Inverse scattering is the designation for mathematical methods which are used to obtain information about an object from scattered wave fields measured outside the object. Inverse scattering applies to a wide range of areas, such as landmine detection, remote sensing, medical imaging, target identification, geophysical explorations, and non-destructive testing. The object, from which information is desired, is usually inaccessible (visually obscured) or its material properties are unknown such that the application of wave fields is one of the few possible means for exploration.
At Electromagnetic Systems we focus on electromagnetic inverse scattering, i.e., the branch of inverse scattering in which electromagnetic wave fields are employed. Electromagnetic inverse scattering applies to the entire frequency band of wave fields, ranging from static fields over micro waves to optical and X-ray fields. Although we focus on the electromagnetic case our research is based on general principles and the results can therefore without much effort be altered to apply to other types of wave fields.
The form of information that can be obtained about an object by application of electromagnetic inverse scattering depends on the type of object under consideration. For instance, a three-dimensional (3-D) perfect electrical conductor (PEC) is uniquely described by its 2-D surface, and an inverse scattering scheme for a PEC will therefore provide a description of this surface. 3-D penetrable objects, on the other hand, are in general given by three 3-D functions describing the spatial variation of the constitutive parameters, that is, permittivity, permeability, and conductivity. It is therefore apparent that an inverse scattering scheme for such a general penetrable object is much more complicated than that for the PEC.
Electromagnetic inverse scattering can be considered as the opposite of forward scattering. In forward scattering one determines, by use of Maxwell's equations, an explicit or implicit relation for the electric or magnetic field outside the object as a function of some properties describing the object. For instance, for a penetrable object the properties are the above-mentioned constitutive parameters. The explicit or implicit relation is referred to as the forward model. The inverse scattering scheme is arrived at by inverting the forward model. This scheme expresses, explicitly or implicitly, the constitutive parameters as a function of the electric or magnetic field. By measuring the electric or magnetic field and using the inverse scattering scheme it is possible to obtain the desired information about the object.
The inversion of the forward model is, however, not a simple task. The reason is that it is ill-posed. In the sense of Hadamard an ill-posed problem fails to fulfill all of the following requirements
- A solution exists.
- The solution is unique (there is only one solution).
- The solution is stable, i.e., it depends continuously on the data.
A problem that satisfies all these requirements is called well-posed. In general, inverse scattering problems do not satisfy any of these requirements. We will comment on this in the following.
Existence of solution
The forward model describes the true scattering process to a certain level of accuracy. The reason is that 1) it is not possible to model all details of the scattering configuration and 2) it is often convenient to introduce approximations in the forward model for simplicity or to transform it from an implicit into an explicit expression. Therefore, there does not necessarily exist a function describing the properties of the object that corresponds to the measured data. Even if the forward model accurately describes the scattering process the measured data will always be influenced by noise which is not accounted for in the forward model. For those reasons one could argue that the question of the existence of a solution to an inverse scattering problem is the wrong one to ask. The right question to ask is whether it is possible to find a stable solution when the measured data does not conform to the forward model. This will be addressed below.
Uniqueness
The forward model of an electromagnetic inverse scattering problem can be expressed in terms of a Fredholm integral equation of the first kind (IFK). The nonuniqueness of electromagnetic inverse scattering is related to the existence of a null space of the IFK. Hence, if the function f is a solution to the IFK and fns is a function belonging to the null space then f+fns is also a solution and uniqueness is therefore lacking. One could then ask: what is the reason for the existence of a null space ? An intuitive explanation is that it is related to the fact that the inverse scattering problem is underdetermined. This means that there is not enough data to uniquely determine the unknown function describing the properties of the object. Since the lack of uniqueness in inverse scattering is a result of lack of data it is seen that one should always try to get as much data as possible. However, in many practical situation the configuration under consideration does not allow the necessary amount of data to be acquired. Therefore, it is necessary to find a way to deal with nonuniqueness. Of course, we could present many solutions f+fns to the electromagnetic inverse scattering problem, but in practice it is more convenient to have only one solution. Therefore, the goal is to choose one solution among the infinitely many solutions. In other words, we have to choose one component of the null space such there is only one solution to our problem. This choice is made on the basis of prior knowledge about the solution. The prior knowledge could be the fact that f is an even function, that it is defined only in a certain interval, that it is a real function, that it has minimum norm, etc. By incorporating the prior knowledge, more information is added to the inverse scattering problem and uniqueness is therefore possible.
Stability
The fact that the solution to an inverse scattering problem is unstable is inherently related to the above-mentioned IFK. A rigorous investigation of this matter requires application of aspects of functional analysis. However, an intuitive explanation is also possible. The core of problem is that for a large class of kernels associated with the IFK the integration process of the IFK has a smoothing effect on f in the sense that high-frequency components of f, like edges and cusps, are smoothed out by the integration. This fact is expressed mathematically by the Riemann-Lebesgue lemma. This lemma states that for the class of kernels under consideration the high-frequency components are damped by the integration. The damping of the high frequencies indicates what we can expect from the reverse process, that is, the determination of f from knowledge of the data g. We can expect that the high-frequency components of g are amplified. This is indeed the case. Since such an amplification is extremely sensitive to noise, the solution will not depend continuously on the data and is therefore unstable. This observation, that the reserve operation is unstable, is indeed in line with the fact that differentiation is a roughness operator. The stability problem can be overcome by using regularization.
[Back to Research projects]