By [various contributors], Junfei Dai, Xianfeng David Gu, Feng Luo
This quantity introduces readers to a few of the present subject matters of study within the geometry of polyhedral surfaces, with functions to special effects. the most characteristic of the quantity is a scientific advent to the geometry of polyhedral surfaces in accordance with the variational precept. The authors concentrate on utilizing analytic tools within the research of a few of the basic effects and difficulties of polyhedral geometry: for example, the Cauchy pressure theorem, Thurston's circle packing theorem, stress of circle packing theorems, and Colin de Verdiere's variational precept. this is often the 1st entire therapy of the huge, and expansively constructed, box of polyhedral geometry.
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Extra resources for Variational Principles for Discrete Surfaces: Theories and Algorithms
Example text
In the reconstruction or synthesis side, the coefficients are up-sampled by inserting a 0 between each sample and then convolved with the dual filters h˜ and g; ˜ the resulting coefficients are summed, and the result is multiplied by 2. The procedure is iterated up to the smallest scale, as depicted in the lower part of Fig. 5. Compared to the CWT, we have far fewer scales because we consider only dyadic scales, that is, scales a j that are a power of 2 of the initial scale a0 (a j = 2 j a0 ). Therefore, for a discrete signal X [n] with N samples, one would typically use J = log2 N scales; the indexing here is such that j = 1 corresponds to the finest scale (high frequencies).
3. a = a + a . ), the discretized CWT of X. If discrete computations assume periodic boundary conditions, the discrete convolution ψ¯ a X can be performed in the Fourier domain: ψ¯ a X = IFFT(FFT(ψ¯ a )FFT(X)), where FFT and IFFT denote the fast Fourier transform and its inverse, respectively. In this case, if the number of voices per octave is set to 12, it is easy to check that the computation of the CWT requires O(12N log2 N)2 operations (Rioul and Duhamel 1992). If the convolution is done in the original (sample) 21 22 The Wavelet Transform domain, we can choose other ways to deal with the boundaries.
The inclusion of the wavelet transform in JPEG 2000, the recent still-picture compression standard, testifies to this lasting and significant impact. 7 (bottom right) shows the decompressed image for a compression ratio of 40, and as can be seen, the result is near-perfect. 6 NONDYADIC RESOLUTION FACTOR Feauveau (1990) introduced quincunx analysis, based on Adelson’s work (Adelson et al. 1987). This analysis √ is not dyadic and allows an image decomposition with a resolution factor equal to 2. The advantage is that only one nonseparable wavelet is needed.