Smoothing Splines on Unit Ball Domains with Application to Corneal Topography

Optical coherence tomography (OCT) is a non-invasive imaging technique used to study and understand internal structures of biological tissues such as the anterior chamber of the human eye. An interesting problem is the reconstruction of the shape of the biological tissue from OCT images, that is not only a good fit of the data but also respects the smoothness properties observed in the images. A similar problem arises in Magnetic Resonance Imaging (MRI). We cast the problem as a penalized weighted least squares regression with a penalty on the magnitude of the second derivative (Laplacian) of the surface. We present a novel algorithm to construct the Kimeldorf-Wahba solution for unit ball domains. Our method unifies the ad-hoc approaches currently in the literature. Application of the theory to data from an anterior segment optical coherence tomographer is presented. A detailed comparison of the reconstructed surface using different approaches is presented.