Fast Visualization by Shear-Warp using Spline Models for Data Reconstruction

This work concerns oneself with the rendering of huge three-dimensional data sets. The target thereby is the development of fast algorithms by also applying recent and accurate volume reconstruction models to obtain at most artifact-free data visualizations. In part I a comprehensive overview on the state of the art in volume rendering is given. Part II is devoted to the recently developed trivariate (linear,) quadratic and cubic spline models defined on symmetric tetrahedral partitions directly obtained by slicing volumetric partitions of a three-dimensional domain. This spline models define piecewise polynomials of total degree (one,) two and three with respect to a tetrahedron, i.e. the local splines have the lowest possible total degree and are adequate for efficient and accurate volume visualization. The following part III depicts in a step by step manner a fast software-based rendering algorithm, called shear-warp. This algorithm is prominent for its ability to generate projections of volume data at real time. It attains the high rendering speed by using elaborate data structures and extensive pre-computation, but at the expense of data redundancy and visual quality of the finally obtained rendering results. However, to circumvent these disadvantages a further development is specified, where new techniques and sophisticated data structures allow combining the fast shear-warp with the accurate ray-casting approach. This strategy and the new data structures not only grant a unification of the benefits of both methods, they even easily admit for adjustments to trade-off between rendering speed and precision. With this further development also the 3-fold data redundancy known from the original shear-warp approach is removed, allowing the rendering of even larger three-dimensional data sets more quickly. Additionally, real trivariate data reconstruction models, as discussed in part II, are applied together with the new ideas to onward the precision of the new volume rendering method, which also lead to a one order of magnitude faster algorithm compared to traditional approaches using similar reconstruction models. In part IV, a hierarchy-based rendering method is developed which utilizes a wavelet decomposition of the volume data, an octree structure to represent the sparse data set, the splines from part II and a new shear-warp visualization algorithm similar to that presented in part III. This thesis is concluded by the results centralized in part V

Option Pricing under Multifactor Black-Scholes Model Using Orthogonal Spline Wavelets

The paper focuses on pricing European-style options on several underlying assets under the Black-Scholes model represented by a nonstationary partial differential equation. The proposed method combines the Galerkin method with $L^2$-orthogonal sparse grid spline wavelets and the Crank-Nicolson scheme with Rannacher time-stepping. To this end, we construct an orthogonal cubic spline wavelet basis on the interval satisfying homogeneous Dirichlet boundary conditions and design a wavelet basis on the unit cube using the sparse tensor product. The method brings the following advantages. First, the number of basis functions is significantly smaller than for the full grid, which makes it possible to overcome the so-called curse of dimensionality. Second, some matrices involved in the computation are identity matrices, which significantly simplifies and streamlines the algorithm, especially in higher dimensions. Further, we prove that discretization matrices have uniformly bounded condition numbers, even without preconditioning, and that the condition numbers do not depend on the dimension of the problem. Due to the use of cubic spline wavelets, the method is higher-order convergent. Numerical experiments are presented for options on the geometric average.Comment: 43 pages, 10 figure

Ellipse-preserving Hermite interpolation and subdivision

We introduce a family of piecewise-exponential functions that have the Hermite interpolation property. Our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses. We prove that the proposed Hermite functions form a Riesz basis and that they reproduce prescribed exponential polynomials. We present a method based on Green's functions to unravel their multi-resolution and approximation-theoretic properties. Finally, we derive the corresponding vector and scalar subdivision schemes, which lend themselves to a fast implementation. The proposed vector scheme is interpolatory and level-dependent, but its asymptotic behaviour is the same as the classical cubic Hermite spline algorithm. The same convergence properties---i.e., fourth order of approximation---are hence ensured