Norm-preserving constraint in the Fisher–Rao registration and its application in signal estimation

Abstract

Registration of temporal observations is a fundamental problem in functional data analysis. Various frameworks have been developed over the past two decades where registrations are conducted based on optimal time warping between functions. Comparison of functions solely based on time warping, however, may have limited application, in particular when certain constraints are desired in the registration. In this paper, we study registration with norm-preserving constraint. A closely related problem is on signal estimation, where the goal is to estimate the ground-truth template given random observations with both compositional and additive noises. We propose to adopt the Fisher–Rao framework to compute the underlying template, and mathematically prove that such framework leads to a consistent estimator. We then illustrate the constrained Fisher–Rao registration using simulations as well as two real data sets. It is found that the constrained method is robust with respect to additive noise and has superior alignment and classification performance to conventional, unconstrained registration methods.

Keywords:

• registration
• alignment with constraint
• functional data analysis
• time warping
• signal estimation

AMS Subject Classifications:

• 62G05
• 62G20

Acknowledgements

The authors would like to thank the Associate Editor and two reviewers for their comments and suggestions that significantly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by NSF grant DMS 1208959.