Journal
JOURNAL OF TOPOLOGY AND ANALYSIS
Volume 4, Issue 1, Pages 49-70Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S1793525312500057
Keywords
Persistent homology; barcodes; Betti numbers; Euler characteristic; random fields; Gaussian processes; Gaussian kinematic formula
Categories
Funding
- Israel Academy of Sciences and Humanities
- US-Israel Binational Science Foundation [2008262]
- National Science Foundation [DMS-0852227]
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In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.
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