# A Framework for Circular Multilevel Systems in the Frequency Domain

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### The Basic Definition

## 3. The Circular-Shape Fourier Transform (CFT)

#### 3.1. The Pseudo-Polar Grid

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

#### 3.3. The Choice of Weights ${w}_{c}$

## 4. The Construction of Multilevel System in Frequency Domain

#### 4.1. 2D Basic Harmonic Function

**Definition 1**.

**Definition**

**1.**

**Definition**

**2.**

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

**Definition**

**3.**

**Theorem**

**1.**

## 5. Quantitative Test Measures

- Isometry of $CFT$:
- (a)
- Closeness to tight: ${M}_{clo}={\mathrm{max}}_{i=1,\dots ,5}\frac{\parallel {\mathcal{F}}_{p}^{\ast}{\mathcal{F}}_{p}{u}_{i}-{u}_{i}{\parallel}_{2}}{\parallel {u}_{i}{\parallel}_{2}},$
- (b)
- Quality of preconditioning. ${M}_{qua}=\frac{{\lambda}_{\mathrm{max}}\left({\mathcal{F}}_{p}^{\ast}{\mathcal{F}}_{p}\right)}{{\lambda}_{\mathrm{min}}\left({\mathcal{F}}_{p}^{\ast}w{\mathcal{F}}_{p}\right)}.$

- Tight Frame Property: The operator norm $\parallel {\mathcal{P}}^{\ast}{\mathcal{P}-I\parallel}_{op}$, which is defined as ${M}_{tig}={\mathrm{max}}_{i=1,\dots ,5}\frac{\parallel {\mathcal{P}}^{\ast}\mathcal{P}{u}_{i}-{u}_{i}{\parallel}_{2}}{\parallel {u}_{i}{\parallel}_{2}}.$
- Robustness:
- (a)
- Thresholding: Let u be the regular sampling of a Gaussian function with mean 0 and variance 512 on ${\left[257\right]}^{2}$ generating an $512\times 512$ image. Two types of robustness are considered, for $k=1,2$, and ${M}_{{p}_{k}}=\frac{\parallel {\mathcal{P}}^{\ast}{T}_{{p}_{k}}{\mathcal{P}u-u\parallel}_{2}}{{\parallel u\parallel}_{2}}$.
- ${T}_{{p}_{1}}$:
- ${T}_{{p}_{1}}$ discards $100(1-{2}^{-{p}_{1}})$ percent of coefficient, with ${p}_{1}=[2:2:20]$.
- ${T}_{{p}_{2}}$:
- ${T}_{{p}_{2}}$ keeps the absolute value of coefficients bigger than $m/{2}^{{p}_{2}}$ with m is the maximal absolute value of all coefficients, where ${p}_{2}=[0.5:0.5:5]$.

- (b)
- Quantization: The quality measure is given as ${M}_{p}=\frac{\parallel {\mathcal{P}}^{\ast}{Q}_{q}{\mathcal{P}u-u\parallel}_{2}}{{\parallel u\parallel}_{2}},$ where ${Q}_{q}\left(c\right)=round(c/(m/{2}^{q}))\xb7(m/{2}^{q})$, and $q\in [5:-0.5:0.5]$.

## 6. Test Results

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CPG | Circular-polar grid |

MRA | Multiresolution analysis |

CFT | Circular-shape Fourier transform |

CMS | Circular-shape directional multilevel system |

PHMS | Polar harmonic multilevel system |

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**Figure 1.**The harmonic wavelet function. (

**a**) the even part ${H}_{e}\left(\omega \right)$; (

**b**) the odd part ${H}_{o}\left(\omega \right)$; (

**c**) the harmonic wavelet function $H\left(\omega \right)$.

**Figure 2.**The harmonic scaling function. (

**a**) the even part ${S}_{e}\left(\omega \right)$; (

**b**) the odd part ${S}_{o}\left(\omega \right)$; (

**c**) the scaling function $S\left(\omega \right)$.

**Figure 4.**The polar harmonic multilevel system (PHMS) in the Fourier domain $j\le 2$, ${\ell}_{j}\in Z$ and $-{2}^{j}\le {\ell}_{j}<{2}^{j}$.

${\mathit{M}}_{\mathit{clo}}$ | ${\mathit{M}}_{\mathit{qua}}$ | ${\mathit{M}}_{\mathit{tig}}$ |
---|---|---|

0.00946092 | 1.9342627 | 0.1036731 |

${\mathit{M}}_{{\mathit{p}}_{1}}$ | 3.6 × 10${}^{-6}$ | 1.7 × 10${}^{-5}$ | 5.9 × 10${}^{-3}$ | 2.1 × 10${}^{-2}$ | 0.9 × 10${}^{-2}$ |

${\mathit{M}}_{{\mathit{p}}_{\mathbf{2}}}$ | 0.009 | 0.063 | 0.103 | 0.172 | 0.197 |

${\mathit{M}}_{\mathit{p}}$ | 0.051 | 0.065 | 0.083 | 0.126 | 0.143 |

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**MDPI and ACS Style**

Sun, G.; Leng, J.; Cattani, C.
A Framework for Circular Multilevel Systems in the Frequency Domain. *Symmetry* **2018**, *10*, 101.
https://doi.org/10.3390/sym10040101

**AMA Style**

Sun G, Leng J, Cattani C.
A Framework for Circular Multilevel Systems in the Frequency Domain. *Symmetry*. 2018; 10(4):101.
https://doi.org/10.3390/sym10040101

**Chicago/Turabian Style**

Sun, Guomin, Jinsong Leng, and Carlo Cattani.
2018. "A Framework for Circular Multilevel Systems in the Frequency Domain" *Symmetry* 10, no. 4: 101.
https://doi.org/10.3390/sym10040101