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Variational Bayesian Approach in Model-Based Iterative Reconstruction for 3D X-Ray Computed Tomography with Gauss-Markov-Potts Prior^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Variational Bayesian Approach

**g**are called the projections and are connected to volume

**f**, of size N, by the linear forward model taking uncertainties into account [16]

**H**is called the projection operator. Its adjoint ${\mathit{H}}^{T}$ is the backprojection operator [14]. Since both the data and the volume are huge, matrix

**H**, which is size $M\times N$, is not storable in memory. Consequently, successive projections and backprojections in MBIR methods are computed on-the-fly [14,15]. Uncertainties $\mathit{\zeta}$ are zero-mean Gaussian [16]

## 3. Computation of diagonal coefficients

## 4. Results

## 5. Conclusions and Perspectives

## Funding

## References

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**Figure 1.**Diagonal coefficients of ${\mathit{H}}^{T}\mathit{H}$ (

**a**) and $\mathit{H}{\mathit{H}}^{T}$ (

**b**).

Parameters | K | ${\mathit{\gamma}}_{0}$ | ${\mathit{v}}_{0}$ | ${\mathit{\alpha}}_{{\mathit{\zeta}}_{0}}$ | ${\mathit{\beta}}_{{\mathit{\zeta}}_{0}}$ | ${\mathit{\alpha}}_{0}$ | ${\mathit{\beta}}_{0}$ |

Values | 5 | 6 | 1 | ${10}^{-4}$ | ${10}^{-2}$ | ${10}^{-6}$ | ${10}^{-2}$ |

Algorithm | ${\mathcal{L}}_{2}$-Relative Error | Computation Time |
---|---|---|

PDFW | 6.0 % | 126.3 s |

JMAP | 9.1 % | 751.6 s |

VBA | 13.5 % | 150.0 s |

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**MDPI and ACS Style**

Chapdelaine, C.; Mohammad-Djafari, A.; Gac, N.; Parra, E.
Variational Bayesian Approach in Model-Based Iterative Reconstruction for 3D X-Ray Computed Tomography with Gauss-Markov-Potts Prior. *Proceedings* **2019**, *33*, 4.
https://doi.org/10.3390/proceedings2019033004

**AMA Style**

Chapdelaine C, Mohammad-Djafari A, Gac N, Parra E.
Variational Bayesian Approach in Model-Based Iterative Reconstruction for 3D X-Ray Computed Tomography with Gauss-Markov-Potts Prior. *Proceedings*. 2019; 33(1):4.
https://doi.org/10.3390/proceedings2019033004

**Chicago/Turabian Style**

Chapdelaine, Camille, Ali Mohammad-Djafari, Nicolas Gac, and Estelle Parra.
2019. "Variational Bayesian Approach in Model-Based Iterative Reconstruction for 3D X-Ray Computed Tomography with Gauss-Markov-Potts Prior" *Proceedings* 33, no. 1: 4.
https://doi.org/10.3390/proceedings2019033004