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Orienteering and Coverage
Published in Yasmina Bestaoui Sebbane, Multi-UAV Planning and Task Allocation, 2020
The Reeb graph can be used as input to the CPP to calculate a Eulerian circuit, which consists of a closed path traversing every cell at least once. The Reeb graph is a construction that originated in Morse theory to study a real-valued function defined on a topological space. The structure of a Morse function can be made explicit by plotting the evolution of the component of the level set. The Reeb graph is a fundamental data structure that encodes the topology of a shape. It is obtained by contracting to a point the connected components of the level-sets (also called contours) of a function defined on a mesh. Reeb graphs can determine whether a surface has been reconstructed correctly, can indicate problem areas and can be used to encode and animate a model. The Reeb graph has been used in various applications to study noisy data, which creates a desire to define a measure of similarity between these structures. A Eulerian circuit can be achieved by doubling selected edges of the Reeb graph, although no edge needs to be duplicated more than once. The Eulerian circuit is the solution of the linear programming problem: Minimizez=∑e∈Ece⋅xe
Orienteering and Coverage
Published in Yasmina Bestaoui Sebbane, Intelligent Autonomy of Uavs, 2018
The Reeb graph can be used as input to the Chinese postperson problem to calculate a Eulerian circuit, which consists of a closed path traversing every cell at least once. The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. The structure of a Morse function can be made explicit by plotting the evolution of the component of the level set. The Reeb graph is a fundamental data structure that encodes the topology of a shape. It is obtained by contracting to a point the connected components of the level-sets (also called contours) of a function defined on a mesh. Reeb graphs can determine whether a surface has been reconstructed correctly, indicate problem areas, and can be used to encode and animate a model. The Reeb graph has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. A Eulerian circuit can be achieved by doubling selected edges of the Reeb graph, although no edge needs to be duplicated more than once. The Eulerian circuit is the solution of the linear programming problem: Minimizez=∑e∈Ece.xe $$ \begin{aligned} \text{ Minimize} z = \sum _{e \in E}{c_e.x_e} \end{aligned} $$
Automated retrieval and comparison of sheet metal parts
Published in International Journal of Computer Integrated Manufacturing, 2023
Yang Yang, Srichand Hinduja, Oladele O Owodunni
Comparing 3D shapes is a challenging task. One of the simplest methods to convert the geometry of an object into a single variable continuous function is a Reeb graph (Reeb 1946), geodesic distance being one such variable (Hilaga et al. 2001). The advantages of the Reeb graph are that it is insensitive to transformations and is quick to compute. However, the Reeb graph fails to capture the finer geometric details. An improvement on this graph is the medial axis which converts the 3D geometrical shape into a central spine or stick-like figure (Sundar et al. 2003; Iyer et al. 2005). Apart from the high computational cost, the shape of the medial axis/surface is dependent on the voxel resolution; furthermore, difficulty may be encountered when matching medial axis diagrams, e.g. a small cylinder is likely to be matched with a bigger object containing cylindrical features (Sundar et al. 2003). Another simple approach for matching 3D shapes is to represent the geometry of a 3D object by a shape function and then use the resulting probability distribution of the function for retrieval. Several functions (e.g. A3, D1, D2, D3 (Osada et al. 2002)) have been suggested, with D2, the distance between any two points on the surface, as the most popular function (Cheng et al. 2011; Huang et al. 2015; Hong, Lee, and Kim 2006; Rea et al. 2005; Li, Zhou, and Liu 2015; Chu, Lo, and Cheng 2017; Li, Zhou, and Liu 2018). With specific sampling numbers, several distances can be obtained by D2 functions; then, these distances are allocated into different intervals, resulting in a shape distribution of the analysed model. Although these shape functions are simplistic in nature and are geometry dependent, it is difficult to detect local differences between two objects.