The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that particular sets of PDs can differentiate between different shapes. This trait is desirable because it provides a method of representing complex shapes using finite sets of descriptors. The problem of choosing such a set of representative PDs and then using them to uniquely determine the shape is referred to as reconstruction. In this paper, we present an algorithm for reconstructing embedded graphs in R^d (plane graphs in R^2) with n vertices from n^2 - n + d + 1 directional PDs. Lastly , we empirically validate the correctness and time-complexity of our algorithm in R^2 on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm.
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes using a finite set of descriptors. A related inverse problem is the following: given a set of PDs (or an oracle we can query for persistence diagrams), what is underlying geometric simplicial complex? In this paper, we present an algorithm for reconstructing embedded graphs in R^d (plane graphs in R^2) with n vertices from n^2 - n + d + 1 directional (augmented) PDs. Additionally , we empirically validate the correctness and time-complexity of our algorithm in R^2 on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm.
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different shapes. This trait is desirable because it provides a method of
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representative PDs and then using them to uniquely determine the shape is referred to as reconstruction.
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(augmented) PDs. Additionally