## 2 Mathematical Formalism
According to Equations (3) and (4), since \(0\leq x^{\prime}<\Delta x\) and \(0\leq y^{\prime}<\Delta y\), the variables \(x\) and \(y\) are bounded in each element of the partition (\(m=0,1,2,\ldots,M-1\); \(n=0,1,2,\ldots,N-1\)) as follows:  
\[m\Delta x-\frac{(M-1)\Delta x}{2}\leq x<(1+m)\Delta x-\frac{(M-1) \Delta x}{2} \tag{11}\] \[n\Delta y-\frac{(N-1)\Delta y}{2}\leq y<(1+n)\Delta y-\frac{(N-1 )\Delta y}{2} \tag{12}\]If we denote \(x_{c,m}\) and \(y_{c,n}\) as the center of each element of the partition, then these centers can be computed by the average of the boundary values of \(x\) and \(y\) in each element of the partition as follows:  
\[x_{c,m} =\frac{(1+2m)\Delta x}{2}-\frac{(M-1)\Delta x}{2}\] \[y_{c,n} =\frac{(1+2n)\Delta y}{2}-\frac{(N-1)\Delta y}{2}\]  
Thus, let us define these values as \(E_{0x}^{\text{ref},d}[m,n]\):  
\[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]0  
Substituting Equation (13) into Equation (10) leads to  
\[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]1 \[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]2 \[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]3 \[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]4  
where we have taken into account that \(\int_{0}^{\Delta y}\int_{0}^{\Delta x}e^{jk_{0}(ux^{\prime}+vy^{\prime})}dx^{ \prime}dy^{\prime}=\Delta x\Delta y\cdot\text{sinc}\Big{(}\frac{k_{0}u\Delta x }{2}\Big{)}\)\(e^{-j\frac{k_{0}}{2}\left[u(M-1)\Delta x+v(N-1)\Delta y\right]}\). If the values of variables \(u\) and \(v\) are the discrete values given by Equations (7) and (8), respectively, then this leads to  
\[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]5 \[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]6 \[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]7 \[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]8  
where \(\theta[p,q]\) and \(\phi[p,q]\) are computed by Equations (5) and (6). The double summation in Equation (15) can be easily identified with the 2D inverse discrete Fourier transform [29]. Since the 2D unverse discrete Fourier transform is an isomorphism, the set of matrices \(\left(E_{0x}^{\text{ref}}\right)=\left(E_{0x}^{\text{inc}}e^{j\varphi}\right)\) and the set of matrices \(\left(E_{\text{cp}}^{\text{rad}}\right)\) are isomorphic. As stated in [9], to determine the separation between the projected point and pre-projected point within sets \(\mathcal{R}\) and \(\mathcal{M}\), a matrix \(\left(E_{\text{cp}}^{\text{rad}}\right)\) with \(M\times N\) entries was considered. In [9], the norm \(\|E_{\text{cp}}^{\text{rad}}\|^{2}\) was defined as the Frobenius norm of the matrix \(\left(E_{\text{cp}}^{\text{rad}}\right)\). In this way, through Theorem 1, we can state the following theorem:  
Theorem 2: _If the conditions of Theorem 1 are satisfied, then the Frobenius norm of the matrix \(\left(E_{\text{cp}}^{\text{rad}}\right)\) is proportional to the Frobenius norm of the matrix \(\left(E_{0x}^{\text{inc}}e^{j\varphi}\right)\)._Proof of Theorem 2.: Let us define \(\widetilde{E}_{0x}^{\text{ref}}[p,q]\) as follows:  
\[v[q] =\frac{2\pi}{N\Delta yk_{0}}(q-\frac{N}{2});q=0,1,2,\ldots,N-1 \tag{8}\]9  
Let us consider a matrix \(\left(\widetilde{E}_{0x}^{\text{ref}}\right)\) which contains the value of \(\widetilde{E}_{0x}^{\text{ref}}[p,q]\) as the \(pq\)th entry and a matrix \(\left(E_{0x}^{\text{ref},d}\right)\) which contains the value of \(E_{0x}^{\text{ref},d}[m,n]\) as the \(mn\)th entry.