Smoothing Splines on Ball Domains with Application to Optometry and Ophthalmology
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The identification of objective criteria to correctly diagnose ectatic diseases of the cornea or to detect early stages of corneal ectasia is of great interest in ophthalmology and optometry. A precise understanding of the shape of the cornea is necessary for accurate diagnosis of these ectatic conditions and for designing customized contact lenses with aspherical back surface for those with corneal ectasia. Due to the recent advances in corneal imaging technology cross sectional visualization of the cornea and the anterior chamber is now possible. For example, Anterior Segment Optical coherence tomography (AS-OCT) is a non-invasive imaging technique used to study and understand internal structures of the anterior chamber of the human eye. An important problem in optometry and ophthalmology is the reconstruction of the corneal surface that is not only a good fit of the data but also respects the smoothness properties observed in the images. A key feature of eye shape reconstruction problems is that one has to match the global shape of the eye while faithfully reproducing local shape changes that indicate corneal disease. The data obtained from the devices is subject to measurement error, which we consider to be independent and identical Gaussian distribution. The data points are along lines separated by 45 degrees on a disk domain. The mean squared residual error is a natural fitting criterion in both parametric and nonparametric approaches to the surface reconstruction. The use of parametric models such as a basis of Zernike polynomials is known to lead to the computational problem of ill-posedness when too many basis functions are used or models with high bias with too few bases functions. On the other hand, nonparametric models such as radial basis functions lead to nonlinear estimation problems, or difficult to resolve model selection problems. Our approach reconstructs the global shape and local deformation in two orthogonal subspaces of the Sobolev space W^2,2 (B_γ (0)), where B_γ (0) is the ball of radius γ centered at the origin 0. In this dissertation, a novel smoothing spline method for shape reconstruction from noisy data on ball domains is presented. We cast the problem as the minimization of the mean squared residual error with a penalty on the smoothness of the solution. This approach leads to surfaces that are natural generalization of smoothing splines on the interval to the disk domain. By the Kimeldorf-Wahba theorem, the smoothing spline is the sum of two functions that belong in different orthogonal subspaces of H^1. One of the functions matches the global shape using a basis of harmonic functions, while the other matches the local deformation using a basis of Green's functions for the biharmonic operator that are adapted to the data. Thus, our method unifies the ad-hoc approaches presented in literature as the global-local decomposition is the solution to an optimization problem. We also present detailed statistical comparison of the reconstructed surface obtained using our and other published methods. We present an application of our method of corneal shape reconstruction to the accurate diagnosis of ectatic corneal diseases, and pre-screening for subclinical ectasia before corneal surgery. Metrics for diagnosis typically employed are curvature maps (axial/sagittal, tangential); elevation map of the anterior surface of the cornea with respect to a reference sphere; and pachymetry (thickness) map of the cornea. We present evidence that these curvature maps do not represent the actual curvatures (principal or mean) in a human cornea. In this dissertation, we present a method to compute the true mean curvature at every point of a central region of the cornea. This computation is based on the quartic smoothing spline algorithm for the corneal surface reconstruction. We show that the true mean curvature can accurately identify the location of the ectasia. We also present a method for the simultaneous computation of elevation maps for anterior and posterior corneal surfaces, pachymetry map along with the true mean curvature map. The input to the method is data from a single measurement from imaging devices such as an AS-OCT or a Scheimpflug imager. We show that the true mean curvature and elevation map of the anterior surface taken together are useful for the diagnosis of existing ectasia, while pachymetry and elevation map of the posterior surface taken together are useful for the diagnosis of subclinical ectasia. We compare our results with existing algorithms both theoretically and computationally, and present applications to a normal cornea, a cornea displaying forme fruste keratoconus, and a cornea with advanced keratoconus.