*3.2. The Circular-Polar Grid (CPG)*

In this section, the circular-polar grid (CPG) is designed (see Figure 3c), which is defined as

$$\mathbb{C}\_R = \mathbb{C}\_R^0 \cup \mathbb{C}\_{R'}^\sharp \, : \tag{16}$$

where *C*0*R* = {(0, 0)} and *CR* = *C*1*R* ∪ *C*2*R*

$$\begin{split} \mathcal{C}\_{\mathbb{R}}^{1} &= \{ (r \cos(\frac{\ell \pi}{m\_{0}}), r \sin(\frac{\ell \pi}{m\_{0}})) : 1 \le |r| \le R, |\ell| \le \frac{m\_{0}}{2} \}, \\ \mathcal{C}\_{\mathbb{R}}^{2} &= \{ (r \sin(\frac{\ell \pi}{m\_{0}}), r \cos(\frac{\ell \pi}{m\_{0}})) : 1 \le |r| \le R, |\ell| \le \frac{m\_{0}}{2} \}, \end{split} \tag{17}$$

where *m*0 is the sampling number in each circle.

As can be seen from Figure 3c, the nod of *C*1*R* is on a *solid line* and the nod on a *dotted line* belongs to *<sup>C</sup>*2*R*. In addition, *r* in (17) serves as the radius and - serves as the parameter of angle. *m*0 = 16 is the sampling number. In the *CPG* coordinates, the nod has the following characteristics, for

$$\mathbb{C}\_{\mathbb{R}}^{1}(\omega\_{\mathbf{x}},\omega\_{\mathbf{y}}) = (r^{1},\theta^{1}), \quad \mathbb{C}\_{\mathbb{R}}^{2}(\omega\_{\mathbf{x}},\omega\_{\mathbf{y}}) = (r^{2},\theta^{2}), \tag{18}$$

where

$$\begin{aligned} r^1 &= k\_{1\prime} & r^2 &= k\_{2\prime} \\ \theta^1 &= \ell\_1 \pi / m\_0; \quad \theta^2 = \ell\_2 \pi / m\_0. \end{aligned} \tag{19}$$

*ki* = 0, ...*R*; *i* = 1, 2 and *i* = −*<sup>m</sup>*0/2, ...*<sup>m</sup>*0/2; *i* = 1, 2. For each fixed angle *θ*, the samples of the *CPG* are equally spaced in the radial direction, and, for each fixed radius *r*, the grid possesses the same angle. Formally,

$$\begin{aligned} \Delta r^1 &\stackrel{\Delta}{=} (k\_1 + 1) - k\_1 = 1; & \Delta r^2 \stackrel{\Delta}{=} (k\_2 + 1) - k\_2 = 1, \\ \Delta \theta^1 &\stackrel{\Delta}{=} (\ell\_1 + 1)\pi/m\_0 - \ell\_1 \pi/m\_0 = \pi/m\_0, \\ \Delta \theta^2 &\stackrel{\Delta}{=} (\ell\_2 + 1)\pi/m\_0 - \ell\_2 \pi/m\_0 = \pi/m\_0. \end{aligned} \tag{20}$$

where *r*1,*r*<sup>2</sup> and *θ*1, *θ*2 are given by (19).

> For an *N* × *N* image *u*, the *CFT* of *u*ˆ on *CPG* holds

$$\sum\_{\mathbf{x},\mathbf{y}=-N/2}^{N/2-1} |u(\mathbf{x},\mathbf{y})|^2 = \sum\_{(\omega\_{\mathbf{x}},\omega\_{\mathbf{y}}) \in \mathbb{C}\_R} w\_{\mathbf{c}}(\omega\_{\mathbf{x}\prime},\omega\_{\mathbf{y}}) |\mathfrak{il}(\omega\_{\mathbf{x}\prime},\omega\_{\mathbf{y}})|^2,\tag{21}$$

and

$$\mathfrak{M}\_{\mathbb{C}\_{R}}(\omega\_{x},\omega\_{y}) = \sum\_{\mathbf{x},\mathbf{y}=-N/2}^{N/2-1} \mathfrak{u}(\mathbf{x},\mathbf{y}) e^{-\frac{2\pi i}{Rm\_{0}+1}(\mathbf{x}\omega\_{x}+\mathbf{y}\omega\_{y})}.\tag{22}$$

Using operator notation, we denote the refined *CFT* of an image *u* as <sup>F</sup>*p*, where

$$(\mathcal{F}\_p u)(r, \ell) \triangleq \mathfrak{a}\_{\mathbb{C}\_{\mathbb{R}}}(r, \ell), \tag{23}$$

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

with *r* = <sup>−</sup>*R*, ..., *R*, - = −*<sup>m</sup>*0/2, ..., *m*0/2. Now, our goal is to choose weight *wc*, such that *wc* satisfies (21), and we have

∑ (*<sup>ω</sup>x*,*ωy*)∈*CR wc*(*<sup>ω</sup>x*, *<sup>ω</sup>y*)|*u*<sup>ˆ</sup>(*<sup>ω</sup>x*, *<sup>ω</sup>y*)|<sup>2</sup> = ∑ (*<sup>ω</sup>x*,*ωy*)∈*CR wc*(*<sup>ω</sup>x*, *<sup>ω</sup>y*)| *N*/2−1 ∑ *<sup>x</sup>*,*y*<sup>=</sup>−*N*/2 *<sup>u</sup>*(*<sup>x</sup>*, *y*)*E*(*<sup>x</sup>*, *y*)|<sup>2</sup> = ∑ (*<sup>ω</sup>x*,*ωy*)∈*CR wc*(*<sup>ω</sup>x*, *<sup>ω</sup>y*)[ *N*/2−1 ∑ *<sup>x</sup>*,*y*<sup>=</sup>−*N*/2 *N*/2−1 ∑ *<sup>x</sup>*,*y*<sup>=</sup>−*N*/2 *<sup>u</sup>*(*<sup>x</sup>*, *y*)*E*(*<sup>x</sup>*, *y*)*u*(*x*, *y*)*E*(*x*, *y*)] = ∑ (*<sup>ω</sup>x*,*ωy*)∈*CR wc*(*<sup>ω</sup>x*, *<sup>ω</sup>y*) *N*/2−1 ∑ *<sup>x</sup>*,*y*<sup>=</sup>−*N*/2 |*u*(*<sup>x</sup>*, *y*)|<sup>2</sup> + ∑ (*<sup>x</sup>*,*y*)=(*x*,*y*) *<sup>u</sup>*(*<sup>x</sup>*, *y*)*u*(*x*, *y*)[ ∑ (*<sup>ω</sup>x*,*ωy*)∈*CR wc*(*<sup>ω</sup>x*, *<sup>ω</sup>y*)*E*(*<sup>x</sup>*, *y*)*E*(*x*, *y*)], (24)

where *<sup>E</sup>*(*<sup>x</sup>*, *y*) - *e*<sup>−</sup> 2*πi Rm*0 (*xωx*+*y<sup>ω</sup>y*). Compared with the left of equation (21), the weights *wc* holds

$$\sum\_{(\omega\_{\mathcal{X}},\omega\_{\mathcal{Y}})\in\mathbb{C}\_{\mathcal{R}}} w\_{\varepsilon}(\omega\_{\mathcal{X}},\omega\_{\mathcal{Y}})e^{-\frac{2\pi i}{\mathcal{R}m+1}(\mathbf{x}\omega\_{\mathcal{X}}+\mathbf{y}\omega\_{\mathcal{Y}})}=\delta(\mathbf{x},\mathbf{y});\tag{25}$$

with −*N*/2 ≤ *x*, *y* ≤ *N*/2 − 1.
