## 2 Derivation of the Coherence Relationship
As shown in Equation (13), the object can be deemed as a secondary radiation source which is stimulated by \(\overrightarrow{\mathbf{E}}\) illuminating the object. Unfortunately, \(\overrightarrow{\mathbf{E}}\) is dependent on not only the incident field \(\overrightarrow{\mathbf{E}}_{\text{Tx}}\) but also the object itself, making the imaging model a nonlinear problem. To settle the nonlinearity, researchers have proposed many iterative approaches, such as Born iterative method [6]. However, in most MCI applications, the interaction can be neglected by applying Born approximation without obviously deteriorating the imaging quality due to the short wavelength of microwaves (the applicability of Born approximation is another big issue that seems to go beyond the scope of this paper; relevant detailed discussion can be found in Reference [23]). Therefore, Born approximation, which ignores the perturbation of the object and replaces \(\overrightarrow{\mathbf{E}}\) with \(\overrightarrow{\mathbf{E}}_{\text{Tx}}\), is widely applied in MCI. Born approximation is also adopted in our analysis. Hence, neither the measurement nor the coherence is affected by the object. Under this approximation, the measurement mode can be formulated by \(\overrightarrow{\mathbf{E}}_{\text{Tx}}\) and \(\overrightarrow{\mathbf{E}}_{\text{Rx}}\):  
\[h\left(\overrightarrow{\mathbf{r}}\right)=-\frac{\left|\text{k}_{0}\right.}{Z_{0 }}\overrightarrow{\mathbf{E}}_{\text{Tx}}\left(\overrightarrow{\mathbf{r}}\right) \cdot\overrightarrow{\mathbf{E}}_{\text{Rx}}\left(\overrightarrow{\mathbf{r}}\right) \tag{14}\]  
In discrete form, the model can be represented by a matrix equation: \(\mathbf{g}=\mathbf{H}\mathbf{x}\), where \(\mathbf{g}\) is an echo vector, \(\mathbf{H}=\left[ \begin{array}{cc}h_{1}^{\text{T}}&\cdots&\mathbf{h}_{M}^{\text{ T}}\end{array} \right]^{\text{T}}\) is the measurement matrix which is composed of the measurement modes in multiple measurements, \(M\) and \(N\) are the numbers of the measurements and the grid of the ROI respectively and \(\mathbf{x}\) is the contrast function of the object. The matrix \(\mathbf{H}\) transforms the space of the object to the measurement space spanned by incoherent measurement modes, encoding the object's spatial information into the echoes. Hence, the object can be retrieved via the inverse of \(\mathbf{H}\). However, this inverse problem always possess natural ill-posedness due to deficient rank of \(\mathbf{H}\) and the presence of noise. In practice, the number of effective bases supporting the measurement space is always inadequate to support the space of the object, which leads to inevitable losses of spatial information during the above-mentioned transformation. Thus, it is critical to construct the bases in the measurement space as more as possible, which requires reducing the mutual coherences in \(\mathbf{H}\). As \(\mathbf{H}\) is generated by the radiations of the equivalent radiation sources, the mutual coherences in \(\mathbf{H}\) results from the mutual coherences of the sources. In order to formulate the relationship between the two sets of mutual coherences, we start with row correlations of the matrix \(\hat{\mathbf{H}}\) which is the matrix \(\mathbf{H}\) with \(\ell_{2}\) normalized rows. Analogously to Gramian matrix, we define the average mutual coherence \(\mu^{\text{T}}\) that is calculating in rows and \([\hat{\mathbf{H}}\hat{\mathbf{H}}^{\ast}]_{ij}\) which is the mutual coherence between measurement #i and #j:  
\[