# Differential Ultra-Wideband Microwave Imaging: Principle Application Challenges
## 1 Introduction  
Microwave imaging may be considered as a method to reconstruct the spatial distribution of matter (bodies, objects) based on the electromagnetic sounding of an observation space of interest. Figure 1 demonstrates the principle. The observation space may be completely or partially surrounded by antennas. The antennas emit electromagnetic fields into the observation space and receive the fields scattered at the surface of the bodies due to the different propagation parameters at both sides of the boundaries. Such propagation parameters are either the permittivity \(\varepsilon\) and the permeability \(\underline{\mu}\) of the involved substances or the propagation speed \(\underline{\varepsilon}=1/\sqrt{\underline{\varepsilon}\underline{\mu}}\) and the intrinsic impedance \(\underline{Z}=\sqrt{\underline{\mu}/\underline{\varepsilon}}\). Typically, every substance is characterized by a specific set of such parameters, which makes different substances distinguishable by electromagnetic sounding. Obviously, by referring to Figure 1, one can measure all combinations between the antenna feeding signals \(a_{i}\); \(i=1\cdots K\) and the received signals \(b_{j}\); \(j=1\cdots K\), where the received signals are affected by the inner structure of the observation space. Hence, from the measurement, we get a set of response functions \(S_{\hat{p}}\); \(i,j=1\cdots K\)  
\[\left[ \begin{array}{c}\underline{b}_{1}\\ \underline{b}_{2}\\ \vdots\\ \underline{b}_{K}\end{array} \right]=\left[ \begin{array}{cccc}\underline{S}_ {11}&\underline{S}_{12}&\cdots&\underline{S}_{1K}\\ \underline{S}_{21}&\underline{S}_{22}&\cdots&\underline{S}_{2K}\\ \vdots&\vdots&\ddots&\vdots\\ \underline{S}_{K1}&\underline{S}_{K2}&\cdots&\underline{S}_{KK}\end{array}  \right]\cdot\left[ \begin{array}{c}\underline{a}_{1}\\ \underline{a}_{2}\\ \vdots\\ \underline{a}_{K}\end{array} \right]\rightarrow\underline{\mathbf{b}}(f)= \underline{\mathbf{S}}(f)\,\underline{\mathbf{a}}(f) \tag{1}\]  
or  
\[\left[ \begin{array}{c}b_{1}\\ b_{2}\\ \vdots\\ b_{K}\end{array} \right]=\left[ \begin{array}{cccc}S_{11}&S_{12}&\cdots&S_{1K }\\ S_{21}&S_{22}&\cdots&S_{2K}\\ \vdots&\vdots&\ddots&\vdots\\ S_{K1}&S_{K2}&\cdots&S_{KK}\end{array} \right]\ast\left[ \begin{array}{c}a_{1} \\ a_{2}\\ \vdots\\ a_{K}\end{array} \right]\rightarrow\mathbf{b}(t)=\mathbf{S}(t)\ast\mathbf{a}(t),\]  
which carry the information (encoded by [PERSON]'s equations and the boundary conditions) about the observation space of interest. Note that the matrices of the response functions are symmetric due to reciprocity (i.e., \(\underline{\mathbf{S}}=\underline{\mathbf{S}}^{T}\,\mathrm{or}\,\mathbf{S}= \mathbf{S}^{T}\), where \({}^{T}\) means transpose). The underscored quantities in the upper equation refer to complex valued frequency functions, which are gained from sinewave measurements. The lower equation deals with real valued time functions resulting from measurements with arbitrary-shaped ultra-wideband (UWB) signals, where the symbol \(\ast\) means convolution [1]. Finally, the aim of the imaging procedure is to \"decode\" the internal structure of the observation space of interest from the measured quantities. For that purpose, the equations describing wave propagation and scattering have to be inverted, using a method called \"inverse scattering\". A comprehensive introduction in this technique and an example for ground penetrating radar are, for example, given in [2,3]. The solution of inverse problems is typically ill-conditioned and numerically quite expensive. Hence, one relies in many cases on some prior knowledge about the observation space of interest and on simplifications of the inversion procedure (e.g., by omitting multipath propagation (often assigned as Born approximation), which is however only applicable in low contrast scenarios).