Challenges in Reconstructing Shapes from Euler Characteristic Curves

Shape recognition and classification is a problem with a wide variety of applications. Several recent works have demonstrated that topological descriptors can be used as summaries of shapes and utilized to compute distances. In this abstract, we explore the use of a finite number of Euler Characteristic Curves (ECC) to reconstruct plane graphs. We highlight difficulties that occur when attempting to adopt approaches for reconstruction with persistence diagrams to reconstruction with ECCs. Furthermore, we highlight specific arrangements of vertices that create problems for reconstruction and present several observations about how they affect the ECC-based reconstruction. Finally, we show that plane graphs without degree two vertices can be reconstructed using a finite number of ECCs.