Local, smooth, and consistent Jacobi set simplification
The relation between two Morse functions defined on a smooth, compact, and orientable 2-manifold can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the two functions are aligned. Both the Jacobi set itself as well as the segmentation of the domain it induces, have shown to be useful in various applications. In practice, unfortunately, functions often contain noise and discretization artifacts, causing their Jacobi set to become unmanageably large and complex. Although there exist techniques to simplify Jacobi sets, they are unsuitable for most applications as they lack fine-grained control over the process, and heavily restrict the type of simplifications possible. In this paper, we introduce a new framework that generalizes critical point cancellations in scalar functions to Jacobi set in two dimensions. We present a new interpretation of Jacobi set simplification based on the perspective of domain segmentation. Generalizing the cancellation of critical points from scalar functions to Jacobi sets, we focus on simplifications that can be realized by smooth approximations of the corresponding functions, and show how these cancellations imply simultaneous simplification of contiguous subsets of the Jacobi set. Using these extended cancellations as atomic operations, we introduce an algorithm to successively cancel subsets of the Jacobi set with minimal modifications to some user-defined metric. We show that for simply connected domains, our algorithm reduces a given Jacobi set to its minimal configuration, that is, one with no birth–death points (a birth–death point is a specific type of singularity within the Jacobi set where the level sets of the two functions and the Jacobi set have a common normal direction).
- Research Organization:
- Univ. of Utah, Salt Lake City, UT (United States)
- Sponsoring Organization:
- USDOE National Nuclear Security Administration (NNSA)
- Grant/Contract Number:
- AC52-07NA27344; NA0002375
- OSTI ID:
- 1556399
- Alternate ID(s):
- OSTI ID: 1221807; OSTI ID: 1360637
- Report Number(s):
- DOE-UTAH-PASCUCCI-0011; S0925772114001163; PII: S0925772114001163
- Journal Information:
- Computational Geometry, Journal Name: Computational Geometry Vol. 48 Journal Issue: 4; ISSN 0925-7721
- Publisher:
- ElsevierCopyright Statement
- Country of Publication:
- Netherlands
- Language:
- English
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