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Conclusions ➤ B-splines are a great tool for interpolation and approximation: link continuous and discrete! • short support; analytical expression; tunable degree • fundamentally linked to differential operators ➤ Shift-invariant spaces due to uniform sampling brings along • fast algorithms (filtering, FFT-based,…) • powerful <strong>the</strong>oretical results (error kernel) ➤ Multi-resolution • m-scale relation for pyramids and wavelets ➤ Multi-dimensional extensions and variations • tensor-product, hex-splines, box-splines (see later) 45
And finally ➤ Many thanks go to Michael Unser Thierry Blu Philippe Thévenaz ➤ Papers, demonstrations, source code: http://bigwww.epfl.ch/ ➤ The Wavelet Digest: (22000+ subscribers) http://www.wavelet.org/ © Annette Unser 46
Page 1 and 2: Point Lattices in Computer Graphics
Page 3 and 4: Notations ➤ one-sided ➤ Fourier
Page 5 and 6: Polynomial B-splines ➤ B-spline o
Page 7 and 8: Fundamental B-spline properties ➤
Page 9 and 10: B-spline differential property “p
Page 11 and 12: Spline fitting ➤ How to find the
Page 13 and 14: Spline interpolation ➤ Discrete B
Page 15 and 16: Spline interpolation ➤ Interpolat
Page 17 and 18: Spline fitting: (2) smoothing splin
Page 19 and 20: Spline fitting: (3) least-square sp
Page 21 and 22: Least-square spline ➤ Special cas
Page 23 and 24: Quantitative approximation quality
Page 25 and 26: High-quality image interpolation pe
Page 27 and 28: High-quality isosurface rendering
Page 29 and 30: Wavelets ➤ Admissible scaling fun
Page 31 and 32: Haar wavelet transform revisited +
Page 33 and 34: Wavelets ➤ Wavelets act as differ
Page 35 and 36: Hexagonal lattices r 2 Voronoi cell
Page 37 and 38: Hex-splines ➤ Basis functions for
Page 39 and 40: Hex-splines ➤ Basis functions for
Page 41 and 42: Hex-splines versus B-splines ➤ Ke
Page 43: Hex-splines versus B-splines ➤ Cl

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