# A Novel Iterative MLEM Image Reconstruction Algorithm Based on Beltrami Filter: Application to ECT Images

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- 1.
- Start with estimate ${x}^{\left(0\right)}$, where ${x}^{\left(0\right)}>0$ for $i=1,2,3,\dots ,I$.
- 2.
- If ${x}^{\left(n\right)}$ denotes the estimate of x at the ${n}^{th}$ iteration, define a new estimate ${n}^{(n+1)}$ using Equation (6).
- 3.
- If the required accuracy for the numerical convergence has been achieved, then stop; else, return to (2).

#### 2.1. Noise Reduction Method Based on Geometric Flow

#### 2.2. Combining MLEM Algorithm and Beltrami Image Flow Filtering

Algorithm 1: filtered MLEM Algorithm |

## 3. Performance Evaluation

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Conventional MLEM reconstructions. (

**a**–

**c**) row: iterations 20, 40, and 60. (

**d**–

**f**) row: iterations 80, 100, and 120.

**Figure 3.**Conventional MLEM reconstructions. (

**a**–

**c**) row: iterations 20, 40, and 60. (

**d**–

**f**) row: iterations 80, 100, and 120.

**Figure 4.**f-MLEM Intermediate MLEM reconstructions. (

**a**–

**c**) row: iterations 20, 40, and 60. (

**d**–

**f**) row: iterations 80, 100, and 120.

**Figure 5.**f-MLEM Intermediate MLEM reconstructions. (

**a**–

**c**) row: iterations 20, 40, and 60. (

**d**–

**f**) row: iterations 80, 100, and 120.

**Figure 6.**ROIs of Huffman Brain Phantom reconstruction achieved using various algorithms as shown under magnification: (

**a**,

**d**) the ground truth; (

**b**,

**e**) MLEM; and (

**c**,

**f**) f-MLEM.

**Figure 7.**ROIs of standard medical image of abdomen reconstructed using various algorithms as shown under magnification: (

**a**,

**d**) the ground truth; (

**b**,

**e**) MLEM; and (

**c**,

**f**) f-MLEM.

**Table 1.**Signal-to-noise ratio (SNR) by varying number of iterations for f-MLEM and MLEM algorithms. Test image: Huffman Brain.

10 | 20 | 40 | 60 | 80 | 100 | |
---|---|---|---|---|---|---|

MLEM | 15.00 | 16.50 | 18.00 | 18.20 | 18.45 | 19.00 |

f-MLEM | 16 | 19.00 | 23.30 | 25.10 | 26.02 | 26.30 |

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**MDPI and ACS Style**

Boudjelal, A.; Elmoataz, A.; Attallah, B.; Messali, Z. A Novel Iterative MLEM Image Reconstruction Algorithm Based on Beltrami Filter: Application to ECT Images. *Tomography* **2021**, *7*, 286-300.
https://doi.org/10.3390/tomography7030026

**AMA Style**

Boudjelal A, Elmoataz A, Attallah B, Messali Z. A Novel Iterative MLEM Image Reconstruction Algorithm Based on Beltrami Filter: Application to ECT Images. *Tomography*. 2021; 7(3):286-300.
https://doi.org/10.3390/tomography7030026

**Chicago/Turabian Style**

Boudjelal, Abdelwahhab, Abderrahim Elmoataz, Bilal Attallah, and Zoubeida Messali. 2021. "A Novel Iterative MLEM Image Reconstruction Algorithm Based on Beltrami Filter: Application to ECT Images" *Tomography* 7, no. 3: 286-300.
https://doi.org/10.3390/tomography7030026