# Restoration for Intensity Nonuniformities with Discontinuities in Whole-Body MRI

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## Abstract

**:**

## 1. Introduction

## 2. Patient and Data Description

#### 2.1. Study Design and Ethical Approval

#### 2.2. Patients and Data Description

- A ${T}_{1}$-weighted (${T}_{1}$w) sequence in a Turbo Spin Echo (TSE) or a Fast Field Echo (FFE) technique,
- A Short Tau Inversion Recovery (STIR): This is an inverted pulse fat suppression method that, apart from the fat suppression, prolongs the ${T}_{1}$w relaxation, and thus reduces the ${T}_{1}$w signal. It furthermore prolongs the ${T}_{2}$w relaxation, thus enhancing the ${T}_{2}$w signal. All in all, STIR has an additive effect of enhancing the ${T}_{2}$w signal. Thus, the STIR sequence offers ${T}_{1}$ weighted (${T}_{1}$w) and ${T}_{2}$ weighted (${T}_{2}$w) images, (${T}_{1}$w + ${T}_{2}$w) images.

#### 2.3. Preparation of the Data

## 3. Methods

#### 3.1. Spatial and Statistical Image Representation

#### 3.1.1. Spatial Image Representation

#### 3.1.2. Statistical Image Representation

#### 3.2. Effect of Spatial Intensity Nonuniformities in Co-Occurrences

#### 3.2.1. Distortion in Co-Occurrences from Spatial Intensity Nonuniformity

#### 3.2.2. Restoration from Spatial Non-Uniformity in Co-Occurrences

#### 3.3. Bayesian Posterior Expectation for the Restoration

#### 3.4. Spatial Image Restoration

#### 3.4.1. Back-Projection of Restoration to Space

#### 3.4.2. Anisotropic Smoothing of the Restoration Field

#### 3.5. Iterative and Stable Estimation of Cumulative Intensity Restoration

#### 3.5.1. Incremental Nonuniformity Correction Field

#### 3.5.2. Iterative Estimation of Cumulative Intensity Restoration

#### 3.5.3. End Condition for the Iterations

#### 3.6. Valid Domains in Image Space and Statistics

#### 3.6.1. Valid Signal Region of Images

#### 3.6.2. Valid Dynamic Co-Occurrence Ranges in Statistics

#### 3.6.3. Stability of Valid Co-Occurrence Ranges along the Iterations

## 4. Experimental Results

#### 4.1. Data Quality

#### 4.2. Implementation

#### 4.2.1. Parameters of the Method and Their Values

#### 4.2.2. Computational Complexity of the Method

- Computation of the co-occurrence and the joint co-occurrence statistics:$3\left|I\right|{\left(\frac{\rho}{\u2206\rho}\right)}^{2}\approx 6\times {10}^{6}\times {\left(\frac{\rho}{\u2206\rho}\right)}^{2}$.
- Deconvolution of the statistics: ${3\left({\eta}_{i}^{max}\right)}^{2}{\left({\sigma}_{\eta}\right)}^{2}=0.75\times {10}^{6}\times {\left({\sigma}_{\eta}\right)}^{2}$.
- Back-projection to the image: $3\left|I\right|{\left(\frac{\rho}{\u2206\rho}\right)}^{2}\approx 6\times {10}^{6}\times {\left(\frac{\rho}{\u2206\rho}\right)}^{2}$.
- Gaussian smoothing of spatial nonuniformity: $2\left|I\right|{\left({\sigma}_{s,cum}\right)}^{2}\approx 4\times {10}^{6}\times {\left({\sigma}_{s,cum}\right)}^{2}$.
- MDL smoothing of spatial nonuniformity: $2\left|I\right|{3}^{3}{K}_{cum}\approx 108\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}{\times \phantom{\rule{3.33333pt}{0ex}}K}_{cum}$, where ${3}^{3}$ is the connectedness in MDL smoothing.
- Multiplication of incremental spatial nonuniformity with cumulative spatial nonuniformity: $\left|I\right|\approx 2\times {10}^{6}$.

#### 4.3. Experiments and Validation with Whole-Body Images

**Figure 2.**First example of joint restoration of ${T}_{1}$w TSE and a ${T}_{1}$w + ${T}_{2}$w STIR. The cumulative restoration fields account for the spatial variations of the coil sensitivities, their different overall sensitivities, and the junctions between them. (

**a**) Initial ${T}_{1}$w image TSE; (

**b**) Cumulative restoration field for TSE; (

**c**) Restored ${T}_{1}$w image TSE; (

**d**) Initial ${T}_{1}$w + ${T}_{2}$w STIR image; (

**e**) Cumulative restoration field for STIR; (

**f**) Restored ${T}_{1}$w + ${T}_{2}$w STIR image.

**Figure 3.**Co-Occurrence statistics of original and restored images of the first example in Figure 2. The restored statistics are sharper. The different distribution of the tissues are better shown in the restored joint-co-occurrence statistics in (

**b**). (

**a**) Original joint-co-occurrence statistics; (

**b**) Restored joint-co-occurrence statistics; (

**c**) Original ${T}_{1}$w TSE co-occurrence statistics; (

**d**) Restored ${T}_{1}$w TSE co-occurrence statistics; (

**e**) Original ${T}_{2}$w STIR co-occurrence statistics; (

**f**) Restored ${T}_{2}$w STIR co-occurrence statistics.

**Figure 4.**Second example of joint restoration of ${T}_{1}$w TSE and a ${T}_{1}$w + ${T}_{2}$w STIR. The cumulative restoration fields account for the spatial variations of the coil sensitivities, their different overall sensitivities, and the junctions between them. (

**a**) Initial ${T}_{1}$w image TSE; (

**b**) Cumulative restoration field for TSE; (

**c**) Restored ${T}_{1}$w image TSE; (

**d**) Initial ${T}_{1}$w + ${T}_{2}$w STIR image; (

**e**) Cumulative restoration field for STIR; (

**f**) Restored ${T}_{1}$w + ${T}_{2}$w STIR image.

**Figure 5.**Co-Occurrence statistics of original and restored images of the second example in Figure 4. The restored joint-co-occurrence statistics better show the distributions of the different tissues. The tissues distributions are shown improved in the restored co-occurrence statistics as well. (

**a**) Original joint-co-occurrence statistics; (

**b**) Restored joint-co-occurrence statistics; (

**c**) Original ${T}_{1}$w TSE co-occurrence statistics; (

**d**) Restored ${T}_{1}$w TSE co-occurrence statistics; (

**e**) Original ${T}_{2}$w STIR co-occurrence statistics; (

**f**) Restored ${T}_{2}$w STIR co-occurrence statistics.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PSF | Point Spread Function |

RF | Radio-Frequency |

IRB | Institutional Review Board |

${T}_{1}$w | ${T}_{1}$-weighted |

${T}_{2}$w | ${T}_{2}$-weighted |

WB-MRI | Whole-Body Magnetic Resonance Imaging |

RHS | Right Hand Side |

ICH GCP | International Conference on Harmonization—Good Clinical Practice |

TSE | Turbo Spin Echo |

FFE | Fast Field Echo |

STIR | Short Tau Inversion Recovery |

y.o. | Years old |

MDL | Minimum Description Length |

MAP | Maximum A Posteriori |

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**Figure 1.**Overview of the non-parametric Bayesian coring derivation of the posterior conditional expectation. The conditional expectation is further expanded with Bayes’ rule. The expressions for the prior and the likelihood are then substituted. It provides the intermediate vector field $\widehat{u}=E\left(u\right|v,x)$ for image restoration.

Parameters ∖ | Sequence, Static Field | ${\mathit{T}}_{1}$w TSE 1.5 T | STIR 1.5 T | ${\mathit{T}}_{1}$w TSE 3.0 T | STIR 3.0 T |
---|---|---|---|---|---|

Number of slices | 66 | 66 | 60 | 60 | |

Voxel size mm ^{2} | $1.17\times 1.17$ | $0.95\times 0.95$ | $1.02\times 1.02$ | $1.04\times 1.04$ | |

Slice hickness (mm) | 3 | 3 | 3 | 3 | |

Spacing (slice gap) (mm) | 0.3 | 0.3 | 1 | 1 | |

Matrix size | $215\times 188$ | $226\times 152$ | $233\times 180$ | $151\times 148$ |

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

−0.46 | 0.23 | −0.52 | −0.89 | −0.02 |

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**MDPI and ACS Style**

Hadjidemetriou, S.; Malich, A.; Rossknecht, L.D.; Ferrarini, L.; Papageorgiou, I.E.
Restoration for Intensity Nonuniformities with Discontinuities in Whole-Body MRI. *Signals* **2023**, *4*, 725-745.
https://doi.org/10.3390/signals4040040

**AMA Style**

Hadjidemetriou S, Malich A, Rossknecht LD, Ferrarini L, Papageorgiou IE.
Restoration for Intensity Nonuniformities with Discontinuities in Whole-Body MRI. *Signals*. 2023; 4(4):725-745.
https://doi.org/10.3390/signals4040040

**Chicago/Turabian Style**

Hadjidemetriou, Stathis, Ansgar Malich, Lorenz Damian Rossknecht, Luca Ferrarini, and Ismini E. Papageorgiou.
2023. "Restoration for Intensity Nonuniformities with Discontinuities in Whole-Body MRI" *Signals* 4, no. 4: 725-745.
https://doi.org/10.3390/signals4040040