# Enhanced Machine Learning Approach for Accurate and Fast Resolution of Inverse Scattering Problem in Breast Cancer Detection

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

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- ${E}^{t}\left(\mathit{r}\right)$ is the total electric field;
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- ${E}^{i}\left(\mathit{r}\right)$ is the incident electric field;
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- ${E}^{s}\left(\mathit{r}\right)$is the scattered field on the measurement surface S;
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- ${k}_{b}=\omega \sqrt{{\epsilon}_{b}{\mu}_{0}}$ is the wavenumber of the background medium;
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- $J\left(\mathit{r}\right)$ is the contrast current density, defined as $J\left(\mathit{r}\right)=\chi \left(\mathit{r}\right){E}^{t}\left(\mathit{r}\right)$, where $\chi \left(\mathit{r}\right)\mathrm{is}$the contrast function containing the body permittivity and is defined as:$$\chi \left(\mathit{r}\right)=\frac{{\epsilon}_{\mathit{r}}\left(\mathit{r}\right)-1}{{\epsilon}_{{r}_{b}}}-j\frac{\left(\sigma \left(\mathit{r}\right)-{\sigma}_{b}\right)}{\omega {\epsilon}_{b}}$$
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- $g\left(\mathit{r},{\mathit{r}}^{\prime}\right)$ is the 2D free-space Green’s function, which is given in terms of Hankel function of second kind [25], namely:$${G}_{2D}\left(\mathit{r},{\mathit{r}}^{\prime}\right)=-\frac{j{k}_{b}^{2}}{4}{H}_{0}^{\left(2\right)}\left({k}_{b}\left|\mathit{r}-{\mathit{r}}^{\prime}\right|\right)$$

**r**), iterative methods should be adopted, especially when dealing with strong scatterers [11].

## 3. Method

#### 3.1. Quadratic Programming Approach to BIM

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- ${E}_{m}^{s}$ is the scattered electric field at position ${\mathit{r}}_{m}$ on the measurement domain S;
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- ${g}_{mn}$ is the discretization of Green’s function, ${a}_{n}=\sqrt{\Delta x\Delta y/\pi ,}$ ${J}_{1}$ is the Bessel function of the first kind, and ${\mathit{r}}_{n}$is the vector position of the n-th pixel;
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- ${\chi}_{n}$ is the contrast value at ${\mathit{r}}_{n}$;
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- ${E}_{n}^{t}$is the total electric field at ${\mathit{r}}_{n}$;
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- $M$ gives the number of receiving antennas.

#### 3.2. Machine Learning Procedure

## 4. Numerical Results

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- ${\epsilon}_{r\left(n\right)}^{t}$is the value of the true relative permittivity corresponding to the n-th pixel;
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- ${\epsilon}_{r\left(n\right)}^{r}$is the value of the reconstructed relative permittivity corresponding to the n-th pixel;
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- ${N}_{p}$ is the total number of pixels.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Relative errors in training, validation, and testing as a function of the number of training epochs.

**Figure 6.**Results for permittivity (upper) and conductivity (lower) using the quadratic programming-based BIM and the CNN method (

**a**) Test 1; (

**b**) Test 2; (

**c**) Test 10; (

**d**) Test 28.

**Table 1.**Sample statistics of the relative error (Equation (9)) to compare the BIM with the quadratic approach and the proposed model.

Relative Error Quadratic BIM (%) | Relative Error Quadratic BIM+CNN (%) | Accuracy Quadratic BIM+CNN (%) | |
---|---|---|---|

minimum | 24.7 | 2 | 97 |

mean | 27 | 2.4 | 97.6 |

median | 26.9 | 2.4 | 97.6 |

maximum | 30.7 | 3 | 98 |

**Table 2.**Sample statistics of the relative error (Frobenius norm) to compare the BIM with quadratic approach and the proposed model.

Relative Error Quadratic BIM [%] | Relative Error Quadratic BIM+CNN [%] | Accuracy Quadratic BIM+CNN [%] | |
---|---|---|---|

minimum | 63.4 | 4.6 | 91.2 |

mean | 68.1 | 6.2 | 93.7 |

median | 68.1 | 6 | 94 |

maximum | 72 | 8.7 | 95.4 |

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**MDPI and ACS Style**

Costanzo, S.; Flores, A.
Enhanced Machine Learning Approach for Accurate and Fast Resolution of Inverse Scattering Problem in Breast Cancer Detection. *Electronics* **2022**, *11*, 2308.
https://doi.org/10.3390/electronics11152308

**AMA Style**

Costanzo S, Flores A.
Enhanced Machine Learning Approach for Accurate and Fast Resolution of Inverse Scattering Problem in Breast Cancer Detection. *Electronics*. 2022; 11(15):2308.
https://doi.org/10.3390/electronics11152308

**Chicago/Turabian Style**

Costanzo, Sandra, and Alexandra Flores.
2022. "Enhanced Machine Learning Approach for Accurate and Fast Resolution of Inverse Scattering Problem in Breast Cancer Detection" *Electronics* 11, no. 15: 2308.
https://doi.org/10.3390/electronics11152308