Wavelet-Based Multi-resolution Digital Geometry Processing

The rapid development of 3D digital geometry model acquisition techniques hasstimulated a new wave of researches on digital geometry processing (DGP) theoriesand applications. Based on the lifting scheme, this dissertation focuses on studingnew wavelet construction methods for curves and surfaces, especially for subdivisionsurfaces with arbitrary topology. Furthermore, based on the wavelet analysis tools,several new multiresolution DGP algorithms are developed. The main contributions ofthis dissertation are summarized as follows:1) By utilizing the global lifting scheme, this dissertation proposes a new wayto construct B-spline wavelets under user-specified constraints, i.e., B-spline waveletswhich fix positions, tangents and/or high order derivatives at some user-specified pa-rameter values, thus extends the capability of B-spline wavelets. Using constrainedB-spline wavelets, this dissertation solves the problem of constrained curve/surfacesmoothing, seamlessly representing a curve/surface with multiple segments/patchesand constrained multiresolution curve/surface editing.2) By expanding the local lifting scheme for constructing fast subdivision sur-face wavelets with local operators, this dissertation introduces a fast Loop subdivisionsurface wavelet transform, which eliminates one shortcoming of traditional subdivi-sion wavelets—the slow wavelet-decomposition speed. Furthermore, a new feature-preserving Loop subdivision surface wavelet scheme is presented, which supports thewavelet construction on subdivision surfaces with sharp features (including boundaries,corners, creases and darts etc.). A systematical optimization strategy for free parame-ters introduced in the wavelet construction process, is proposed.3) By combining the merits of both normal meshes and subdivision surface wave-let hierarchies, this dissertation proposes a new hierarchical surface representationnamed"normal wavelet meshes". A normal wavelet mesh is a special subdivisionsurface wavelet hierarchy in which all wavelet coefficients are in the local normal di-rections only and hence can be stored as scalar values. A two-stage geometric opti-mization algorithm, which serves as the basis for the normal wavelet mesh conversionalgorithm, is introduced. Based on the idea of two-stage geometric optimization, aLoop subdivision surface fitting algorithm is also developed.4) This dissertation tests the fast Loop subdivision wavelets and normal meshesfor 3D geometry data compression. The experimental results show that using normalwavelet meshes can improve the compression performance considerably, comparedwith the traditional subdivision surface wavelets. By means of subdivision wavelets,this dissertation presents geometric signal processing techniques with high pass, stop-band filtering and detail enhancement effects. Some applications of such fast subdivi-sion wavelets to multiresolution geometry editing and multiresolution speedup meth-ods for physics-based deformable modeling are also investigated in this dissertation.To conclude, this dissertation presents several new curve/surface wavelet con-struction methods and multiresolution DGP algorithms in a systematical way, whichconstitutes a powerful extension to wavelet-based multiresolution DGP techniques.