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

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

Wavelet-transform-based strategy for generating new Chinese fonts

Author name used in this publication: Chi K. TseRefereed conference paper2002-2003 > Academic research: refereed > Refereed conference paperVersion of RecordPublishe

Reversing subdivision rules: local linear conditions and observations on inner products

AbstractIn a previous work (Samavati and Bartels, Comput. Graphics Forum 18 (1998) 97–119) we investigated how to reverse subdivision rules using global least-squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finite-dimensional Cartesian vector space. We produced simple and sparse reconstruction filters, but had to appeal to matrix factorization to obtain an efficient, exact decomposition. We also made some observations on how the inner product that defines the semiorthogonality influences the sparsity of the reconstruction filters. In this work we carry the investigation further by studying biorthogonal systems based upon subdivision rules and local least-squares fitting problems that reverse the subdivision. We are able to produce multiresolution structures for some common univariate subdivision rules that have both sparse reconstruction and decomposition filters. Three will be presented here – for quadratic and cubic B-spline subdivision and for the four-point interpolatory subdivision of Dyn et al. We observe that each biorthogonal system we produce can be interpreted as a semiorthogonal system with an inner product induced on the multiresolution that is quite different from that normally used. Some examples of the use of this approach on images, curves, and surfaces are given

Boundary element based multiresolution shape optimisation in electrostatics

We consider the shape optimisation of high-voltage devices subject to electrostatic field equations by combining fast boundary elements with multiresolution subdivision surfaces. The geometry of the domain is described with subdivision surfaces and different resolutions of the same geometry are used for optimisation and analysis. The primal and adjoint problems are discretised with the boundary element method using a sufficiently fine control mesh. For shape optimisation the geometry is updated starting from the coarsest control mesh with increasingly finer control meshes. The multiresolution approach effectively prevents the appearance of non-physical geometry oscillations in the optimised shapes. Moreover, there is no need for mesh regeneration or smoothing during the optimisation due to the absence of a volume mesh. We present several numerical experiments and one industrial application to demonstrate the robustness and versatility of the developed approach.Web of Science29759858

Boundary element based multiresolution shape optimisation in electrostatics

We consider the shape optimisation of high-voltage devices subject to electrostatic field equations by combining fast boundary elements with multiresolution subdivision surfaces. The geometry of the domain is described with subdivision surfaces and different resolutions of the same geometry are used for optimisation and analysis. The primal and adjoint problems are discretised with the boundary element method using a sufficiently fine control mesh. For shape optimisation the geometry is updated starting from the coarsest control mesh with increasingly finer control meshes. The multiresolution approach effectively prevents the appearance of non-physical geometry oscillations in the optimised shapes. Moreover, there is no need for mesh regeneration or smoothing during the optimisation due to the absence of a volume mesh. We present several numerical experiments and one industrial application to demonstrate the robustness and versatility of the developed approach.We gratefully acknowledge the support provided by the EU commission through the FP7 Marie Curie IAPP project CASOPT (PIAP-GA-2008-230224). K.B. and F.C. thank for the additional support provided by EPSRC through #EP/G008531/1. J.Z. thanks for the support provided by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070) and by the project SPOMECH – Creating a Multidisciplinary R&D Team for Reliable Solution of Mechanical Problems, reg. no. CZ.1.07/2.3.00/20.0070 within the Operational Programme ‘Education for Competitiveness’ funded by the Structural Funds of the European Union and the state budget of the Czech Republic. Special thanks to Andreas Blaszczyk from the ABB Corporate Research Center Switzerland for fruitful discussions and for providing the industrial applications.This is the final version of the article. It first appeared from Elsevier via http://dx.doi.org/10.1016/j.jcp.2015.05.01

Multiresolution curve editing with linear constraints

The use of multiresolution control toward the editing of freeform curves and surfaces has already been recognized as a valuable modeling too

Dyadic Splines

Dyadic splines are a simple and efficient function representation that supports multiresolution design and analysis. These splines are defined as limits of a process that alternately doubles and perturbs a sequence of points, using B-spline subdivision to smoothly perform the doubling. An interval-query algorithm is presented that efficiently and flexibly evaluates a limit function for points and intervals. Methods are given for fitting these functions to input data, and for minimizing the energy and redundancy of the representation. Several methods are given for designing dyadic splines by controlling the perturbations of the limit process. Several applications are explored, including shape design, synthesis of terrain and other natural forms, and compression