UM ETheses Collection (澳門大學電子學位論文庫)
 Title

PFST(SE) 000 (SAMPLE) Curlfree and divergencefree wavelets for computer graphics
 English Abstract

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Abstract Over the last three decades, wavelet theory has been developed as a powerful mathematical tool to extract information from many kinds of data such as audio signal and images. Compared to global bases such as the Fourier base, wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. Another main advantage is that wavelets can form a multiresolution analysis (MRA) of interested function spaces. This means that we can perform the multiresolution analysis for data. Wavelets have been widely used to process scalar fields in the fields of Computer Graphics, Computer Vision and Image Processing, such as image processing (denoising, compression and fusion), geometry processing (denoising, compression), and rendering (adaptive rendering, compression). However, there are very few works that apply wavelets to process vector fields of 2D and 3D. In Computer Graphics, there are many important problems associated with vector fields such as gradientdomain image processing, surface reconstruction from oriented points, and fluid simulation. If we still use traditional wavelet methods to process vector fields by representing them as scalar fields, we would miss some nice properties of vector fields. According to the famous HelmholtzHodge decomposition theorem, any vector field can be decomposed into the sum of a curlfree field, a divergencefree field and a harmonic field (both curlfree and divergencefree). We know that the set of each kind of the vector fields forms a subspace. It would be better to construct wavelets for those subspaces rather than using wavelets for scalar fields. In this thesis, we develop two kinds of powerful vectorvalued wavelets for processing vector fields called curlfree and divergencefree wavelets. The constructed curlfree wavelets (resp. divergencefree wavelets) form a basis of the curlfree subspace (resp. divergencefree subspace). With the two kinds of wavelets, we design efficient algorithms for solving three important problems in Computer Graphics: Gradientdomain image compositing, Surface reconstruction from oriented points, and Fluid simulation. For the problem of Gradientdomain image compositing, we develop an efficient approximate wavelet algorithm for gradientdomain image compositing on both the CPU an GPU. The algorithm has O(n) time complexity and 2n memory cost for an npixels composition and outperforms stateoftheart methods both on the CPU and GPU. For the problem of Surface reconstruction, we develop a new biorthogonal wavelet approach to creating a watertight surface from oriented point sets. The approach supports streaming implementations for processing very large datasets. For the problem of Fluid simulation, we develop a new multiresolution wavelet algorithm for solving the HelmholtzHodge decomposition problem and apply it to smoke simulation and turbulence synthesis. Keywords: Computer Graphics, Curlfree Wavelets, Divergencefree Wavelets, Surface Reconstruction, Gradientdomain image processing, Fluid Simulation, Turbulence Processing
 Issue date

2019.
 Author

Ren, Xiao Hua
 Faculty

Faculty of Science and Technology
 Department

Department of Computer and Information Science
 Degree

Ph.D.
 Subject
 Supervisor

Wu, Enhua
 Location
 1/F Zone C
 Library URL
 991008148399706306