Representation of Videokeratoscopic Height Data Using a Set of Discrete Tchebichef Orthogonal Polynomials

The continuous orthogonal polynomials, such as Zernike and pseudo-Zernike, are often used as an expansion of corneal height data. However, the use of continuous polynomials has some limitations due to the discretization. It is because that the integrals are usually approximated by discrete summations, and this process not only leads to numerical errors, but also severely affects some analytical properties such as rotation invariance, orthogonality, etc. To overcome these drawbacks, this paper presents a methodology for decomposing corneal height data into discrete orthogonal Tchebichef polynomials. Tchebichef polynomials, which are a product of angular functions and radial Tchebichef polynomials, are orthogonal in the discrete coordinate. Therefore, the approximation error caused by discretization can be avoided, and the analytical property can be well preserved. Examples of modeling corneal elevation are provided for simulation corneas, real normal corneas, and real abnormal corneas. The experimental results show that the proposed discrete Tchebichef polynomials have better surface representation capability than Zernike polynomials or pseudo-Zernike polynomials, and have more robust fitting for the level of noise found in current videokeratoscopes, so that they can be used as an alternative to fit the corneal surface.