## 4 Regularized Inversion  
According to a recent revisit of (4), accounting for the mutual coupling between elementary antennas, the brightness temperature distribution \(T(\xi)\) has to be substituted with the difference \(T(\xi)-T_{r}\), where \(T_{r}\) is the physical temperature of the receivers [28]. In any case, this constant term does not affect the zero-spacing baselines, and it can be removed by measuring a flat target [29]. As a consequence, without any loss of generality, after the discretization of the integral found in (4) over an appropriate sampling grid in the direction of the cosine domain [30], the relationship between the complex visibilities and the brightness temperature distribution of the scene under observation can be written in algebraic form:  
\[\widetilde{V}=\widetilde{\mathbf{G}}\;T \tag{5}\]  
where \(\widetilde{\mathbf{G}}\) is the linear modeling matrix of the imaging radiometer using SAI in the far-field approximation. Likewise, the general relationship (3) can be written in a very similar algebraic form:  
\[V=\mathbf{G}\;T \tag{6}\]  
where \(\mathbf{G}\) is the general linear modeling matrix of the imaging radiometer using SAI, contrary to what is implied in the title of [31] as the difference \(r_{q}(\xi)-r_{p}(\xi)\) is there expended in Taylor series, thus resulting in an approximated linear modeling matrix, not a general one.  
It is now well established that these modeling matrices are rank-deficient [32] as a consequence of a number of unknowns (the values of the brightness temperature distribution over the sampling grid) being larger than the number of data (the complex visibilities). In both paradigms, the effective rank of the modeling matrix is equal to the number of Fourier frequencies in the experimental frequency coverage \(\mathcal{H}\) of the antenna array. Indeed, such imaging radiometers are band-limiting devices as the extent of the baselines \(\mathbf{b}_{pq}\) is limited to \(\mathcal{H}\). As a consequence, the corresponding inverse problems have to be regularized in order to provide unique solutions to the linear systems (5) and (6). For obvious reasons of comparison, in this study, the same regularization is implemented in either case. Otherwise, the differences that might be observed could be attributed to the regularization itself and not to the modeling matrix (and, finally, not to the concerned electromagnetic region). Among the many regularization methods that can be found in the literature, the minimum-norm one is widely used in SAI [33]:  
\[\widetilde{T}_{r}=\min_{T}\|T\|^{2}\;\text{s.t.}\;\widetilde{\mathbf{G}}T= \widetilde{V}. \tag{7}\]  
The extension to the general case does not raise any difficulty:  
\[T_{r}=\min_{T}\|T\|^{2}\;\text{s.t.}\;\mathbf{G}T=V. \tag{8}\]  
Numerical implementations are available in many algorithmic libraries and with many programming languages. In both cases, the reconstructed map is obtained through the computation of the pseudo-inverse [34] of the corresponding modeling matrix:  
\[\widetilde{T}_{r}=\widetilde{\mathbf{G}}^{+}\widetilde{V}, \tag{9}\]  
\[T_{r}=\mathbf{G}^{+}V. \tag{10}\]  
In either case, the pseudo-inverse is computed with the aid of a truncated singular value decomposition [35], where only the largest significant singular values are kept during the inversion.  
Finally, in order to filter out the Gibbs effects [36] caused by the sharp cut-off in the Fourier domain due to the limited extent of the frequency coverage \(\mathcal{H}\), these maps have to be damped by an appropriate apodization (or windowing) function \(W\)[37]:  
\[\widetilde{\mathcal{T}}_{r}=W\star\widetilde{T}_{r}, \tag{11}\]  
\[\mathcal{T}_{r}=W\star T_{r}, \tag{12}\]  
where \(\star\) is the convolution operator. These maps have to be compared to  
\[\mathcal{T}=W\star T, \tag{13}\]  
which is the \"ideal\" temperature map to be reconstructed and apodized with the same window \(W\). Indeed, they cannot be compared to \(T\) because they are not at the same angular resolution.