# A Model-Based Optimization Framework for Iterative Digital Breast Tomosynthesis Image Reconstruction

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## Abstract

**:**

## 1. Introduction

#### 1.1. A Short Review on Iterative Methods for CT

#### 1.2. Aim and Contribution of the Paper

- We propose three different models inside a unique optimization framework, combining a data-fitting function (identified in the least squares function) and a regularization TV-like term. We solve the minimization problem with efficient iterative algorithms, converging to the global minimum of the problem, and we analyze and compare the results achieved from real noisy and large-size data sets. Two of these models have been presented in our previous works on low-sampled CT image reconstruction [26,27], but they were applied to small simulated data sets with different geometries (not limited angles). This optimization approach is new for DBT applications.
- We propose a user independent and computationally effortless rule to set and adapt the regularization parameter at each iteration of the algorithms.
- In order to assess our proposals, we implement the methods and test them on real large-size and noisy projection data from both a breast accreditation phantom and a human patient. We analyze the algorithms performance in recovering the breast tumor objects of interest (i.e., masses and microcalcifications clusters), by means of measures of merits and visual inspection, at different stages of the Iterative Reconstruction process. We analyze the volume via its recovered slices, both perpendicularly and along the Z direction (see Figure 1).

## 2. The Optimization Framework in a Model-Based Formulation

## 3. Iterative Optimization Methods

- a fast error decreasing in the initial algorithm execution, in order to obtain a good image in few iterations;
- a low computational cost per iteration (which is mainly determined by the number of matrix-vector products), to efficiently run the solver in short time;
- a limited request of memory, to solve real-size problems on commercially affordable hardware.

#### 3.1. Scaled Gradient Projection Algorithm

Algorithm 1 Scaled Gradient Projection algorithm (SGP) |

Input:$M,b,\lambda $- 1:
**Initialize:**${x}^{\left(0\right)}\ge 0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\gamma ,\sigma \in (0,1),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0<{\alpha}_{min}\le {\alpha}_{max},$- 2:
- k = 0
- 3:
**while**not convergence**do**- 4:
- Compute ${g}^{\left(k\right)}=2({M}^{T}M{x}^{\left(k\right)}-{M}^{T}b)+\lambda \nabla T{V}_{\beta}\left({x}^{\left(k\right)}\right)$
- 5:
- Compute ${S}_{k}\in {S}_{{\rho}_{k}}$ as in (14)
- 6:
- Define ${\alpha}_{k}\in [{\alpha}_{min},{\alpha}_{max}]$ with alternate BB rules
- 7:
- ${d}^{\left(k\right)}={\mathcal{P}}_{+}\left(\right)open="("\; close=")">{x}^{\left(k\right)}-{\alpha}_{k}{S}_{k}{g}^{\left(k\right)}$
- 8:
- ${\eta}_{k}=1$
- 9:
**while**$f({x}^{\left(k\right)}+{\eta}_{k}{d}^{\left(k\right)})>f\left({x}^{\left(k\right)}\right)+\sigma {\eta}_{k}{\left({g}^{\left(k\right)}\right)}^{T}{d}^{\left(k\right)}$**do**- 10:
- ${\eta}_{k}=\gamma {\eta}_{k}$
- 11:
- ${x}^{(k+1)}={x}^{\left(k\right)}+{\eta}_{k}{d}^{\left(k\right)}$
- 12:
- k = k+1
Output:${x}^{\left(k\right)}$ |

#### 3.2. The Fixed Point Algorithm

Algorithm 2 Lagged diffusivity Fixed Point algorithm (FP) |

Input:$M,b,\lambda ,maxiter$- 1:
**Initialize:**${x}^{\left(0\right)}\ge 0$- 2:
**for**$k=0$ to $maxiter-1$**do**- 3:
- Compute ${g}^{\left(k\right)}=2({M}^{T}M{x}^{\left(k\right)}-{M}^{T}b)+\lambda \nabla T{V}_{\beta}\left({x}^{\left(k\right)}\right)$
- 4:
- Solve the linear system ${H}_{k}{d}^{\left(k\right)}=-{g}^{\left(k\right)}$, where ${H}_{k}={M}^{T}M+\lambda L\left({x}^{\left(k\right)}\right)$, with the Conjugate Gradient method.
- 5:
- ${x}^{(k+1)}={x}^{\left(k\right)}+{d}^{\left(k\right)}$
Output:${\mathcal{P}}_{+}\left({x}^{(k+1)}\right)$ |

#### 3.3. The Chambolle–Pock Algorithm

- compute ${y}^{(k+1)}$ as $pro{x}_{\sigma}\left[{F}^{*}\right]({y}^{\left(k\right)}+\sigma K{\overline{x}}^{\left(k\right)})$;
- compute ${x}^{(k+1)}$ as $pro{x}_{\tau}\left[G\right]({x}^{\left(k\right)}-\tau {K}^{T}{y}^{(k+1)})$;
- define ${\overline{x}}^{(k+1)}$ with an extrapolation step: ${\overline{x}}^{(k+1)}={x}^{(k+1)}+\theta ({x}^{(k+1)}-{x}^{\left(k\right)})$ and $\theta >0$.

Algorithm 3 Chambolle–Pock algorithm (CP) |

Input:: $M,b,\u03f5,maxiter$- 1:
**Compute:**$\mathsf{\Gamma}$ as an approximation of ${\parallel K\parallel}_{2}$- 2:
**Initialize:**$\tau =\sigma =\frac{1}{\mathsf{\Gamma}}>0$, $\theta \in [0,1]$- 3:
**Initialize:**${x}^{\left(0\right)}\in {\mathbb{R}}^{{N}_{v}}$$({x}^{\left(0\right)}\ge 0)$, ${\overline{x}}^{\left(0\right)}\in {\mathbb{R}}^{{N}_{v}},{y}^{\left(0\right)}\in {\mathbb{R}}^{{N}_{d}}$ and ${w}^{\left(0\right)}\in {\mathbb{R}}^{3\xb7{N}_{v}}$ to zeros-vectors- 4:
**for**$k=0$ to $maxiter-1$**do**- 5:
- ${\overline{y}}^{\left(k\right)}={y}^{\left(k\right)}+\sigma (M{\overline{x}}^{\left(k\right)}-b)$
- 6:
- ${y}^{(k+1)}=max(\parallel {\overline{y}}^{\left(k\right)}{\parallel}_{2}-\sigma \u03f5,0)\frac{{\overline{y}}^{\left(k\right)}}{\parallel {\overline{y}}^{\left(k\right)}{\parallel}_{2}}$
- 7:
- ${\overline{w}}^{\left(k\right)}={w}^{\left(k\right)}+\sigma ({\nabla}_{x};{\nabla}_{y};{\nabla}_{z}){\overline{x}}^{\left(k\right)}$
- 8:
- $\overline{\overline{{w}^{\left(k\right)}}}=\left(\right|{\overline{w}}^{\left(k\right)}|,|{\overline{w}}^{\left(k\right)}|,|{\overline{w}}^{\left(k\right)}\left|\right)$
- 9:
- ${w}^{(k+1)}={\overline{w}}^{\left(k\right)}.\ast (\lambda ./max(\lambda ,\overline{\overline{{w}^{\left(k\right)}}})$
- 10:
- ${x}^{(k+1)}={x}^{\left(k\right)}-\tau ({M}^{T}{y}^{(k+1)}+{({\nabla}_{x};{\nabla}_{y};{\nabla}_{z})}^{T}{w}^{(k+1)})$
- 11:
- ${x}^{(k+1)}={\mathcal{P}}_{+}\left({x}^{(k+1)}\right)$
- 12:
- ${\overline{x}}^{(k+1)}={x}^{(k+1)}+\theta ({x}^{(k+1)}-{x}^{\left(k\right)})$
Output:${x}^{(k+1)}$ |

#### 3.4. User-Independent Choice of the Regularization Parameter

- Set ${\lambda}_{0}=0$ to initialize the algorithm and run the first iteration (labeled with k = 0) to compute ${x}^{\left(1\right)}$;
- Set ${\lambda}_{1}=\frac{\sqrt{LS\left({x}^{\left(1\right)}\right)}}{TV\left({x}^{\left(1\right)}\right)}$ and use it to compute ${x}^{\left(2\right)}$;
- For each $k\ge 2$, set$${\lambda}_{k}=\frac{1}{k}{\lambda}_{1}$$

#### 3.5. The Projection Matrix Algorithm

## 4. Materials

#### 4.1. DBT System Configuration

#### 4.2. Data Sets

#### 4.3. Measure and Graphics of Merits

## 5. Numerical Results and Discussion

#### 5.1. Methods Comparison for Early Reconstructions

#### 5.2. SGP Algorithm Insights

#### 5.3. Experiments on a Human Data Set

#### 5.4. Experiments with a Variable Regularization Parameter

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Scheme of the Digital Breast Tomosynthesis (DBT) reconstruction process. On the left, a draft of the frontal (coronal) section of a DBT system acquiring projection images of the breast; in the center, a chart representing the kth iteration of the algorithm computing the sequence ${\left\{{x}^{\left(k\right)}\right\}}_{k}$ of approximate solutions by solving the model-based minimization problem; on the right, the evaluation of reconstructed volumes by inspection of cancer objects of interest.

**Figure 2.**Schematic draw representing the Distance Driven approach to compute the system matrix. (

**a**) View on the $YZ$ plane of an X-ray projection onto a single pixel, from a fixed angle. The intersection of the X-ray beam with the volume is highlighted in magenta. (

**b**) The magenta area represents the backward projection of the blue recording unit onto a volume slice parallel to the $XY$ plane.

**Figure 3.**Reconstructions of microcalcification cluster number 3 in BR3D phantom obtained with Scaled Gradient Projection (SGP), Fixed Point (FP) and Chambolle–Pock (CP) methods. In the upper row, reconstructions in 5 iterations; in the bottom row, reconstructions in 15 iterations.

**Figure 4.**Plots of the Plane Profile on the left and of the Artifact Spread Function (ASF) vectors on the right, taken over one microcalcification of cluster number 3 in BR3D phantom obtained. In all the plots the red line corresponds to SGP method, the blue line to FP method and the green line to CP method.

**Figure 5.**Objective function values vs. the iteration number for the SGP execution on the phantom test. The convergence has been reached after 44 iterations by satisfying condition (29). The red labels outline the function values at 5, 15 and 30 iterations.

**Figure 6.**SGP results on BR3D phantom. (

**a**–

**c**) Reconstructions of MC cluster number 5 obtained after 5, 15 and 30 iterations. (

**d**–

**f**) Reconstructions of mass number 2 obtained after 5, 15 and 30 iterations.

**Figure 7.**SGP results on BR3D phantom. Plots of the Plane Profile on the left and of the ASF vectors on the right, taken over one microcalcification of cluster 5 (

**upper row**) and over mass number 2 (

**bottom row**). In all the plots the black line corresponds to 5 iterations, red line to 15 iterations and blue line to 30 iterations.

**Figure 8.**Results obtained after 5, 15 and 30 SGP iterations on a human breast data set. (

**a**–

**c**) Reconstructions of a 440 × 400 pixels region presenting both a spherical mass (pointed by the arrow) and a microcalficication (identified by the circle). (

**d**,

**e**) Plane profiles on the mass and on the microcalcification. In the plots: black line corresponds to 5 iterations and blue line to 30 iterations.

**Figure 9.**Results obtained after 5, 15 and 30 SGP iterations on a human breast data set. The reported 558 × 480 pixels crops present two speculated masses.

**Figure 10.**Plot of the set of values ${\lambda}_{k}$ versus the iteration k (blue line) in the SGP execution on the phantom test. The red straight line represents the constant value $\lambda =0.005$ used in SGP for the experiments presented in the previous sections.

**Figure 11.**Plane Profiles on one microcalcification of cluster number 3 of the phantom, obtained with SGP with different regularization parameters, in 5 and 15 iterations. In all the plots the red line corresponds to fix parameter and the blue line to the adaptive choice of $\lambda $.

**Table 1.**Diameters of the microcalcifications (MC) and of the masses (MS) in the BR3D phantom as reported in [42]. Measures are in micrometers (m).

1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|

MC | 400 | 290 | 230 | 196 | 165 | 130 |

MS | 6300 | 4700 | 3900 | 3100 | 2300 | 1800 |

5 it. | 15 it. | 30 it. | |
---|---|---|---|

MC cluster 3 | 24.21 | 33.34 | 38.00 |

MC cluster 5 | 10.03 | 19.00 | 28.00 |

MC cluster 6 | 7.27 | 11.02 | 17.00 |

MS 2 | 0.82 | 1.07 | 1.66 |

MS 4 | 0.87 | 1.00 | 1.33 |

FWHM | w (µm) | |||||
---|---|---|---|---|---|---|

5 it. | 15 it. | 30 it. | 5 it. | 15 it. | 30 it. | |

MC cluster 3 | 4.77 | 3.32 | 2.70 | 430 | 299 | 243 |

MC cluster 5 | 3.52 | 2.65 | 2.32 | 317 | 238 | 209 |

MC cluster 6 | - | 2.05 | 1.52 | - | 185 | 137 |

**Table 4.**Valuexs of the of CNR and FWHM indices computed on the mass and the microcalcification observable in Figure 8a–c.

CNR | FWHM | |||||
---|---|---|---|---|---|---|

5 it. | 15 it. | 30 it. | 5 it. | 15 it. | 30 it. | |

MS | 0.239 | 0.381 | 0.558 | - | - | - |

MC | 8.78 | 16.59 | 16.49 | 8.57 | 7.81 | 7.29 |

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## Share and Cite

**MDPI and ACS Style**

Loli Piccolomini, E.; Morotti, E.
A Model-Based Optimization Framework for Iterative Digital Breast Tomosynthesis Image Reconstruction. *J. Imaging* **2021**, *7*, 36.
https://doi.org/10.3390/jimaging7020036

**AMA Style**

Loli Piccolomini E, Morotti E.
A Model-Based Optimization Framework for Iterative Digital Breast Tomosynthesis Image Reconstruction. *Journal of Imaging*. 2021; 7(2):36.
https://doi.org/10.3390/jimaging7020036

**Chicago/Turabian Style**

Loli Piccolomini, Elena, and Elena Morotti.
2021. "A Model-Based Optimization Framework for Iterative Digital Breast Tomosynthesis Image Reconstruction" *Journal of Imaging* 7, no. 2: 36.
https://doi.org/10.3390/jimaging7020036