# On the Use of Focused Incident Near-Field Beams in Microwave Imaging
## 2 Motivation
### Sufficient Measured Data  
In the previous section, we discussed suppressing the effect of undesired scatterers to avoid including them as unknowns in the inversion algorithm. As noted earlier, if we decide to include undesired scatterers as part of the unknowns in the inversion process, we need to ensure that sufficient measured data are collected to reconstruct not only the OI but also the undesired scatterers. We now rely on Section 2.1 to further discuss this in an intuitive fashion. To this end, let us first note that the equivalent inverse source problems corresponding to the MWI problem (see Figure 1) are, in fact, antenna characterization problems which we often encounter in NF antenna measurement techniques. (In NF antenna measurement techniques, antenna characterization or diagnostics deals with finding the equivalent currents of the antenna under test from NF antenna measurements.) This is due to the fact that each set of these contrast sources can be thought as an unknown _antenna_ radiating the measured scattered fields. From the theory of cylindrical and spherical NF antenna measurements [42], it is known that the number of required measured data points depends on both the wavelength of operation (\(\lambda\)) and the electrical size of the antenna. Noting the relation between the MWI and inverse source problems, it can be understood that the required sampling resolution for MWI will similarly depend on the size of the OI and the wavelength of operation. We also note that, in NF antenna measurements, the above criteria is for finding the fields _outside_ the measurement domain (i.e., for performing NF to FF transformation), which is easier than reconstructing the fields interior to the measurement domain. For example, the required angular and vertical data sampling resolutions for a cylindrical NF antenna measurement system is [43]  
\[\Delta\varphi=\frac{\lambda}{2(a+\lambda)}\quad\text{ and }\quad\Delta z= \frac{\lambda}{2}, \tag{4}\]  
where \(a\) is the radius of the smallest cylinder that encloses the antenna under test. It is worthwhile to note that the above sampling resolution requires collecting two orthogonal field components at each measurement location, which are tangential to the measurement surface, i.e., \(E_{\varphi}\) and \(E_{z}\) in the cylindrical coordinates where the axis of the measurement cylinder is assumed to be on the \(z\)-axis. (In MWI, often one field component is collected at each measurement location, in most implementation \(E_{z}\).) Based on the above discussion and the similarity of the inverse scattering and inverse source problems, the parameter \(a\) in MWI can be thought as the smallest cylinder that encloses the OI and undesired scatterers. Therefore, it can be easily understood that the larger \(a\) with respect to the wavelength, the more measured data are necessary for performing successful inversion. (We re-emphasize that this approach is only being used to develop some intuitive understanding about the MWI data sampling resolution, and should not be considered as an exact governing formula for the MWI problem. For a more detailed discussion on retrievable information, see [44].) If we are not able to provide the inversion algorithm with sufficient information content (e.g., due to the physical size of the antennas or strong mutual coupling between them), an alternative option will be to reduce the number of unknowns, e.g., by _not_ seeing some parts of the original imaging domain. Therefore, although 3D full-vectorial inverse scattering algorithms are the most accurate option for inverting the measured data as practical systems and objects are all 3D structures, it is still important to simplify the inversion process for the cases where the amount of measured data cannot support the retrieval of many unknowns associated with 3D full-vectorial inversion. For example, each _voxel_ of a discretized imaging domain will have four complex unknowns in 3D full vectorial inversion: \(E_{x}\), \(E_{y}\), \(E_{z}\), and the OI's complex permittivity.