Fast algorithm for directional time-scale analysis using wavelets

Fast algorithms performing time-scale analysis of multivariate functions are discussed. The algorithms employ univariate wavelets and involve a directional parameter, namely the angle of rotation. Both the rotation steps and the wavelet analysis/synthesis steps in the algorithms require a number of computations proportional to the number of data involved. The rotation and wavelet techniques are used for the segregation of wanted and unwanted components in a seismic signal. As an illustration, the rotation and wavelet methods are applied to a synthetic shot record

Subdivision Surface based One-Piece Representation

Subdivision surfaces are capable of modeling and representing complex shapes of arbi-trary topology. However, methods on how to build the control mesh of a complex surfaceare not studied much. Currently, most meshes of complicated objects come from trian-gulation and simplification of raster scanned data points, like the Stanford 3D ScanningRepository. This approach is costly and leads to very dense meshes.Subdivision surface based one-piece representation means to represent the final objectin a design process with only one subdivision surface, no matter how complicated theobject\u27s topology or shape. Hence the number of parts in the final representation isalways one.In this dissertation we present necessary mathematical theories and geometric algo-rithms to support subdivision surface based one-piece representation. First, an explicitparametrization method is presented for exact evaluation of Catmull-Clark subdivisionsurfaces. Based on it, two approaches are proposed for constructing the one-piece rep-resentation of a given object with arbitrary topology. One approach is to construct theone-piece representation by using the interpolation technique. Interpolation is a naturalway to build models, but the fairness of the interpolating surface is a big concern inprevious methods. With similarity based interpolation technique, we can obtain bet-ter modeling results with less undesired artifacts and undulations. Another approachis through performing Boolean operations. Up to this point, accurate Boolean oper-ations over subdivision surfaces are not approached yet in the literature. We presenta robust and error controllable Boolean operation method which results in a one-piecerepresentation. Because one-piece representations resulting from the above two methodsare usually dense, error controllable simplification of one-piece representations is needed.Two methods are presented for this purpose: adaptive tessellation and multiresolutionanalysis. Both methods can significantly reduce the complexity of a one-piece represen-tation and while having accurate error estimation.A system that performs subdivision surface based one-piece representation was im-plemented and a lot of examples have been tested. All the examples show that our ap-proaches can obtain very good subdivision based one-piece representation results. Eventhough our methods are based on Catmull-Clark subdivision scheme, we believe they canbe adapted to other subdivision schemes as well with small modifications

Multiresolution analysis as an approach for tool path planning in NC machining

Wavelets permit multiresolution analysis of curves and surfaces. A complex curve can be decomposed using wavelet theory into lower resolution curves. The low-resolution (coarse) curves are similar to rough-cuts and high-resolution (fine) curves to finish-cuts in numerical controlled (NC) machining.;In this project, we investigate the applicability of multiresolution analysis using B-spline wavelets to NC machining of contoured 2D objects. High-resolution curves are used close to the object boundary similar to conventional offsetting, while lower resolution curves, straight lines and circular arcs are used farther away from the object boundary.;Experimental results indicate that wavelet-based multiresolution tool path planning improves machining efficiency. Tool path length is reduced, sharp corners are smoothed out thereby reducing uncut areas and larger tools can be selected for rough-cuts

Development of an adaptive multi-resolution method to study the near wall behavior of two-dimensional vortical flows

In the present investigation, a space-time adaptive multiresolution method is developed to solve evolutionary PDEs, typically encountered in fluid mechanics. The new method is based on a multiresolution analysis which allows to reduce the number of active grid points significantly by refining the grid automatically in regions of steep gradients, while in regions where the solution is smooth coarse grids are used. The method is applied to the one-dimensional Burgers equation as a classical example of nonlinear advection-diffusion problems and then extended to the incompressible two-dimensional Navier-Stokes equations. To study the near wall behavior of two-dimensional vortical flows a recently revived, dipole collision with a straight wall is considered as a benchmark. After that an extension to interactions with curved walls of concave or convex shape is done using the volume penalization method. The space discretization is based on a second order central finite difference method with symmetric stencil over an adaptive grid. The grid adaptation strategy exploits the local regularity of the solution estimated via the wavelet coefficients at a given time step. Nonlinear thresholding of the wavelet coefficients in a one-to-one correspondence with the grid allows to reduce the number of grid points significantly. Then the grid for the next time step is extended by adding a safety zone in wavelet coefficient space around the retained coefficients in space and scale. With the use of Harten's point value multiresolution framework, general boundary conditions can be applied to the equations. For time integration explicit Runge-Kutta methods of different order are implemented, either with fixed or adaptive time stepping. The obtained results show that the CPU time of the adaptive simulations can be significantly reduced with respect to simulations on a regular grid. Nevertheless the accuracy order of the underlying numerical scheme is preserved