# A Novel Kernel-Based Regularization Technique for PET Image Reconstruction

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Acquisitions and Sinograms

#### 2.2. PET Data Corrections

#### 2.3. Maximum-Likelihood Expectation-Maximization Algorithm

- Start with an initial estimate ${x}_{j}^{\left[0\right]}$ satisfying ${x}_{j}^{\left[0\right]}>0,j=1,2,\cdots N$,
- If ${x}_{j}^{\left[k\right]}$ denotes the estimate of ${x}_{j}$ at the ${k}^{\mathrm{th}}$ iteration, define a new estimate ${x}_{j}^{[k+1]}$ by using Equation (11),
- If the required accuracy for the numerical convergence has been achieved, then stop.

#### 2.4. Regularization

#### 2.4.1. The Non-Local Means Filter

#### 2.4.2. The Anisotropic filtering

#### 2.4.3. Proposed Kernel-Based Exponentially-Modified Gaussian Regularization

## 3. Computer Simulation

#### 3.1. Simulated PET Data (Setup)

#### 3.2. Quality Evaluation

## 4. Results and Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Kernel feature extraction, calculating the box-wise multiplication with the image box and convolution kernel.

**Figure 4.**Reconstructed images of the Hoffman brain phantom by different algorithms with 150 iterations. MLEM: maximum-likelihood expectation-maximization; $\kappa $-MLEM: the experimental kernel-based MLEM; ADF-MLEM: the penalized likelihood reconstruction regularized by anisotropic diffusion filter; Pre-NLM: MLEM precondition reconstruction algorithm that was regularized by non-local means; Post-NLM: post-reconstruction de-noising methods applied post-MLEM.

**Figure 5.**Comparative analysis of reconstructing a Hoffman phantom using various reconstruction methods by varying iteration number. MSE: mean square error; NCC: normalized cross-correlation; PSNR: peak signal-to-noise ratio.

**Figure 6.**Comparative analysis of reconstructing a Hoffman phantom using various reconstruction methods by varying the iteration number.

**Figure 9.**Zoom-in of the tumors of the Hoffman phantom reconstruction achieved with various algorithms.

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Boudjelal, A.; Messali, Z.; Elmoataz, A. A Novel Kernel-Based Regularization Technique for PET Image Reconstruction. *Technologies* **2017**, *5*, 37.
https://doi.org/10.3390/technologies5020037

**AMA Style**

Boudjelal A, Messali Z, Elmoataz A. A Novel Kernel-Based Regularization Technique for PET Image Reconstruction. *Technologies*. 2017; 5(2):37.
https://doi.org/10.3390/technologies5020037

**Chicago/Turabian Style**

Boudjelal, Abdelwahhab, Zoubeida Messali, and Abderrahim Elmoataz. 2017. "A Novel Kernel-Based Regularization Technique for PET Image Reconstruction" *Technologies* 5, no. 2: 37.
https://doi.org/10.3390/technologies5020037