Abstract
This paper presents one of the techniques for solving inverse source problems involving waves. We studied the effects of frequency of waves on resolving images obtained via detectors. It is well known that inverse source problems present a challenge in medical imaging. The difficulty is ill-posedness: the answer is arbitrarily sensitive to measurement data. In this study we presented a step-by-step way of solving inverse source problems using singular value decomposition (SVD). The goal was to select the appropriate regularization factor and thus constrain the solution to realistic values. This is a common method used in breast cancer detection by microwave tomography. We concluded that the regularization factor is dependent on the noise level that arises in measuring the data.
Introduction
Medical imaging is a field which is receiving increasing attention as medicine strives for ways to non-invasively image the human body for diagnosis and functional study. Several popular imaging technologies require the solution of an “inverse problem” in order to obtain an image. Examples of such medical imaging techniques include electrical impedance tomography, diffuse optical tomography, X-ray tomography, microwave scattering, reflection tomography, and inverse source problems like electroencephalogram (EEG)(1,2,3,4).
An inverse problem involves using results of an observation to determine the parameters of a system. It is therefore the reverse of the “forward problem” which involves predicting observations given the parameters of the system. Inverse problems are, thus, much more complicated than forward problems. A common setback with inverse problems is the concept of ill-posedness. A problem is defined as ill-posed if its answer is arbitrarily sensitive to measurement data. In the real world, measurements are limited to certain accuracies. Noise always perturbs the data, causing changes in the solution. With an ill-posed problem, small changes in data can lead to arbitrarily large changes in solutions which may overwhelm the desired image.
In this introductory paper, we discuss the mathematics of solving a type of inverse problem known as inverse source problems: we seek to identify amplitudes of the sources of waves and thereby construct an image, given only the total fields measured at a set of detectors. The sources are supposedly inside the body, and the detectors are necessarily outside for non-invasive imaging. We study a simple two-dimensional single frequency Helmholtz model problem that illustrates the essential practical issues arising in more complicated real-world imaging problems.
Given a set of known detector locations and unknowns (source locations and strengths), the goal is to detect the internal properties of the sources via the amplitudes at the detectors. Some of the problems of interest include whether we can distinguish nearby sources from one another.
One major way to approach this problem of ill-posedness is via “regularization.” By regularizing, we constrain the solution to realistic values, thus maintaining image quality even in the presence of noise.
Mathematical statement of problem
We consider n sources in the plane at locations zi, for i=1… n, and detectors at yj, j = 1…m, (See Figure 1).
Scalar waves at fixed frequency, ω, satisfy the Helmholtz equation. The fundamental solution which is the field due to a point source, involves the Hankel function which only depends on the distance from sources,
(1)
Thus the field at a point yi due to source at zj is:
(2)
The total field at yi is the weighted sum of the above,