*4.1. 2D Basic Harmonic Function*

First, we define the 2D basic harmonic wavelet functions in the Fourier domain. For deriving convenience, the wavelet function *H* and scaling function *S* can be normalized in the form given by **Definition** 1.

**Definition 1.** *The 2D harmonic basic functions are defined as*

$$\begin{aligned} H(r,\theta) & \triangleq H(2\pi|r|\cos(|\theta|))S(2\pi|r|\sin(|\theta|)),\\ S(r,\theta) & \triangleq S(2\pi|r|\cos(|\theta|))S(2\pi|r|\sin(|\theta|)),\end{aligned} \tag{31}$$

*with r* ∈ [−*R*, *<sup>R</sup>*]*, θ* ∈ [0, *<sup>π</sup>*]*.*

> Then, the support of *<sup>H</sup>*(*<sup>r</sup>*, *θ*) and *<sup>S</sup>*(*<sup>r</sup>*, *θ*) are investigated, according to (3) and (6),

$$H(2\pi|r|\cos(|\theta|)) \neq 0, \quad S(2\pi|r|\sin(|\theta|)) \neq 0 \tag{32}$$

hold simultaneously; therefore,

$$1 \le |r| \cos(|\theta|) \le 2, \quad 0 \le |r| \sin(|\theta|) \le 1. \tag{33}$$

Thus, the support of *<sup>H</sup>*(*<sup>r</sup>*, *θ*) is given as

$$
\sqrt{2} \le |r| \le 2, \quad |\theta| \le \pi/4. \tag{34}
$$

Similarly,

$$\text{supp}\mathbf{S}(r,\theta) = \{(r,\theta) : 0 \le |r| \le 1, |\theta| \le \pi/4\}.\tag{35}$$

Next, the 2D scaling and shifting of *<sup>H</sup>*(*<sup>r</sup>*, *θ*) and *<sup>S</sup>*(*<sup>r</sup>*, *θ*) are defined.

**Definition 2.** *The 2D scaling and shifting of harmonic basic functions in the frequency domain are defined as*

$$\begin{split} &H\_{j,\ell}(r,\theta): \stackrel{\scriptstyle \triangleq}{=} H(2\pi 2^{-j}|r|\cos(2^{j}|\theta-\ell|))S(2\pi 2^{-j}|r|\sin(2^{j}|\theta-\ell|)),\\ &S\_{j,\ell}(r,\theta): \stackrel{\scriptstyle \triangleq}{\simeq} S(2\pi 2^{-j}|r|\cos(2^{j}|\theta-\ell|))S(2\pi 2^{-j}|r|\sin(2^{j}|\theta-\ell|)),\\ &H\_{j,\ell}^{\*}(r,\theta): \stackrel{\scriptstyle \triangleq}{=} H(2\pi 2^{-j}|r|\sin(2^{j}|\theta-\ell|)S(2\pi 2^{-j}|r|\cos(2^{j}|\theta-\ell|))),\\ &S\_{j,\ell}^{\*}(r,\theta): \stackrel{\scriptstyle \triangleq}{=} S(2\pi 2^{-j}|r|\sin(2^{j}|\theta-\ell|-\frac{\pi}{2}))S(2\pi 2^{-j}|r|\sin(2^{j}|\theta-\ell|)),\end{split}$$

*with j*, - ∈ *R.*

*4.2. The Polar Harmonic Multilevel System in the Frequency Domain (PHMS) on* CPG

In this section, we give the definition of the polar harmonic multilevel system (PHMS) defined on *CPG*.

**Definition 3.** *The 2D* PHMS *on* CPG *is defined as*

$$\text{PHMS} : \stackrel{\Delta}{=} \mathcal{H}\_{\text{j},\ell\_{\text{j}}}(\mathbf{r},\boldsymbol{\theta}) \cup \mathbb{S}\_{\text{j},\ell\_{\text{j}}}(\mathbf{r},\boldsymbol{\theta}) \cup \mathcal{H}\_{\text{j},\ell\_{\text{j}}}^{\*}(\mathbf{r},\boldsymbol{\theta}) \cup \mathbb{S}\_{\text{j},\ell\_{\text{j}}}^{\*}(\mathbf{r},\boldsymbol{\theta}),\tag{37}$$

*where <sup>H</sup>j*,-*, <sup>H</sup>*<sup>∗</sup>*j*,-*, <sup>S</sup>j*,- *and S*∗*j*,- *are given in* (36)*. The level parameter j* ≤ [log2 *<sup>R</sup>*]*, the shifting parameter is related to j, and we defined the j* = -2−*j π*4 *with* |-| ≤ 2*j*, - ∈ *Z.*

From |-| ≤ 2*j*, we have 2(2*j*+<sup>1</sup> + 1) subbands in each level *j*, in order to reduce the overlap, we choose 2(2*j*+<sup>1</sup>) subbands; then, the *PHMS* constructs a partition of the Fourier domain. We displayed the *PHMS* structure in Figure 4.

**Figure 4.** The polar harmonic multilevel system (PHMS) in the Fourier domain *j* ≤ 2, *j* ∈ *Z* and −2*j* ≤ *j* < 2*j*.

For a signal or image *u*, the corresponding *PHMS* transform P(*u*) in the frequency domain can be defined as

$$\begin{split} \mathcal{P}(u) \triangleq & \mathfrak{h}\_{\mathbb{C}\_{R'}} PHMS > = \sum\_{j=0}^{I} \sum\_{\substack{\ell\_{j}=-\ell(2^{-j}\frac{\pi}{4^{j}}) \ \ell=-2^{j}}}^{\ell(2^{-j}\frac{\pi}{4^{j}})} \sum\_{\ell=-2^{j}}^{2^{j}} (\mathcal{H}\_{j,\ell\_{j}} \* \mathfrak{h}\_{\mathbb{C}\_{R}} + \mathcal{S}\_{\mathbb{J},\ell\_{j}} \* \mathfrak{h}\_{\mathbb{C}\_{R}} \\ & + \mathcal{H}\_{j,\ell\_{j}}^{\*} \cdot \* \mathfrak{h}\_{\mathbb{C}\_{R}} + \mathcal{S}\_{j,\ell\_{j}}^{\*} \cdot \* \mathfrak{h}\_{\mathbb{C}\_{R}}), \end{split} \tag{38}$$

with *j* = -2−*j π*4 , |-| ≤ 2*j*, - ∈ *Z*, where *<sup>u</sup>*<sup>ˆ</sup>*CR* is the *CPFT* of *u*, defined in (22). In addition, .∗ is the dot product, and the matrix *M*1. ∗ *M*2 is defined as

$$(M\_1.\*M\_2)\_{i,j} = (M\_1)\_{i,j}(M\_2)\_{i,j}.\tag{39}$$

**Theorem 1.** *The discrete polar harmonic multilevel system* PHMS *defined on* CPG *forms a framelet of <sup>L</sup>*<sup>2</sup>(*R*<sup>2</sup>)*.*

According to the framelet defined in [18], for the signal *U* in (38),

$$\|\|\mathbf{U}\|\|^2 \le \|<\mathbf{U}, \text{PHMS} > \|\|^2 \le c \|\|\mathbf{U}\|\|^2,\tag{40}$$

where *c* < +∞ is the constant; therefore, *PHMS* forms a framelet of *<sup>L</sup>*<sup>2</sup>(*R*<sup>2</sup>).

Then two denoising reconstruction tests of *PHMS* are shown in Figure 5.

**Figure 5.** Recovery results by *PHMS* with scale *j* = 4, the random noise level *σ* = 20.
