**2.2 Software**

While much has been made of a range of non-linear iterative schemes and various stochastic approaches for image reconstruction approaches, we have opted to use the robust and well-regarded Gauss-Newton algorithm. **Figure 6** shows examples of the magnitude contour plots for 2D wave propagation from different antennas surrounding the medium with a high contrast object. The differences between the values measured at the receive antenna locations for the situation where an object is present minus that for the case for the homogeneous bath becomes the measurement data for the algorithm. Note that the techniques are completely translatable to 3D but are more easily understood in the context of 2D.

For the untransformed algorithm, the minimization statement examines the differences between the measured field values and the associated computed values at each receive antenna and for all views of the object from each iteration:

$$\min \quad \left\| \mathbf{E}^{\mathrm{m}} - \mathbf{E}^{\mathrm{c}} \left( \mathbf{k}^{2} \right) \right\|^{2} \tag{1}$$

*2D magnitude contour plots within an imaging zone containing a high contrast scatterer for point source illuminations from different directions.*

*Theoretical Premises and Contemporary Optimizations of Microwave Tomography DOI: http://dx.doi.org/10.5772/intechopen.103011*

where Em and Ec are the complex measured and computed electric field vectors, respectively, and k2 is the wavenumber squared which is essentially the image and embeds both the permittivity and conductivity, respectively. The associated logtransformed equation is [49]:

$$\min \quad \left\| \Gamma^{\mathrm{m}} - \Gamma^{\mathrm{c}}(\mathbf{k}^{2}) \right\|^{2} + \left\| \Phi^{\mathrm{m}} - \Phi^{\mathrm{c}}(\mathbf{k}^{2}) \right\|^{2} \tag{2}$$

where Γm, Γ<sup>c</sup> , Φm, and Φ<sup>c</sup> are the measured and computed log-magnitude and phase vectors, respectively. The two equations are essentially the same except for the fact that the phase and magnitude need to be accounted for explicitly. In addition, both the measured and computed phases need to be properly unwrapped. The image is then updated at each iteration by:

$$\mathbf{k}\_{i}^{2} = \mathbf{k}\_{i-1}^{2} + \mathbf{S}\Delta\mathbf{k}^{2} \tag{3}$$

where k2 <sup>i</sup> and k<sup>2</sup> <sup>i</sup>�<sup>1</sup> are the wavenumbers squared vectors at iterations i and i + 1, Δk2 is the image update at each iteration and S is the iteration step size. In all cases, the initial property estimate of the image is that of the homogeneous coupling bath which is the least biased estimate possible. The stopping criteria is generally met when the changes in the image (Δk2 ) are small—determined empirically. For all cases, regularization is necessary to stabilize the solution. For this situation, we have found both Tikhonov and Marquardt–Levenberg techniques to be suitable [55, 56].

For applications such as MR, phase unwrapping is normally performed as a function of position—i.e. the phase at a specific location is set as the reference phase and the phases at neighboring locations are compared with it [57]. If the differences between the values at new locations differ from the reference ones by more than 180°, the values for the new ones are adjusted by multiples of 360° until the differences are minimized. This process is continued until the entire spatial domain has been unwrapped. For the microwave measured values, this is impossible since the phases can only be known at the receive antenna locations—i.e. the phase can vary more than 180° between antennas. Our alternative is to unwrap the phases as a function of frequency. A baseline assumption implies that the scattered phases should be within the bounds of +/� 180° for the lowest frequency. The measured phases are forced to be within the baseline Riemann sheet, i.e. �180 to +180°. The values at the next frequency are unwrapped against these, followed by those at the next frequency being unwrapped with respect to the second set and so on. This is described in more detail in Meaney et al. [50]. For the computed phases, it would be possible to unwrap the values as a function of position. However, this is tedious, complicated and slow. We previously introduced a novel technique to unwrap the phases as a function of image reconstruction iteration [50]. As long as the phases do not change dramatically between iterations of the reconstruction process, it can be assumed that their differences will be less than 180°. To ensure this criterion, we deliberately force the step size (Eq. (3)) to be small during the earliest iterations so that the image, and hence the phases, differ only slightly between iterations. In this way, the phases only need to be computed at the antenna locations, and the values for the previous iteration need to be stored for comparison purposes. This technique is described in more detail in Meaney et al. [50].

Finally, the most substantial computational cost for these types of algorithms are for the forward field solutions which need to be computed for each transmit antenna at each image reconstruction iteration. Different techniques have been proposed to reduce this time, most notably finite element (FEM) and finite difference time domain (FDTD) modeling [26, 58]. However, these can be quite slow, especially as the forward problem grows in size, and often requires multi-processor computers and GPU processors to keep the computation times modest. While our 2D FEM-based approach can usually produce images in roughly 5 minutes, we have explored ways to further reduce the time. In particular, we have been experimenting with the discrete dipole approximation (DDA) to improve efficiency [59]. This technique is generally not suitable when there are large, high contrast scatterers in the field of view especially other bulky antennas; however, because our monopole antennas only slightly perturb the fields, and almost not at all when submerged in a lossy bath, this algorithm is ideal for our approach. The primary notion is that it utilizes Gaussiterative based forward solver techniques whose most substantial time cost is a matrix–vector multiplication that needs to be performed repeatedly (O(N<sup>2</sup> )) where N is the vector length). However, it can be easily shown that the associated matrix can be formulated to be a symmetric, block Toeplitz matrix. Each block matrix can then be easily converted to a circulant matrix. Circulant matrix–vector multiplication can be performed efficiently (O(NlogN)) by use of the convolution theorem and the use of the fast Fourier transform (FFT). We have shown that this can reduce the forward solution time by a factor of 20–50 times with respect to efficient commercial software —i.e. full reconstructions in 6 seconds or less. This technique is described in detail in Hosseinzadegan et al. [59, 60].
