Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics

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1 Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group Overview! B-splines: the right tool for interpolation fundamental properties spline fitting interpolation; smoothing; least-squares quantitative approximation quality! A primer to the wavelet transform multi-resolution, semi-orthogonal wavelets! Multi-dimensional extensions: hex-splines hexagonal versus Cartesian lattice IEEE Visualization Tutorial (Image Processing 2D) 2 Notations Polynomial B-splines! one-sided! B-spline of degree n! Fourier transform! Z-transform! Symmetric B-spline! Key properties compact support piecewise polynomial positivity smoothness (continuity) [Schoenberg, 1946] 3 4

2 B-spline representation Fundamental B-spline properties! The link between continuous and discrete analog signal in in the B-spline coefficients continuous domain in in the discrete domain! Partition of unity reproduction of the constant! Riesz basis stability: small perturbation of coefficients results into small change of spline signal unambiguity: each representation is unique! m-scale relation! Cardinal setting = unit spacing and! points 5 6 B-spline Fourier expression B-spline differential property poor man s derivative (finite difference) exact derivative Fourier transform of basic element:! Link between discrete and exact derivatives discrete filtering spline degree reduction 7 8

3 Generalized fractional B-splines Spline fitting! Definition in the Fourier domain! How to find the spline coefficients? [Unser&Blu2000,Blu&Unser2003] 9 10 Spline fitting: (1) spline interpolation Spline interpolation! Spline interpolation (exact, reversible) discrete input filtering such that! Discrete B-spline kernels! Smoothing spline! Least square splines (approximation between spline spaces)! Satisfying interpolation condition: inverse filter!! Efficient recursive implementation: cascade of causal and anti-causal filters e.g., cubic spline interpolation causal anti-causal 11 12

4 Spline interpolation Spline interpolation! Generic C-code main recursion void ConvertToInterpolationCoefficients ( double c[ ], long DataLength, double z[ ], long NbPoles, double Tolerance) { double Lambda = 1.0; long n, k; if (DataLength == 1L) return; for (k = 0L; k < NbPoles; k++) Lambda = Lambda * (1.0 - z[k]) * ( / z[k]); for (n = 0L; n < DataLength; n++) c[n] *= Lambda; for (k = 0L; k < NbPoles; k++) { c[0] = InitialCausalCoefficient(c, DataLength, z[k], Tolerance); for (n = 1L; n < DataLength; n++) c[n] += z[k] * c[n - 1L]; c[datalength - 1L] = (z[k] / (z[k] * z[k] - 1.0)) * (z[k] * c[datalength - 2L] + c[datalength - 1L]); for (n = DataLength - 2L; 0 <= n; n--) c[n] = z[k] * (c[n + 1L]- c[n]); } }! Interpolating or fundamental B-spline initialization double InitialCausalCoefficient ( double c[ ], long DataLength, double z, double Tolerance) { double Sum, zn, z2n, iz; long n, Horizon; Horizon = (long)ceil(log(tolerance) / log(fabs(z))); if (DataLength < Horizon) Horizon = DataLength; zn = z; Sum = c[0]; for (n = 1L; n < Horizon; n++) {Sum += zn * c[n]; zn *= z;} return(sum); } Spline interpolation Spline fitting: (2) smoothing spline! The fundamental spline converges to sinc as the degree goes to infinity! Spline interpolation! Smoothing spline discrete and noisy input filtering subject to regularization! Least square splines (approximation between spline spaces)! Shannon s theory appears as a particular case 15 16

5 Smoothing spline Spline fitting: (3) least-square spline! The solution (among all functions) of the smoothing spline problem is a cardinal spline of degree 2m-1. Its coefficients can be obtained by suitable digital filtering of the input samples:! Spline interpolation! Smoothing spline! Least-square spline (approximation between spline spaces) spline model resampling & filtering! Related to: MMSE (Wiener filtering); splines form optimal space!!!! Special case: the draftman s spline Minimum curvature interpolant is obtained for = cubic spline! such that error [Unser&Blu2005] Least-square spline Least-square spline! Minimize quadratic error between splines with! Special case: surface projection first-order B-splines on source and target grid weight of sample = overlap between B-splines support determine ; e.g., by spline interpolation resample using with obtain samples of new spline representation prefilter resampling postfilter [Unser et al. 1995]

6 Quantitative approximation quality Quantitative approximation quality! Best approximation in a space? analog input sampling & filtering analysis function at scale a fixed support W=4! Orthogonal projection with error kernel [BluUnser,1999] 21 [Thévenaz et al., 2000] 22 B-spline interpolation in 2D High-quality image interpolation! 2D separable model! Geometric transformations 2D filtering 2D resampling performance! Applications zooming, rotation, resizing, warping 23 cost [Thévenaz et al., 2000] 24

7 Interpolation benchmark High-quality isosurface rendering! Cumulative interpolation experiment: the best algorithm wins! 3D B-spline representation of volume data! Isosurface analytical knowledge of normal vectors bilinear windowed sinc cubic spline 25 [Thévenaz et al., 2000] 26 Multi-resolution approximation Wavelets! m-scale relation! Pyramid or tree algorithms ( ) fast evaluation of binomial filter for high n ~ Gaussian filter n=1! Admissible scaling function ( father wavelet ) Riesz basis conditions partition of unity two-scale relation! B-splines are perfect candidates! Then there exists a wavelet such that forms a Riesz basis of L 2 27 [Mallat-Meyer, 1989] 28

8 Haar wavelet transform revisited Haar wavelet transform revisited! Signal representation basis function: + - Multi-scale signal representation + - multi-scale basis function: Wavelet: Semi-orthogonal wavelets Wavelets! Scaling and wavelet spaces! Wavelets act as differentiators! Semi-orthogonality conditions cubic orthogonal B-spline wavelet Effect on transient features: 1) locality 2) sparsity (vanishing moments) 31 32

9 Wavelets and differentiation Hexagonal lattices! Fundamental property: multiscale differentiator! Responsible for vanishing moments decorrelation! Very successful for coding applications r 2 r 1 Voronoi cell = best tessellation:! Six equivalent neighbours! Twelve-fold symmetry! High isotropy JPEG2000 lattice matrix: Hex-splines! Basis functions for hexagonal grids Hex-splines! Basis functions for hexagonal grids First order Second order 35 36

10 Hex-splines! Basis functions for hexagonal grids Hex-splines! Basis functions for hexagonal grids Third order Fourth order Hex-splines Hex-splines versus B-splines! B-spline-like construction algorithm: generating functions localization operators! B-spline-like properties: convolution property (by construction) positivity, partition of unity, compact support convergence to Gaussian! But no two-scale relation! Keep sampling density equal: det(r)=! Hex-splines on hexagonal grid r 2 r 1 B-splines on orthogonal grid r 2 r

11 Hex-splines versus B-splines Hex-splines versus B-splines! Extra samples so approximation quality B-splines equals that one of hex-splines sampling gain (surface area) p C B" spline C hex" spline p #$ % %# 8& ! Classical result: isotropic band-limited signals are better approximated on hexagonal lattices [Mersereau, 1979]! Here, result for non-bandlimited signals first order (nearest neighbor) on hexagonal lattices does not pay second order (linear-like) is already close to optimal; still has easy analytical characterization! Great, but hex-splines do not have a two-scale relation! order [Van De Ville et al., 2005] Conclusions And finally! B-splines are a great tool for interpolation and approximation 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 theoretical 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)! Many thanks go to Michael Unser Thierry Blu Philippe Thévenaz! Papers, demonstrations, source code: The Wavelet Digest: Annette Unser 43 44