# Restoration of Bi-Contrast MRI Data for Intensity Uniformity with Bayesian Coring of Co-Occurrence Statistics

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## Abstract

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## 1. Introduction

## 2. General Bayesian Formulation

#### 2.1. Spatial and Statistical Image Representation

#### 2.2. Posterior Expectation for Voxelwise Intensity Restoration

#### 2.3. Back-Projection of Intensity Restoration to the Images

## 3. Methods

#### 3.1. Spatial and Statistical Image Representation

#### 3.2. Statistical Representation of Intensity Non-Uniformities

#### 3.3. Non-Stationary Restoration of the Co-Occurrence Statistics

#### 3.4. Back-Projection of the Co-Occurrences Restoration to the Images

#### 3.5. Estimation of the Cumulative Intensity Restoration

#### 3.6. Validity of Image Domains and Stability of the Images’ Dynamic Ranges

## 4. Results

#### 4.1. Implementation and Efficiency

#### 4.2. Description of the Phantom BrainWeb Brain Images

#### 4.3. Validation Measures for the Phantom BrainWeb Brain Images

#### 4.4. Experiments with the Phantom BrainWeb Brain Images

#### 4.5. Description of the Real Images

#### 4.6. Validation Measure for the Real Brain Images

#### 4.7. Experiments with Brain Images of the Human Connectome Project

#### 4.8. Experiments with the Brain Images of Parkinson’s Disease Patients

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Derivation of the PSF, ${\mathit{P}}_{\mathit{b}}$, of the Statistical Distortion

## References

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**Figure 2.**Overview of the non-parametric Bayesian formulation for image restoration. The restoration is repeated iteratively.

**Figure 3.**Overview of the series of steps in the implementation of the Bayesian formulation for image restoration. The restoration is joint for two images and iterative.

**Figure 4.**The auto-co-occurrence statistics and the joint-co-occurrence statistics of a ${T}_{1}$ and a ${T}_{2}$ image of the BrainWeb phantom without non-uniformity, $b=0\%$, and with noise of $n=5\%$ [37]. The densities in the statistics are displayed in logarithmic scale. The individual distributions of the Gray Mater (GM), White Matter (WM), and Cerebrospinal Fluid (CSF) are apparent.

**Figure 5.**The restoration of a ${T}_{1}$w and a ${T}_{2}$w BrainWeb image pair with non-uniformity of $100\%$ and noise of $5\%$. The restoration makes the cerebellum brighter and the statistics sharper.

**Figure 6.**Example restoration of a ${T}_{1}$w and a ${T}_{2}$w image pair for the HCP LS dataset. The intensities of the white matter and the gray matter in both the ${T}_{1}$w image and in the ${T}_{2}$w image become more uniform. The statistical distributions become sharper.

**Figure 7.**Example restoration of a ${T}_{1}$w and a ${T}_{2}$w image pair of a Parkinson’s disease patient. The intensities of the white matter in both the ${T}_{1}$w image and in the ${T}_{2}$w image become more uniform. The statistical distributions become sharper.

**Table 1.**Validation for BrainWeb phantom ${T}_{1}$w and ${T}_{2}$w images with $CJ{V}_{i}$ of GM and WM tissue regions. A low value indicates improved performance. In parentheses is the ratio of the restored to the original, $CJ{V}_{i}^{ratio}$. Low values and less than unity indicate improved performance.

BrainWeb∖Method | Original | Joint Co–Occurrences | ||
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${\mathit{T}}_{\mathbf{1}}$ | ${\mathit{T}}_{\mathbf{2}}$ | ${\mathit{T}}_{\mathbf{1}}$ | ${\mathit{T}}_{\mathbf{2}}$ | |

n = 0, RF = 0 | 0.581369 | 0.770175 | 0.581369 (1) | 0.770175 (1) |

n = 3, RF = 40 | 0.765254 | 1.1697 | 0.660506 (0.86312) | 1.02978 (0.880374) |

n = 5, RF = 0 | 0.720008 | 1.13181 | 0.720008 (1) | 1.13181 (1) |

n = 5, RF = 20 | 0.735217 | 1.21963 | 0.762611 (1.03726) | 1.24902 (1.02409) |

n = 5, RF = 40 | 0.815382 | 1.37113 | 0.758738 (0.930531) | 1.26971 (0.926038) |

n = 5, RF = 60 | 1.29703 | 2.33456 | 0.843404 (0.650259) | 1.39622 (0.598069) |

n = 5, RF = 80 | 1.29703 | 2.33456 | 0.847001 (0.653031) | 1.39361 (0.59695) |

n = 5, RF = 100 | 1.29703 | 2.33456 | 0.861492 (0.664204) | 1.39666 (0.598253) |

**Table 2.**Validation for BrainWeb phantom ${T}_{1}$w and ${T}_{2}$w images with difference to underlying anatomic images $dif{f}_{t,i}$. A low value indicates improved performance. In parentheses is the ratio of after to before the restoration, $dif{f}_{i}^{ratio}$. Low values and less than unity indicate improved performance.

BrainWeb∖Method | Original | Joint Co–Occurrences | ||
---|---|---|---|---|

${\mathit{T}}_{\mathbf{1}}$ | ${\mathit{T}}_{\mathbf{2}}$ | ${\mathit{T}}_{\mathbf{1}}$ | ${\mathit{T}}_{\mathbf{2}}$ | |

n = 0, RF = 0 | 0.0203729 | 0.0397313 | 0.0203729 (1) | 0.0397313 (1) |

n = 3, RF = 40 | 0.0453967 | 0.0604082 | 0.0253967 (0.55944) | 0.0502752 (0.832258) |

n = 5, RF = 0 | 0.0331167 | 0.0591677 | 0.0331167 (1) | 0.0591677 (1) |

n = 5, RF = 20 | 0.0385987 | 0.0610285 | 0.0325415 (0.843074) | 0.0582324 (0.954184) |

n = 5, RF = 40 | 0.0494876 | 0.0681559 | 0.0326034 (0.658818) | 0.0584461 (0.857535) |

n = 5, RF = 60 | 0.116259 | 0.105483 | 0.0503919 (0.433445) | 0.0655376 (0.621312) |

n = 5, RF = 80 | 0.116255 | 0.105479 | 0.0506253 (0.435466) | 0.0651502 (0.617661) |

n = 5, RF = 100 | 0.116252 | 0.105476 | 0.0523669 (0.450459) | 0.0655749 (0.621706) |

**Table 3.**Statistics of ${H}_{exp}^{ratio}$ for the ${T}_{1}$w and the ${T}_{2}$w images for the 27 HCP LS volunteers. The ${H}_{exp}^{ratio}$ are significantly negative for all images and hence all the restorations are successful.

Mean | Stand. Dev. | Median | Minimum | Maximum | |
---|---|---|---|---|---|

${H}_{ratio}$ for ${T}_{1}$ | $-0.2346099$ | 0.03155122 | $-0.233064$ | $-0.289283$ | $-0.168966$ |

${H}_{ratio}$ for ${T}_{2}$ | $-0.1566847$ | 0.07183654 | $-0.13836$ | $-0.37288$ | $-0.0605923$ |

**Table 4.**Statistics of ${H}_{exp}^{ratio}$ for the ${T}_{1}$w and the ${T}_{2}$w images for the 45 HCP Retest volunteers. The ${H}_{exp}^{ratio}$ are significantly negative for all images and hence all the restorations are successful.

Mean | Stand. Dev. | Median | Minimum | Maximum | |
---|---|---|---|---|---|

${H}_{ratio}$ for ${T}_{1}$ | $-0.251255$ | 0.03185837 | $-0.254848$ | $-0.316611$ | $-0.182293$ |

${H}_{ratio}$ for ${T}_{2}$ | $-0.1476605$ | 0.04146889 | $-0.145265$ | $-0.261554$ | $-0.0680923$ |

**Table 5.**Statistics of ${H}_{exp}^{ratio}$ for the ${T}_{1}$w and the ${T}_{2}$w images for the 60 Parkinson’s disease patients. The ${H}_{exp}^{ratio}$ are significantly negative for all images and hence all the restorations are successful.

Mean | Stand. Dev. | Median | Minimum | Maximum | |
---|---|---|---|---|---|

${H}_{ratio}$ for ${T}_{1}$ | $-0.1381152$ | 0.01930447 | $-0.138554$ | $-0.183722$ | $-0.0961817$ |

${H}_{ratio}$ for ${T}_{2}$ | $-0.224674$ | 0.06584581 | $-0.2302005$ | $-0.337128$ | $-0.0176404$ |

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**MDPI and ACS Style**

Hadjidemetriou, S.; Psychogios, M.N.; Lingor, P.; Von Eckardstein, K.; Papageorgiou, I.
Restoration of Bi-Contrast MRI Data for Intensity Uniformity with Bayesian Coring of Co-Occurrence Statistics. *J. Imaging* **2017**, *3*, 67.
https://doi.org/10.3390/jimaging3040067

**AMA Style**

Hadjidemetriou S, Psychogios MN, Lingor P, Von Eckardstein K, Papageorgiou I.
Restoration of Bi-Contrast MRI Data for Intensity Uniformity with Bayesian Coring of Co-Occurrence Statistics. *Journal of Imaging*. 2017; 3(4):67.
https://doi.org/10.3390/jimaging3040067

**Chicago/Turabian Style**

Hadjidemetriou, Stathis, Marios Nikos Psychogios, Paul Lingor, Kajetan Von Eckardstein, and Ismini Papageorgiou.
2017. "Restoration of Bi-Contrast MRI Data for Intensity Uniformity with Bayesian Coring of Co-Occurrence Statistics" *Journal of Imaging* 3, no. 4: 67.
https://doi.org/10.3390/jimaging3040067