Identifying the position and shape of two-dimensional metal objects using the surface adjustment method

Dissertation of Master's Course in Electrical Engineering-Communication Wave Orientation

Abstract

In this thesis, the identification of the position and shape of two-dimensional metal objects by optimization[1] of the cost function[2] with the help of the level adjustment method[3] and the simulation results are presented. In this method, we were able to obtain the shape and position of the objects by introducing a suitable form of the function's shape change rate [4] and applying it to the Hamilton-Jacobi equation [5] and solving this equation and repeating this process. In the processing process, we used frequency jump [7] to prevent the cost function from being in the local minimum [6]. Level adjustment methods have two important features. One is the lack of basic information about objects and the surrounding environment, and the other is the ability to identify multiple objects in a computing environment [8]. The results show the acceptable identification of the shape of metal objects and their position.

Keyword: optimization, cost function, level adjustment method, frequency jump, Hamilton-Jacobi equation

Chapter 1-Introduction

1-1-Introduction

1-1-1- Direct and inverse problems

Almost any problem in which there is an assumption and ruling can be turned into a new problem by moving the assumption and ruling. In this case, we call the first problem direct and the second one inverse. For example, if we look out from the window of our room and see that it is raining, we ask ourselves what is the cause of this rain? The answer is obvious; Rain clouds in the sky are the cause of rain. But how is the inverse problem expressed? Now the sky is cloudy. In this case, will we have rain? It is easy to see that the second problem is more difficult to diagnose and its solution requires more information. At the same time, the answer to this question is much more useful and exciting. The reverse question can be asked more difficult and more widely used: Will there be rain two days later? Almost no person can be found for whom the answer to this question is not important. In many cases, the answer to this question is directly related to people's financial income. For example, farmers and activists in the field of land, sea and air transportation include checking the weather forecast in their daily and weekly schedule. Therefore, we see that the inverse problem is much more useful in this case. In most cases, finding the answer to the inverse problem is more difficult. But it is so widely used that it is seriously placed on the agenda of researchers.

1-1-2- Well-behaved and ill-behaved problems

Generally, any problem that has the following three characteristics [1] is called well-behaved:

1. The problem has an answer (existence [2])

2. There should be at most one solution for the problem (unique [3])

3. The answer should change continuously with the change of data (stability[4])

The mathematical definition of the above three cases about the well-behaved function is as follows:

Definition: Suppose and are normal spaces and there is a mapping (linear or non-linear) so that we have An equation is well-behaved if it has the following three characteristics:

1. For every there exists at least one such that (existence)

2. For each there is at most one such that (unique)

3. For each sequence, if with , then (stability)

Any problem that is not well-behaved (does not have at least one of the above three characteristics) is called ill-behaved[5].

The most important concern in solving inverse problems is the third case or the stability problem. In the same example of the movement of clouds and rain that was stated in the first part, let's assume that by looking at the meteorological maps and communications of the surrounding countries, we come to the conclusion that, for example, due to the passage of a cloud system from west to east, we will have rain in Tehran for the next three days, in this case, the wind will blow from north to south.In the same example of the movement of clouds and rainfall that was stated in the first part, let's assume that by observing the meteorological maps and the communication of the surrounding countries, we come to the conclusion that, for example, due to the passage of a cloud system from west to east, we will have rain in Tehran for the next three days. In this case, a small change in the input data has led to a major change in the output. Therefore, in solving inverse problems, we must pay special attention to the stability or stabilization of the problem.

1-2-Inverse problems in magnetism

In the field of electromagnetism, direct and inverse problems can also be imagined. Often in electromagnetics, due to its wide application, inverse problems are investigated and classified in the field of scattering. In this way, in the direct problem, we project the field into the environment. So that the gender and position of the object in the environment is known to us. In this case, the calculation of the scattering field [6] is desirable. But in the reverse mode, we radiate a field with a certain amplitude and phase to the environment and collect the scattered fields. In this case, we want to identify the type and position of the scatterers in the environment. Let's examine three cases of misbehavior in the inverse problem. With the assumption that we know the type of scattering object is metal and we are looking for its position.

Existence of the answer: The field received by the receiving antenna may have changed so much that the value it shows is not caused by any kind of scattering metal object.

Uniqueness of the answer: If the observations are limited, for example, the number of receiving antennas is small or it cannot be seen 360 degrees. It received the return and transmission fields in two-dimensional mode, in this case, we may still arrive at an answer due to the reception of noisy or incorrect data, and of course limited, which is caused by two or more types of objects.  

Instability: suppose that the field received by a receiving antenna is equal to or close to zero and the field of the rest of the points has a continuous and slow change in size around the value of 10 voltmeters. For example, the reason is that two waves with close amplitude and 180 degree phase difference interfered before hitting the receiving antenna and neutralized each other's effect at the position of that antenna. In this case, with a slight movement of the antenna, we reach a significant difference. This state is an example of instability in the field of its practical reception.

1-3-Problems of solving inverse scattering problems

One ??of the basic problems in these problems is their non-uniqueness. For example, fading fields caused by lossy environment or parts with very small dimensions will not be detectable. Other problems can be mentioned, including:

1. Data loss: due to limited space and the influence of scattered waves, or duplicate information in data measurement

2. Noisy data: the data taken at the receiving antenna will be affected by random noise.

3. Unobservable data: It means that solving an optimization problem leads to non-physical information. In other words, information that cannot be modeled through a direct model.

4. Inexact method: optimization methods may lead to instability.

1-4-Applications of scattering and inverse scattering

Scattering of acoustic and magnetic waves plays a fundamental role in applied sciences. Some of its uses are as follows:

1. Photographing patients' bodies for medical purposes: such as using magnetic waves to diagnose bone marrow cancer in people

2. Subsurface photography: for applications such as demining, oil exploration, archaeological research, etc.

3. Radar applications: identifying the number, shape and dimensions of moving objects such as airplanes, ships, etc.

4. Conducting non-destructive tests such as detecting cracks inside objects, detecting the presence of dangerous substances such as combustibles inside objects, etc.

1-5-General methods of solving inverse problems

Depending on the need in solving the inverse problem, the question can be adjusted.