Multilinear Wavelets: A Statistical Shape Space for Human Faces

We present a statistical model for $3$D human faces in varying expression, which decomposes the surface of the face using a wavelet transform, and learns many localized, decorrelated multilinear models on the resulting coefficients. Using this model we are able to reconstruct faces from noisy and occluded $3$D face scans, and facial motion sequences. Accurate reconstruction of face shape is important for applications such as tele-presence and gaming. The localized and multi-scale nature of our model allows for recovery of fine-scale detail while retaining robustness to severe noise and occlusion, and is computationally efficient and scalable. We validate these properties experimentally on challenging data in the form of static scans and motion sequences. We show that in comparison to a global multilinear model, our model better preserves fine detail and is computationally faster, while in comparison to a localized PCA model, our model better handles variation in expression, is faster, and allows us to fix identity parameters for a given subject.Comment: 10 pages, 7 figures; accepted to ECCV 201

Haar-LikeWavelets over Tetrahedra

In this paper we define a Haar-like wavelets basis that form a basis for L2(T,S,μ), μ being the Lebesgue measure and S the σ -algebra of all tetrahedra generated from a subdivision method of the T tetrahedron. As 3D objects are, in general, modeled by tetrahedral grids, this basis allows the multiresolution representation of scalar functions defined on polyhedral volumes, like colour, brightness, density and other properties of an 3D object.Facultad de Informátic

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

Lifting-based subdivision wavelets with geometric constraints.

Qin, Guiming."August 2010."Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (p. 72-74).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.5Chapter 1.1 --- B splines and B-splines surfaces --- p.5Chapter 1. 2 --- Box spline --- p.6Chapter 1. 3 --- Biorthogonal subdivision wavelets based on the lifting scheme --- p.7Chapter 1.4 --- Geometrically-constrained subdivision wavelets --- p.9Chapter 1.5 --- Contributions --- p.9Chapter 2 --- Explicit symbol formulae for B-splines --- p.11Chapter 2. 1 --- Explicit formula for a general recursion scheme --- p.11Chapter 2. 2 --- Explicit formulae for de Boor algorithms of B-spline curves and their derivatives --- p.14Chapter 2.2.1 --- Explicit computation of de Boor Algorithm for Computing B-Spline Curves --- p.14Chapter 2.2.2 --- Explicit computation of Derivatives of B-Spline Curves --- p.15Chapter 2. 3 --- Explicit power-basis matrix fomula for non-uniform B-spline curves --- p.17Chapter 3 --- Biorthogonal subdivision wavelets with geometric constraints --- p.23Chapter 3. 1 --- Primal subdivision and dual subdivision --- p.23Chapter 3. 2 --- Biorthogonal Loop-subdivision-based wavelets with geometric constraints for triangular meshes --- p.24Chapter 3.2.1 --- Loop subdivision surfaces and exact evaluation --- p.24Chapter 3.2.2 --- Lifting-based Loop subdivision wavelets --- p.24Chapter 3.2.3 --- Biorthogonal Loop-subdivision wavelets with geometric constraints --- p.26Chapter 3. 3 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.35Chapter 3.3.1 --- Catmull-Clark subdivision and Doo-Sabin subdivision surfaces --- p.35Chapter 3.3.1.1 --- Catmull-Clark subdivision --- p.36Chapter 3.3.1.2 --- Doo-Sabin subdivision --- p.37Chapter 3.3.2 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.38Chapter 3.3.2.1 --- Biorthogonal Doo-Sabin subdivision wavelets with geometric constraints --- p.38Chapter 3.3.2.2 --- Biorthogonal Catmull-Clark subdivision wavelets with geometric constraints --- p.44Chapter 4 --- Experiments and results --- p.49Chapter 5 --- Conclusions and future work --- p.60Appendix A --- p.62Appendix B --- p.67Appendix C --- p.69Appendix D --- p.71References --- p.7