For two embeddings of the same finite simplicial complex, the integrated distance between their Euler Characteristic Transforms is bounded by a constant times the total vertex displacement, and SELECT obeys an analogous bound for simplex-wise monotone fields.
Persistent homology—a survey.Discrete & Com- putational Geometry - DCG, 453, 01 2008
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On the Stability of the Euler Characteristic Transform for a Perturbed Embedding
For two embeddings of the same finite simplicial complex, the integrated distance between their Euler Characteristic Transforms is bounded by a constant times the total vertex displacement, and SELECT obeys an analogous bound for simplex-wise monotone fields.