*3.1. The Pseudo-Polar Grid*

The pseudo-polar grid Ω*R* is given as

$$
\Omega\_R = \Omega\_R^1 \cup \Omega\_{R'}^2 \tag{11}
$$

where

$$\begin{aligned} \Omega^1\_R &= \{ (\frac{-4\ell k}{RN}, \frac{2k}{R}) : |\ell| \le N/2, |k| \le RN/2 \}, \\ \Omega^2\_R &= \{ (\frac{2k}{R}, -\frac{4\ell k}{RN}) : |\ell| \le N/2, |k| \le RN/2 \}, \end{aligned} \tag{12}$$

with *R* = 2 the oversampling parameter. The nod of <sup>Ω</sup>1*R* is on the *solid line* in Figure 3b and the nod of <sup>Ω</sup>2*R*is on *dotted line*.

For an *N* × *N* image *u*, the general discrete Fourier transform *u*ˆ is evaluated on the *N* × *N* Cartesian grid in the form

$$\mathfrak{M}(\omega\_x, \omega\_y) = \sum\_{\mathbf{x}, \mathbf{y} = -N/2}^{N/2 - 1} \mathfrak{u}(\mathbf{x}, \mathbf{y}) e^{-\frac{2\pi i}{N}(\mathbf{x}\omega\_x + y\omega\_y)} \,\_{\mathbf{y}} \tag{13}$$

where {(*<sup>ω</sup>x*, *<sup>ω</sup>y*) : *ωx*, *<sup>ω</sup>y* = −*N*/2, ..., *<sup>N</sup>*/2.}, and

$$\sum\_{\mathbf{x},\mathbf{y}=-N/2}^{N/2-1} \left| u(\mathbf{x},\mathbf{y}) \right|^2 = \frac{1}{N^2} \sum\_{\omega\_x,\omega\_y=-N/2}^{N/2-1} \left| \mathfrak{A}(\omega\_x,\omega\_y) \right|^2. \tag{14}$$

*Symmetry* **2018**, *10*, 101

Analogously, the pseudo-polar Fourier transform is the same as (13) (see [5]), but {(*<sup>ω</sup>x*, *<sup>ω</sup>y*) ∈ <sup>Ω</sup>*R*.}. According to the Plancherel theorem, (14) can be modified by introducing the weighting function *w*

$$\sum\_{\mathbf{x},\mathbf{y}=-N/2}^{N/2-1} |u(\mathbf{x},\mathbf{y})|^2 = \sum\_{\{\omega\_{\mathbf{x}},\omega\_{\mathbf{y}}\} \in \Omega\_R}^{N/2-1} w(\omega\_{\mathbf{x}}, \omega\_{\mathbf{y}}) |\hat{u}(\omega\_{\mathbf{x}}, \omega\_{\mathbf{y}})|^2. \tag{15}$$
