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"A study, by two of the major contributors to the theory, of the inverse scattering transform and its application to problems of nonlinear dispersive waves that arise in fluid dynamics, plasma physics, nonlinear optics, particle physics, crystal lattice theory, nonlinear circuit theory and other areas. A soliton is a localized pulse-like nonlinear wave that possesses remarkable stability properties. Typically, problems that admit soliton solutions are in the form of evolution equations that describe how some variable or set of variables evolve in time from a given state. The equations may take a variety of forms, for example, PDEs, differential difference equations, partial difference equations, and integrodifferential equations, as well as coupled ODEs of finite order. What is surprising is that, although these problems are nonlinear, the general solution that evolves from almost arbitrary initial data may be obtained without approximation. For such exactly solvable problems, the inverse scattering transform provides the general solution of their initial value problems. It is equally surprising that some of these exactly solvable problems arise naturally as models of physical phenomena. Simply put, the inverse scattering transform is a nonlinear analog of the Fourier transform used for linear problems. Its value lies in the fact that it allows certain nonlinear problems to be treated by what are essentially linear methods."

"Pencitraan gelombang mikro sudah banyak digunakan di dunia dalam berbagai bidang yang berbeda, salah satunya digunakan pada bidang kesehatan. Penerapan dari gelombang mikro dapat digunakan untuk mendiagnosis tumor/kanker payudara dengan menggunakan sensitivitas yang tinggi untuk mendeteksi jaringan abnormal payudara yang memiliki kontras dielektrik yang rendah, dibandingkan dengan jaringan normal lainnya. Skripsi ini merancang sistem validasi algoritma sensitivity-maps pencitraan gelombang mikro yang telah dilakukan dipenelitian sebelumnya. Metode ini memanfaatkan pengukuran dua jenis objek sebagai kalibrasi sistem, yaitu: objek referensi sebagai latar yang tidak terdapat penghambur dan objek kalibrasi berupa objek kecil sebagai penghambur (scattering). Objek yang diuji terdiri dari objek dengan kontras dielektrik rendah yang terbuat dengan menggunakan material Polyurethane Foam, Balsa Wood, dan Expanded Polystyrene. Rekonstruksi dilakukan pada tiga jenis data pengukuran S-Parameter yaitu S11, S21, dan gabungan keduanya. S-parameter diukur pada dua frekuensi, yaitu 3 dan 10 GHz. Hasil pengukuran akan direkonstruksi menggunakan MATLAB untuk dijadi sebuah citra. Selian itu, parameter relative root mean squared error (RRMSE) dan structural similarity index (SSIM) digunakan untuk menganalisis citra secara kuantitatif. Hasil rekonstruksi menunjukkan pengukuran gabungan ( S11 dan S21) dengan kualitas citra terbaik dengan nilai RRMSE 0.082  dan SSIM 0.477

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Microwave imaging has been widely used in the world in a variety of different fields, one of which is used in the health sector. The application of microwaves can be used to diagnose tumors/breast cancer by using a high sensitivity to detect abnormal breast tissue that has a low dielectric contrast, compared to other normal tissues. This thesis designs a validation system for the sensitivity-maps algorithm for microwave imaging that has been carried out in previous studies. This method utilizes the measurement of two types of objects as system calibration, namely: a reference object as a background where there are no scatterers and a calibration object in the form of small objects as scattering. The objects tested consist of objects with low dielectric contrast made using Polyurethane Foam, Balsa Wood, and Expanded Polystyrene materials. Reconstruction was carried out on three types of S-Parameter measurement data, namely S11, S21, and a combination of both. S-parameters are measured at two frequencies, namely 3 and 10 GHz. The measurement results will be reconstructed using MATLAB to become an image. In addition, the relative root mean squared error (RRMSE) and structural similarity index (SSIM) parameters are used to analyze the image quantitatively. The reconstruction results show the combined measurements (S11 and S21) with the best image quality with an RRMSE value of 0.082 and SSIM 0.477."

"Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems."

"Pencitraan gelombang mikro telah banyak diterapkan pada berbagai bidang, salah satunya adalah pencitraan payudara untuk diagnosis kanker/tumor. Teknik pencitraan ini berpotensi menjadi modalitas komplementer untuk modalitas yang sudah ada, terutama untuk aplikasi pendeteksian dini. Sistem pencitraan gelombang mikro untuk deteksi tumor/kanker payudara memerlukan sensitivitas tinggi untuk mendeteksi jaringan abnormal yang memiliki sedikit kontras pada payudara dengan kepadatan tinggi. Tesis ini mengusulkan simulasi sistem pencitraan gelombang mikro kualitatif berbasis inverse scatteringdengan metode sensitivity-maps. Metode ini memperhitungkan medan yang dihamburkan objek sebagai data pengukuran. Metode ini memanfaatkan pengukuran dua jenis objek sebagai kalibrasi sistem: objek referensi sebagai latar yang tidak terdapat penghambur dan objek kalibrasi berupa objek kecil sebagai penghambur untuk mendapatkan respon impuls dari sistem. Penggunaan objek kalibrasi membuat metode ini memiliki sensitivitas yang tinggi. Objek yang diuji terdiri dari objek dengan kontras dielektrik rendah dan objek dengan banyak kontras dielektrik (multi-kontras). Rekonstruksi dilakukan pada tiga jenis data pengukuran S-Parameter yaitu , , dan gabungan keduanya. S-parameter diukur pada beberapa frekuensi, yaitu 3, 10, 14, 15, 16, 20 GHz, dan penggabungan seluruh frekuensi tersebut (multi-frekuensi). Hasil simulasi pencitraan menunjukkan sistem mampu merekonstruksi objek dielektrik dengan akurat. Selain analisis kualitatif, parameter elative root mean squared error (RRMSE) dan structural similarity index (SSIM) digunakan untuk menganalisis citra hasil rekonstruksi secara kuantitatif. Hasil rekonstruksi menunjukkan pengukuran  dengan multi-frekuensi memiliki kualitas citra terbaik dengan nilai RRMSE 0,1272 dan SSIM 0,9076. Sistem pencitraan yang dirancang juga berhasil merekonstruksi phantom multi-kontras secara akurat dengan RRMSE 0,1434 dan SSIM 0,4609.

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Microwave imaging has been widely applied in various fields, one of which is breast imaging for cancer/tumor diagnosis. This imaging technique has the potential to be a complementary modality to existing modalities, especially for early detection applications. Microwave imaging systems for breast tumor/cancer detection require high sensitivity in order to detect abnormal tissue which has little contrast in high-density breasts. This thesis proposes a simulation of a qualitative microwave imaging system based on inverse scattering using the sensitivity-maps method. This method takes into account the scattered field of the object as measurement data. This method utilizes the measurement of two types of objects as system calibration: a reference object as a scatterer-free background and a calibration object in the form of a small object as a scatterer to obtain an impulse response from the system. The use of calibration objects makes this method has high sensitivity. The object under test (OUT) consists of an object with low dielectric contrast and an object with multiple dielectric contrast (multi-contrast). Reconstruction was carried out on three types of S-Parameter measurement data, namely  and a combination of both. S-parameters are measured at several frequencies which are 3, 10, 14, a15, 16, 20 GHz, and the combination of all these frequencies (multifrequency). Imaging simulation results show the system is capable of reconstructing dielectric objects accurately. In addition to qualitative analysis, the relative root mean squared error (RRMSE) and structural similarity index (SSIM) parameters were used to analyze the reconstructed image quantitatively. The reconstruction results show that the multi-frequency  measurement has the best image quality with RRMSE values of 0.1272 and SSIM of 0.9076. The designed imaging system also successfully reconstructed multi-contrast phantom accurately with RRMSE of 0.1434 and SSIM of 0.4609. "