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<!-- -*- Mode: html-helper -*- --><html> <head><title>Suggestions for Projects</title></head><body><h1>Suggestions for Projects</h1>We are collecting a number of possible projects here for students topick from. While you are encouraged to dream up your own you can alsopick one off of this list. For lack of a better systemization we havegrouped the projects into those which are primarily implementation, i.e.,write code for a small interactive application, and those which areprimarily theory oriented, e.g., study the smoothness of a class ofsubdivision schemes. The latter may at times involve small amounts ofimplementation along the lines of writing a Maple program tomanipulate various Fourier transform expressions of subdivisionmatrices, for example.<p>Note that his list is incomplete and we are adding to it as we goalong.<h1>Implementation Projects</h1><h3>Loop's scheme</h3>Loop's scheme is a subdivision method for surfaces which will work forarbitrary triangular meshes and which generalizes quartic triangularsplines. The main references on this are Loop's thesis from Utah,which we have in hardcopy only and a recent Siggraph paper,``Piecewise Smooth Surface Reconstruction,'' by Hugues Hoppe et al.,Siggraph 1994, pages 295-305.<h3>Peters' C<sup>1</sup> surface scheme</h3>Jörg Peters has recently described a scheme for arbitrarytopology meshes which results in a globally C<sup>1</sup> bygenerating a number of Bezier patches. The scheme is not as simple asLoop's scheme for example, but it results in a finite set of patches,while other generatlization such as Loop's scheme only produce asurface in the subdivision limit around extraordinary points. In factsurprisingly little is known about the smoothness around suchextraordinary points. An intriguing possibility of Peters' scheme isthe fact that it allows for the incorporation of interpolationconstraints. This can possibly be turned into an interpolatingsubdivision scheme for arbitrary topology meshes. The relevant papersare online. The C<sup>1</sup> algorithm is described in <!WA0><ahref="http://www.cs.caltech.edu/~ps/papers/jorg/93ffss.ps">``C<sup>1</sup>Surfaces Splines''</a> and a somewhat restricted but much simplerversion is described in <!WA1><ahref="http://www.cs.caltech.edu/~ps/papers/jorg/94nocut.ps">``SmoothingPolyhedra Made Easy''</a>.<h3>Catmull-Clark and Doo-Sabin schemes</h3>These schemes generalize quadratic and cubic splines over arbitarytopology meshes by modifying the schemes around extraordinarypoints. The schemes were the first introduced and are classic. The aresimple to implement and result in surfaces which are C<sup>1</sup> andC<sup>2</sup> everywhere except at extraordinary points. We havehardcopy of the relevant articles availalbe.<h3>Butterfly scheme</h3>This is the only scheme which is interpolating by design. It is basedon work by Dyn/Gregory/Levin and we have hardcopy of the article. Itworks for triangular meshes. If these are regular the limiting surfacewill be C<sup>1</sup>. For irregular vertices this is not true andthere are a number of possible modifications that would be interestingto explore.<h3>Kobbelt's scheme</h3>This scheme is very similar to the Butterfly scheme except it worksfor meshes which have quadrilateral faces rather than triangularones. The behaviour at extraordinary points appears nice, but no proofis yet given for this. The relevant paper is <!WA2><ahref="http://www.cs.caltech.edu/~ps/papers/kobbelt/four.ps.Z">``InterpolatorySubdivision on Open Quadrilateral Nets with Arbitrary Topology''</a>.<h2>Some general remarks</h2>Any of the above could be oriented more towards theory by focusing onsuch aspects as smoothness of the limit surfaces near extraodinarypoints. This typically involves computing various Fourier transformsof subdivision matrices and examining the eigen values of these. Insome cases this can lead to suggested modifications. Often there aredegrees of freedom left and an interesting exploration would be to seehow the resulting functions change as these degrees of freedom getmanipulated. Such an exploration could easily be done in Maple orMatlab for those how prefer to stick with such tools.<h1>Theory Projects</h1><h3>Deslauriers Dubuc interpolating subdivision</h3>This is a classic scheme for the real line which is interpolating. Thepaper describes how to compute the exact Hölder smoothness of theresulting limit functions. Build a Maple (or Mathematica) toolbox tocompute this smoothness for various order schemes. We have hardcopyof the relevant paper.<h3>Variational design of interpolating subdivisions</h3>Leif Kobbelt <!WA3><ahref="http://www.cs.caltech.edu/~ps/papers/kobbelt/variation.ps">describes</a>and interesting idea: instead of using polynomials to do interpolatingsubdivision compute new points by solving a discrete variationalproblem. This could easily be implemented in Matlab or Maple and itwould be interesting to see just what kinds of curves one can buildwith this.<h3>Differentiating a subdivision curve</h3>There are a number of observations relating differecing of controlpoints and applying a somewhat modified scheme to the differencecoefficients, to the derivative of a given scheme. For example,differentiating the interpolating schemes of Deslauriers-Dubuc leadsto the average interpolating functions of <!WA4><ahref="http://www.cs.caltech.edu/~ps/papers/donoho/blocky.ps">Donoho</a>.This observation generalizes. A nice project would be to pick onefamily of curves, e.g., B-spline curves and work out the variousderivatives in the irrgularly spaced knot setting. Again this ismostly pencil and paper and some Maple/Mathematica work.<hr><b>Copyright © 1995 Jim Arvo and Peter Schröder</b></body> </html>⌨️ 快捷键说明
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