Signal processing and eigenvalue decomposition of polygonal meshes and applications

The continuing trend of increased mesh complexity appears to have reached a point where traditional computed-aided geometric design techniques, especially those based on geometric optimization, are often too expensive computationally. In this thesis, we take a signal processing approach to model and solve problems related to large, irregular 3-D meshes, and provide the first formal treatment of the topic.; We formalize the basic concepts of mesh signal processing by defining and studying several important concepts from image processing, such as convolution, in the 3-D irregular grid setting. We analyze eigenvalue decomposition , which extends the classical Fourier transform (DFT) to ED-transforms for mesh modeling, from a theoretical point of view. We suggest necessary conditions under which ED-transforms resemble the classical DFTs and examine the similarities and differences between them. In particular, we formulate and prove several fundamental results about ED-transforms that are analogous to the basic DFT theorems of 1-D signals.; We also explore the many useful applications of eigenvalue decomposition, such as mesh fairing, feature enhancement, mesh partitioning, spectral compression, and water marking. Specifically, we present new algorithms and techniques we have developed for shape-preserving mesh fairing, e.g., using Butterworth filters, and for shape matching and similarity. Finally, our study has motivated us to design a new and improved linear operator for eigenvalue decomposition.