We have combined methods from volume visualization and data analysis to support better diagnosis and treatment of human retinal diseases. Many diseases can be identified by abnormalities in the thicknesses of various retinal layers captured using optical coherence tomography (OCT). We used a support vector machine (SVM) to perform semi-automatic segmentation of retinal layers for subsequent analysis including a comparison of layer thicknesses to known healthy parameters. We have extended and generalized an older SVM approach to support better performance in a clinical setting through performance enhancements and graceful handling of inherent noise in OCT data by considering statistical characteristics at multiple levels of resolution. The addition of the multi-resolution hierarchy extends the SVM to have "global awareness." A feature, such as a retinal layer, can therefore be modeled.

We present a method for the hierarchical approximation of functions in one, two, or three variables based on the finite element method (Ritz approximation). Starting with a set of data sites with associated function, we first determine a smooth (scattered-data) interpolant. Next, we construct an initial triangulation by triangulating the region bounded by the minimal subset of data sites defining the convex hull of all sites. We insert only original data sites, thus reducing storage requirements. For each triangulation, we solve a minimization problem: computing the best linear spline approximation of the interpolant of all data, based on a functional involving function values and first derivatives. The error of a best linear spline approximation is computed in a Sobolev-like norm, leading to element-specific error values. We use these interval/triangle/tetrahedron-specific values to identify the element to subdivide next. The subdivision of an element with largest error value requires the recomputation of all spline coefficients due to the global nature of the problem. We improve efficiency by 1) subdividing multiple elements simultaneously and 2) by using a sparse-matrix representation and system solver.

We present a method for the construction of hierarchies of single-valued functions in one, two, and three variables. The input to our method is a coarse decomposition of the compact domain of a function in the form of an interval (univariate case), triangles (bivariate case), or tetrahedra (trivariate case). We compute best linear spline approximations, understood in an integral least squares sense, for functions defined over such triangulations and refine triangulations using repeated bisection. This requires the identification of the interval (triangle, tetrahedron) with largest error and splitting it into two intervals (triangles, tetrahedra). Each bisection step requires the recomputation of all spline coefficients due to the global nature of the best approximation problem. Nevertheless, this can be done efficiently by bisecting multiple intervals (triangles, tetrahedra) in one step and by reducing the bandwidths of the matrices resulting from the normal equations.

We present a method for the construction of multiple levels of tetrahedral meshes approximating a trivariate function at different levels of detail. Starting with an initial, high-resolution triangulation of a three-dimensional region, we construct coarser representation levels by collapsing tetrahedra. Each triangulation defines a linear spline function, where the function values associated with the vertices are the spline coefficients. Based on predicted errors, we collapse tetrahedron in the grid that do not cause the maximum error to exceed a use-specified threshold. Bounds are stored for individual tetrahedra and are updated as the mesh is simplified. We continue the simplification process until a certain error is reached. The result is a hierarchical data description suited for the efficient visualization of large data sets at varying levels of detail.