# Human Body Shapes Anomaly Detection and Classification Using Persistent Homology

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## Abstract

**:**

## 1. Introduction

- Extract information from human bodies with interpretation in terms of human anatomy;
- Detect scans anomalies;
- Identify and separate human point clouds by gender;
- Classify male and female morphotypes.

## 2. Methodology

#### 2.1. Dataset

`Spring0001`to

`Spring4800`, and for convenience we refer to

`SpringXXXX`by

`SXXXX`.

#### 2.2. Persistence Diagrams, Landscapes, Silhouettes and Distances

- ${H}_{0}$: The connected components;
- ${H}_{1}$: The non-homotopic loops;
- ${H}_{2}$: The two-dimensional voids.

`S0013`(Figure 1) are given.

#### 2.3. Interpretation of Persistent Homology

- Dimension 0: All ${H}_{0}$ homologies are born when the radius of the balls is zero. For each homology ${H}_{0}$, we choose to display the second point of the pair covered at the birth of the homology as its representative.
- Dimension 1: First, we make an undirected graph containing all the points of a set, where each time a pair of points is covered, as the radius of the balls increases, we connect these points by an edge with a weight equal to the radius of the balls. At the birth of a homology ${H}_{1}$, before adding the edge to our graph, we compute the shortest path connecting these two points, which we display by closing it with the segment connecting these points. The lace displayed is a likely representative of this homology. At the death of this homology, we recover the information of the triangle covered by the balls, and we add it to the display to give a general idea of the evolution of our homology.
- Dimension 2: For each homology ${H}_{2}$, we simply display the triangle covered at its birth and the tetrahedron covered at its death.

`S0013`given in Figure 2, there are 13 different homologies numbered in the persistence barcode from 0 to 12. With this approach, we display each homology in Figure 5 and we can interpret them as follows:

- n°0: ${H}_{2}$ corresponding to the left part of the torso,
- n°1: ${H}_{2}$ corresponding to the right part of the torso,
- n°2: ${H}_{1}$ corresponding to a loop between legs at foot level,
- n°3: ${H}_{1}$ corresponding to a loop between legs from ankles to calves,
- n°4: ${H}_{1}$ corresponding to a loop between legs from knees to calves,
- n°5: ${H}_{2}$ corresponding to the head,
- n°6: ${H}_{2}$ corresponding to the right calf,
- n°7: ${H}_{2}$ corresponding to the left calf,
- n°8: ${H}_{2}$ corresponding to the right foot,
- n°9: ${H}_{2}$ corresponding to the whole body,
- n°10: ${H}_{1}$ corresponding to a loop around the right foot,
- n°11: ${H}_{1}$ corresponding to a loop around the left foot,
- n°12: ${H}_{0}$ of all the connected balls.

#### 2.4. Normalization of Point Clouds by Homothety

## 3. Anomaly Detection

`S2962`.

## 4. Gender Discrimination Index

#### 4.1. Evolution of the GDI Score as a Function of the Number of Clusters

#### 4.2. Restriction to Trunks

## 5. Human Body Shapes Classification

#### 5.1. Male Morphotypes

#### 5.2. Female Morphotypes

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Visual explanation of persistence landscapes. The persistence diagram (

**left**) is tilted so that the diagonal becomes the new horizontal axis (

**top right**). The ${\lambda}_{i}$ are the piecewise linear functions (

**bottom right**).

**Figure 6.**Individuals (

**a**–

**c**) are 1.89 m, 1.93 m and 1.65 m tall, respectively. Among them, the couple (

**a**,

**b**) is the closest before normalization, and the couple (

**b**,

**c**) is the closest after normalization.

**Figure 7.**Dendrogram associated to a complete-linkage hierarchical clustering of the persistence diagrams of male point clouds with the Wasserstein distance. Clusters composed of one individual are presented without parentheses.

**Figure 8.**Anomalies of men scans detected by complete-linkage hierarchical clustering of persistence diagrams.

**Figure 9.**Persistence diagram and barcode of the anomaly of scan

`S2962`. Three particular homologies reflecting the anomaly are highlighted.

**Figure 11.**Dendrogram associated to a complete-linkage hierarchical clustering of the persistence diagrams of female point clouds with the Wasserstein distance.

**Figure 12.**Anomalies of female scans detected by complete-linkage hierarchical clustering of persistence diagrams.

**Figure 13.**GDI score evolution of various clustering algorithms on the persistence diagrams with Wasserstein distance.

**Figure 16.**Comparison of GDI score on the persistence diagrams of the whole body and the trunk with the Wasserstein distance.

**Figure 18.**Dendrogram associated to a Ward-linkage hierarchical clustering of the silhouettes of the persistence diagrams of male point clouds.

**Figure 21.**Dendrogram associated to a Ward-linkage hierarchical clustering of the silhouettes of the persistence diagrams of female point clouds.

**Table 1.**Average GDI scores on the persistence diagrams of the whole body and the trunk with the Wasserstein distance.

Complete | Ward | K-Medoids | |
---|---|---|---|

Body | 0.2 | 0.54 | 0.526 |

Trunk | 0.21 | 0.553 | 0.582 |

Ward | K-Means | K-Medoids | |
---|---|---|---|

Body | 0.738 | 0.73 | 0.737 |

Trunk | 0.765 | 0.767 | 0.827 |

Cluster | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | ${\mathit{C}}_{6}$ | ${\mathit{C}}_{7}$ | ${\mathit{C}}_{8}$ |
---|---|---|---|---|---|---|---|---|

Size | 2 | 4 | 50 | 311 | 125 | 273 | 415 | 332 |

Proportion (in percent) | 0.1 | 0.3 | 3 | 21 | 8 | 18 | 27 | 22 |

Mean distance | 79.6 | 39.4 | 28.6 | 17.4 | 20.3 | 16.3 | 18.9 | 19.4 |

Diameter | 79.6 | 53.9 | 62 | 49.6 | 59.2 | 54.6 | 67.1 | 69.4 |

Distance to the mean | 39.8 | 24.4 | 19.6 | 12.1 | 14.2 | 11.5 | 13.1 | 13.6 |

Distance mean–medoid | 39.8 | 20.1 | 9 | 3.4 | 5.9 | 6.9 | 5.1 | 6.1 |

Clusters | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | ${\mathit{C}}_{6}$ | ${\mathit{C}}_{7}$ |
---|---|---|---|---|---|---|---|

Size | 306 | 263 | 403 | 107 | 122 | 112 | 214 |

Proportion (in percent) | 20 | 17 | 27 | 7 | 8 | 7 | 14 |

Mean distance | 14.2 | 12.7 | 14.1 | 14.3 | 14 | 19 | 21.3 |

Diameter | 36.6 | 32 | 32.1 | 31.9 | 38.2 | 59.3 | 62.4 |

Distance to the mean | 10.1 | 9 | 10.1 | 10.1 | 9.9 | 13.3 | 14.8 |

Distance mean–medoid | 3.4 | 3.4 | 4.5 | 3.3 | 3.6 | 5.9 | 5.2 |

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**MDPI and ACS Style**

de Rose, S.; Meyer, P.; Bertrand, F.
Human Body Shapes Anomaly Detection and Classification Using Persistent Homology. *Algorithms* **2023**, *16*, 161.
https://doi.org/10.3390/a16030161

**AMA Style**

de Rose S, Meyer P, Bertrand F.
Human Body Shapes Anomaly Detection and Classification Using Persistent Homology. *Algorithms*. 2023; 16(3):161.
https://doi.org/10.3390/a16030161

**Chicago/Turabian Style**

de Rose, Steve, Philippe Meyer, and Frédéric Bertrand.
2023. "Human Body Shapes Anomaly Detection and Classification Using Persistent Homology" *Algorithms* 16, no. 3: 161.
https://doi.org/10.3390/a16030161