Regularization of an ill-posed problem in corneal topography

Abstract

In this paper, we use Tikhonov regularization and spectrum cutoff methods to regularize a solution of the inverse problem in a partial differential equation describing the corneal geometry. This solution is closely connected with some intrinsic parameters of cornea like elasticity and intra-ocular pressure which are important in applications. We obtain a stable solution which can give some new insight into biomechanical structure of cornea. Also, we provide an order of convergence estimates and a numerical comparison with the real corneal data. This method gives reasonable results with an error of order of a few per cent.

Keywords:

• corneal topography
• mathematical model
• partial differential equation
• Tikhonov regularization
• ill-posed problem

AMS Subject Classifications:

• 35R30
• 92C99

Acknowledgments

The authors would like to thank Dr Robert Iskander from Institute of Biomedical Engineering and Instrumentation, Wroclaw University of Technology, Poland and School of Optometry, Queensland University of Technology, Australia for access to the data. This research was supported by the Polish Government funds for science: research project NCN 2012/05/N/ST1/02860.