# A Genetic Algorithm for Three-Dimensional Discrete Tomography

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Individuals

**e**.

#### 2.2. Projections

**e**.

#### 2.3. Fitness Function

#### 2.4. Crossover

**e**.

#### 2.5. Mutation

#### 2.6. Crossbreeding

#### 2.7. Cloning

## 3. Results

**e**–

**h**.

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Some of the objects we considered for our experiments (

**a**–

**h**). For a better visualization, they are presented here by small dots. The lower objects (

**e**–

**h**) exhibit hollow spaces and therefore are shown also by means of a couple of their sections (

**e’**–

**h’**). For example, the sphere (

**e**) contains six smaller empty spheres (

**e’**).

**Figure 4.**The twelve directions of the considered projections. The direction along the tunnel indicated by the red dashed line is missing to allow the body move along the tunnel of our mock-up equipment. All arrows point toward the center of the cube.

**Figure 7.**Given two parents (

**a**,

**b**), the offspring (

**c**,

**d**) are obtained by a random cut along one of the twelve directions, which is also randomly selected (${p}_{5}$ in this example). It is usually needed to add or remove some points from the offspring (

**e**,

**f**) to keep their number. These points are highlighted in (

**g**) (which shows the difference between (

**c**,

**e**) and (

**h**) (which shows the difference between (

**d**) and (

**h**)). For interpretation of the colors, the reader is referred to the electronic version of this article.

**Figure 8.**From top to bottom, binary individual vectors of the individuals in Figure 7

**a**,

**b**,

**e**,

**f**. For interpretation of the colors, the reader is referred to the electronic version of this article.

**Figure 9.**The object (

**a**) in Figure 7

**a**and one of its possible random mutations (

**b**). Differences between (

**a**) and (

**b**) are highlighted in (

**c**).

**Figure 10.**To randomly permute the $nc$ individuals in $nd$ demes, we apply the permutation made of $nd$ offset random permutations with length $nc/nd$. Here, we consider $nc=16$ and $nd=4$.

**Figure 11.**Average fitness (red) of the best individual and average standard deviation of the fitness (blue) among the five experiments per object in Figure 2, by using the preset parameters. The genetic diversity, highlighted by the increasing standard deviation, contributed to the convergence towards the correct solutions. As in the case (

**e**), a greater number of generations indicates more complexity in the object to reconstruct. For interpretation of the colors, the reader is referred to the electronic version of this article.

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**MDPI and ACS Style**

Toscano, E.; Valenti, C.
A Genetic Algorithm for Three-Dimensional Discrete Tomography. *Symmetry* **2024**, *16*, 923.
https://doi.org/10.3390/sym16070923

**AMA Style**

Toscano E, Valenti C.
A Genetic Algorithm for Three-Dimensional Discrete Tomography. *Symmetry*. 2024; 16(7):923.
https://doi.org/10.3390/sym16070923

**Chicago/Turabian Style**

Toscano, Elena, and Cesare Valenti.
2024. "A Genetic Algorithm for Three-Dimensional Discrete Tomography" *Symmetry* 16, no. 7: 923.
https://doi.org/10.3390/sym16070923