# A Generalized Construction Model for CT Projection-Wise Filters on the SDBP Technique

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## Abstract

**:**

## 1. Introduction

## 2. Inverse Radon Transform with SDBP Technique

#### 2.1. Radon Transform

**Hypothesis**

**1.**

- $f\left(\overrightarrow{\mathit{x}}\right)$is continuous;
- The double integral$\iint \frac{\left|f\left(\overrightarrow{\mathit{x}}\right)\right|}{\sqrt{{\overrightarrow{\mathit{x}}}^{T}\overrightarrow{\mathit{x}}}}{d}^{2}\overrightarrow{\mathit{x}}$extending over the whole plane, converges;
- For any arbitrary point$\overrightarrow{\mathit{x}}$on the plane, $\underset{\epsilon \to {0}_{+}}{\mathrm{lim}}{\displaystyle {\int}_{0}^{2\pi}f\left(\overrightarrow{\mathit{x}}+\frac{\overrightarrow{\mathit{\phi}}}{\epsilon}\right)}d\phi =0$, where$\overrightarrow{\mathit{\phi}}={\left[\mathrm{cos}\phi ,\mathrm{sin}\phi \right]}^{T}$.

**Definition**

**1.**

#### 2.2. SDBP Technique

**Hypothesis**

**2.**

- The integral${\int}_{\mathbb{R}}\left|\mathcal{R}\left[f\right]\left(r,\phi \right)\right|}dr$converges;
- $\underset{s\to 0}{\mathrm{lim}}{\Delta}_{r}^{2}\left[\mathit{\mathcal{R}}\left[f\right]\left(r,\phi \right)\right]\left(s\right)=0$, where${\Delta}_{r}^{2}\left[\otimes \right]\left(s\right)$of$s\in \mathbb{R}$is the second-order difference operator with respect to variable$r$, and more specifically$${\Delta}_{r}^{2}\left[\mathcal{R}\left[f\right]\left(r,\phi \right)\right]\left(s\right):=\mathcal{R}\left[f\right]\left(r-s,\phi \right)-2\mathcal{R}\left[f\right]\left(r,\phi \right)+\mathcal{R}\left[f\right]\left(r+s,\phi \right);$$
- The first-order partial derivative$\frac{\partial}{\partial r}\mathcal{R}\left[f\right]\left(r,\phi \right)$is continuous, or only has removable discontinuities;
- The second-order partial derivative$\frac{{\partial}^{2}}{\partial {r}^{2}}\mathcal{R}\left[f\right]\left(r,\phi \right)$is continuous, or only has removable or jump discontinuities.

**Theorem**

**1.**

- Calculation of the second-order divided difference of the projection function under a certain projection angle;
- Linear integration of the second-order divided difference at the projection position of the target point, which is calculated in step 1;
- Rotational integration of the linear integral value calculated in step 2, also known as the back projection progress.

**Proof**

**of**

**Theorem**

**1.**

## 3. Computational Model for Filter Expressions

#### 3.1. Projection Data Preprocess

**Hypothesis**

**3.**

- The sampling interval of the detector is$\mathit{d}$, $\mathit{d}\in {\mathbb{R}}_{+}$;
- The projection angles are discrete at equal intervals, and$M$is the number of total projections;
- For$m\in \mathbb{N}$and$n\in \mathbb{Z}$, the projection value is defined as$${P}_{m}\left(n\right):={\displaystyle {\int}_{\mathbb{R}}\mathcal{R}\left[f\right]\left(r,{\phi}_{m}\right)\cdot \delta \left(r-n\mathit{d}\right)dr},$$

**Definition**

**2.**

**Hypothesis**

**4.**

- $k\left(\widehat{s}\right)$is piecewise continuous, and the integral${\int}_{\mathbb{R}}k\left(\widehat{s}\right)d\widehat{s}}=1$;
- $k\left(\widehat{s}\right)\hspace{0.17em}=\hspace{0.17em}k\left(-\widehat{s}\right)$,$k\left(\nu \right)\left|{}_{\nu \ne 0}\right.=0$, and$\underset{t\to \infty}{\mathrm{lim}}k\left(\widehat{s}\right)=0$;
- $k\left(\widehat{s}\right)$is differentiable at any$\widehat{s}=\nu \ne 0$;
- The second-order derivative$\frac{{d}^{2}k\left(\widehat{s}\right)}{d{\widehat{s}}^{2}}$is continuous in the deleted neighborhood of any$\widehat{s}=\nu \ne 0$, and$\frac{{d}^{2}k\left(\widehat{s}\right)}{d{\widehat{s}}^{2}}$converges on both sides of any$\widehat{s}=\nu \ne 0$.

#### 3.2. Conversion Relation between Convolution Kernels and Filter Functions

**Theorem**

**2.**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

## 4. Illustration with Examples

#### 4.1. R-L Filter: $\mathrm{sin}\mathrm{c}\left(\u2022\right)$ Kernel

#### 4.2. S-L Filter: $\Pi \left(\u2022\right)$ Kernel

#### 4.3. Delta Filter: $\delta \left(\u2022\right)$ Kernel

#### 4.4. Imaging Tests

## 5. Analysis of Basic Composition of Filters

#### 5.1. Decomposition of Kernel Functions

**Definition**

**3.**

**Definition**

**4.**

- $\sum _{\nu \in \mathbb{Z}}{\mathfrak{a}}_{\Lambda}\left(\nu \right)}\hspace{0.17em}=\hspace{0.17em}1$;
- ${\mathfrak{a}}_{\Lambda}\left(\nu \right)\hspace{0.17em}=\hspace{0.17em}{\mathfrak{a}}_{\Lambda}\left(-\nu \right)$;
- For a finite fixed value of $\Lambda $, $\underset{\nu \to \infty}{\mathrm{lim}}{\mathfrak{a}}_{\Lambda}\left(\nu \right)\hspace{0.17em}=\hspace{0.17em}0$.

#### 5.2. Filters of Basic Type

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**4.**

#### 5.3. Imaging Tests

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Table of some functional Fourier transform pairs [26].

Rule Number | Function | Fourier Transform |
---|---|---|

0 | $f\left(x\right)$ | ${\mathcal{F}}_{x}\left[f\right]\left(\xi \right)\hspace{0.17em}=\hspace{0.17em}{\displaystyle {\int}_{\mathbb{R}}f\left(x\right)\cdot \mathrm{exp}\left(-2\pi \mathit{i}\xi x\right)dx}$ |

1 | $a\cdot f\left(x\right)+b\cdot g\left(x\right)$ | $a\cdot {\mathcal{F}}_{x}\left[f\right]\left(\xi \right)+b\cdot {\mathcal{F}}_{x}\left[g\right]\left(\xi \right)$ |

2 | $f\left(x-a\right)$ | ${\mathcal{F}}_{x}\left[f\right]\left(\xi \right)\cdot \mathrm{exp}\left(-2\pi \mathit{i}a\xi \right)$ |

3 | $f\left(ax\right)$ | $\frac{1}{\left|a\right|}\cdot {\mathcal{F}}_{x}\left[f\right]\left(\frac{\xi}{a}\right)$ |

4 | ${\mathcal{F}}_{\xi}\left[f\right]\left(x\right)$ | $f\left(-\xi \right)$ |

5 | $\frac{{d}^{n}}{d{x}^{n}}f\left(x\right)$ | ${\left(2\pi \mathit{i}\xi \right)}^{n}\cdot {\mathcal{F}}_{x}\left[f\right]\left(\xi \right)$ |

6 | ${x}^{n}\cdot f\left(x\right)$ | ${\left(\frac{\mathit{i}}{2\pi}\right)}^{n}\cdot \frac{{d}^{n}}{d{\xi}^{n}}{\mathcal{F}}_{x}\left[f\right]\left(\xi \right)$ |

7 | $f\left(x\right)\ast g\left(x\right)$ | ${\mathcal{F}}_{x}\left[f\right]\left(\xi \right)\cdot {\mathcal{F}}_{x}\left[g\right]\left(\xi \right)$ |

8 | $f\left(x\right)\cdot g\left(x\right)$ | ${\mathcal{F}}_{x}\left[f\right]\left(\xi \right)\ast {\mathcal{F}}_{x}\left[g\right]\left(\xi \right)$ |

## Appendix B

**Table A2.**Data set of RMSE and AGM of reconstructed Shepp–Logan images by basic filters with different $\lambda $.

λ Value | RMSE | AGM | |||
---|---|---|---|---|---|

0 | ±0.1 | 0.2824 | 0.2867 | 1.8782 | 1.9003 |

±0.2 | ±0.3 | 0.2890 | 0.2942 | 1.9617 | 2.0801 |

±0.4 | ±0.5 | 0.3221 | 0.3975 | 2.2354 | 2.4578 |

±0.6 | ±0.7 | 0.5801 | 0.8699 | 2.7555 | 3.1952 |

±0.8 | ±0.9 | 1.2499 | 1.6373 | 3.8340 | 4.5736 |

±1.1 | ±1.2 | 1.6345 | 1.2620 | 4.5341 | 3.6529 |

±1.3 | ±1.4 | 0.9700 | 0.8676 | 2.8780 | 2.3502 |

±1.5 | ±1.6 | 0.9142 | 1.0400 | 2.0154 | 1.8432 |

±1.7 | ±1.8 | 1.2166 | 1.4085 | 1.8012 | 1.8297 |

±1.9 | ±2.5 | 1.5649 | 1.1463 | 1.9009 | 1.5652 |

±3.5 | ±4.5 | 1.1787 | 1.1407 | 1.4371 | 1.3034 |

±5.5 | ±6.5 | 1.1055 | 1.0779 | 1.2284 | 1.2093 |

±7.5 | ±8.5 | 1.0533 | 1.0425 | 1.1680 | 1.1519 |

±9.5 | ±10.5 | 1.0359 | 1.0189 | 1.1417 | 1.1248 |

±11.5 | ±12.5 | 1.0056 | 0.9930 | 1.1149 | 1.0965 |

±13.5 | ±14.5 | 0.9813 | 0.9711 | 1.0747 | 1.0532 |

±15.5 | ±16.5 | 0.9614 | 0.9498 | 1.0377 | 1.0167 |

±17.5 | ±18.5 | 0.9431 | 0.9384 | 0.9981 | 0.9874 |

±19.5 | 0.9365 | 0.9754 |

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**Figure 1.**Reconstructed Shepp–Logan phantom images by exemplified filters: (

**a**) R-L filter without noise; (

**b**) S-L filter without noise; (

**c**) Delta filter without noise; (

**d**) R-L filter in noise of σ = 5; (

**e**) S-L filter in noise of σ = 5; (

**f**) Delta filter in noise of σ = 5.

**Figure 2.**Reconstructed Shepp–Logan phantom images by basic filters with different parameters: (

**a**) $\lambda =1/2$; (

**b**) $\lambda =25/2$; (

**c**) $\lambda =49/2$; (

**d**) $\lambda =3/4$; (

**e**) $\lambda =7/8$; (

**f**) $\lambda =15/16$.

**Figure 3.**RMSE and AGM of CT images reconstructed by basic filters with different $\lambda $ ($\lambda =\nu /10$, $\nu \in \mathbb{Z}$ and $\lambda \notin \mathbb{Z}\backslash \left\{0\right\}$ ).

**Figure 4.**RMSE and AGM of CT images reconstructed by basic filters with different $\lambda $ ($\lambda =1/2+\nu $ and $\nu \in \mathbb{Z}$ ).

**Figure 5.**Digital resolution chart images reconstructed by basic filters with different parameters: (

**a**) $\lambda =1/2$; (

**b**) $\lambda =25/2$; (

**c**) $\lambda =49/2$; (

**d**) $\lambda =3/4$; (

**e**) $\lambda =7/8$; (

**f**) $\lambda =15/16$.

**Figure 6.**Local centers of digital resolution chart images reconstructed by basic filters with different parameters: (

**a**) $\lambda =1/2$; (

**b**) $\lambda =3/4$; (

**c**) $\lambda =15/16$.

Noise Density | R-L Filter | S-L Filter | Delta Filter |
---|---|---|---|

$\sigma =0$ | 0.2672 | 0.2508 | 0.2431 |

$\sigma =1$ | 0.3231 | 0.2919 | 0.2784 |

$\sigma =5$ | 0.6544 | 0.5886 | 0.5332 |

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**MDPI and ACS Style**

Jiang, Y.; Zhao, J.; Hu, X.; Zou, J. A Generalized Construction Model for CT Projection-Wise Filters on the SDBP Technique. *Mathematics* **2022**, *10*, 579.
https://doi.org/10.3390/math10040579

**AMA Style**

Jiang Y, Zhao J, Hu X, Zou J. A Generalized Construction Model for CT Projection-Wise Filters on the SDBP Technique. *Mathematics*. 2022; 10(4):579.
https://doi.org/10.3390/math10040579

**Chicago/Turabian Style**

Jiang, Yiming, Jintao Zhao, Xiaodong Hu, and Jing Zou. 2022. "A Generalized Construction Model for CT Projection-Wise Filters on the SDBP Technique" *Mathematics* 10, no. 4: 579.
https://doi.org/10.3390/math10040579