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Admissible Diffusion Wavelets and Their Applications in Space-Frequency Processing
Jan. 2013 (vol. 19 no. 1)
pp. 3-15
Tingbo Hou, Dept. of Comput. Sci., Stony Brook Univ., Stony Brook, NY, USA
Hong Qin, Dept. of Comput. Sci., Stony Brook Univ., Stony Brook, NY, USA
As signal processing tools, diffusion wavelets and biorthogonal diffusion wavelets have been propelled by recent research in mathematics. They employ diffusion as a smoothing and scaling process to empower multiscale analysis. However, their applications in graphics and visualization are overshadowed by nonadmissible wavelets and their expensive computation. In this paper, our motivation is to broaden the application scope to space-frequency processing of shape geometry and scalar fields. We propose the admissible diffusion wavelets (ADW) on meshed surfaces and point clouds. The ADW are constructed in a bottom-up manner that starts from a local operator in a high frequency, and dilates by its dyadic powers to low frequencies. By relieving the orthogonality and enforcing normalization, the wavelets are locally supported and admissible, hence facilitating data analysis and geometry processing. We define the novel rapid reconstruction, which recovers the signal from multiple bands of high frequencies and a low-frequency base in full resolution. It enables operations localized in both space and frequency by manipulating wavelet coefficients through space-frequency filters. This paper aims to build a common theoretic foundation for a host of applications, including saliency visualization, multiscale feature extraction, spectral geometry processing, etc.
Index Terms:
wavelet transforms,feature extraction,geometry,signal reconstruction,signal resolution,smoothing methods,spectral analysis,spectral geometry processing,admissible diffusion wavelet,space-frequency processing,signal processing tool,biorthogonal diffusion wavelet,mathematics,smoothing process,scaling process,multiscale analysis,shape geometry,scalar field,ADW,meshed surface,point cloud,dyadic power,data analysis,signal reconstruction,signal resolution,wavelet coefficient,space-frequency filter,saliency visualization,multiscale feature extraction,Wavelet transforms,Manifolds,Multiresolution analysis,Geometry,Surface waves,Feature extraction,feature extraction,wavelet transforms,feature extraction,geometry,signal reconstruction,signal resolution,smoothing methods,spectral analysis,spectral geometry processing,admissible diffusion wavelet,space-frequency processing,signal processing tool,biorthogonal diffusion wavelet,mathematics,smoothing process,scaling process,multiscale analysis,shape geometry,scalar field,ADW,meshed surface,point cloud,dyadic power,data analysis,signal reconstruction,signal resolution,wavelet coefficient,space-frequency filter,saliency visualization,multiscale feature extraction,Wavelet transforms,Manifolds,Multiresolution analysis,Geometry,Surface waves,Feature extraction,space-frequency processing,Diffusion wavelets,wavelet analysis
Citation:
Tingbo Hou, Hong Qin, "Admissible Diffusion Wavelets and Their Applications in Space-Frequency Processing," IEEE Transactions on Visualization and Computer Graphics, vol. 19, no. 1, pp. 3-15, Jan. 2013, doi:10.1109/TVCG.2012.111
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