# Human Body Shapes Anomaly Detection and Classification Using Persistent Homology
## 1 Introduction  
The separation of human bodies into groups of morphologies is a common issue for garment industries. Rather than targeting a single standard body shape, the discrimination of morphologies helps to improve sizing systems and can reduce production costs for apparel manufacturing. Among the classifications already established, there is one from [1] particularly used by industries, where the authors obtained nine types of female body shapes such as triangle, inverted triangle, hourglass, oval, etc. This is the first work where mathematical criteria, together with the help of experts, have been used to define these groups. In terms of data science, we can approach this problem by clustering algorithms. To this end, different types of data can be extracted from a body such as measurements or anthropometric point clouds. Body measurements can be directly represented in a Euclidean space to use methods from data analysis. In [2], principal component and K-means cluster analyses are performed on measurements and key body locations, and three female lower body shape groups are obtained. This representation in a vector space to perform the clustering is straightforward but has disadvantages. For example, it is not clear that it is appropriate to compare with Euclidean metric measurements of different types such as body lengths, circumferences or individual weight. On the other hand,this requires the choice of the set of measurements extracted from the body, and key morphological characteristics may be omitted. The use of 3D representations of the bodies is suitable for these issues but becomes difficult to implement since we need a way to extract information from anthropometric point clouds and compare them. For example, in [3], the authors use control points and correlation strength principal component analysis of trunks. Reinterpreting these components by averaged shape figures and combining factor loading maps, five female trunk shape groups are defined by a Ward-linkage hierarchical clustering. Different methods from data science have been used to classify human body shapes; see for example [4; 5; 6; 7]. Topological data analysis [8; 9] is a powerful tool to study and understand the shape of data, and thus it naturally applies in this context. In particular, persistent homology [10; 11] can be used to extract relevant topological information from data and point clouds. These extracted features are encoded by diagrams and have stability properties relative to specific distances [12; 13]. Several applications of this theory have been established in different contexts such as time-series data analysis [14], object recognition [15], complex network analysis [16], molecular biology data exploration [17], biomedicine [18], geographical information science [19] and environmental science [20]. Feature extraction for classification is an active research topic in pattern recognition and machine learning; see for example [21; 22] or [23]. In this work, we use persistent theory applied on human point clouds in order to perform the following:  
* 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. More precisely, we compute the persistence diagrams, Wasserstein distance and associated silhouettes on the human point clouds of the CAESAR database [24]. Using graph theory, among other things, approaches by homological degree allow us to interpret persistent homologies and identify them to body areas and limbs. To define morphotypes independently of individuals' height, we normalize the point clouds using three-dimensional homotheties. Then, we show that anomalies of scans are naturally isolated clusters when performing complete-linkage hierarchical clustering on the persistence diagrams of the point clouds using the Wasserstein distance.