned with the reconstruction of the scattered field in the vicinity of the target from the noisy far field pattern. To this aim, the single layer potential representation is exploited such that the far field pattern is modeled as if it is generated by an unknown single layer potential density on a circle which is assumed to cover the inaccessible target with preferably minimum radius. The resulting Fredholm integral equation of first kind is severely ill-posed due to smoothing properties of its kernel. Thus the singular value decomposition is exploited to invert the integral equation in a regularized manner to determine the unknown single layer potential density. At this step, Morozov’s discrepancy principle is utilized to select a proper regularization parameter. Once the potential density is reconstructed, the scattered field can be approximated in the region outside of the minimum circle. Later, to represent the scattered field inside the minimum circle, a Taylor series expansion of the reconstructed scattered field is exploited. In particular, instead of the scattered field on the minimum circle, the field on a larger circle is expanded into Taylor’s series in the radial direction in order to avoid the singularity of the fundamental solution of the Helmholtz equation. Thus the scattered field is analytically continued to the boundary of the target. Together with the boundary condition that the total field on the unknown target must vanish, this series expansion enables to reduce the shape reconstruction problem into the solution of a nonlinear equation which contains the surface contour as unknown. The resulting nonlinear equation is in a polynomial form which includes the higher order derivatives of the reconstructed field as constant. When multiple illuminations are employed, since each far field pattern completely characterizes the unknown shape, a different solution can be reconstructed for each illumination. However as the shape of the inaccessible target is independent from the source configuration, a global solution which uses all the available data simultaneously is looked after. To this aim Gauss – Newton algorithm is exploited for the iterative solution of the nonlinear equations. In order to avoid possible instabilities that may arise due to the numerical errors in the derivatives of the nonlinear equations, a finite dimension solution in terms of a linear combination of predetermined basis functions is sought. The presented method is numerically validated through several simulations and it is observed that the method provides satisfactory reconstruction for both convex and concave targets with star-like shapes. It is concluded that the size of target should be comparable to the operating wavelength when only a single illumination is used. However this limitation regarding to size of the obstacle can be improved with the usage of additional illuminations. Furthermore it is observed that the robustness against the noise on data increases with the usage of multiview data.