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 IEEE Visualization Tutorial (Image Processing 2D)
2 Overview Bsplines: the right tool for interpolation fundamental properties spline fitting interpolation; smoothing; leastsquares quantitative approximation quality A primer to the wavelet transform multiresolution, semiorthogonal wavelets 2D extension: hexsplines on any regular periodic lattice hexagonal versus Cartesian lattice 2
3 Notations onesided Fourier transform Ztransform 3
4 Polynomial splines is a polynomial spline of degree n with knots iff Piecewise polynomial (of degree n) within each interval Higherorder continuity at the knots of Effective degrees of freedom is 1 Cardinal splines : Unit spacing: number of knots 4
5 Polynomial Bsplines Bspline of degree n Symmetric Bspline Key properties compact support piecewise polynomial positivity smoothness (continuity) [Schoenberg, 1946] 5
6 Bspline representation The link between continuous and discrete analog signal in the continuous domain Bspline coefficients in the discrete domain 6
7 Fundamental Bspline properties Partition of unity reproduction of the constant and of polynomials up to degree n Riesz basis stability: small perturbation of coefficients results into small change of spline signal unambiguity: each representation is unique mscale relation (for m integer) 7
8 Bspline Fourier expression Fourier transform of basic element: 8
9 Bspline differential property poor man s derivative (finite difference) exact derivative Link between discrete and exact derivatives discrete filtering spline degree reduction 9
10 Generalized fractional Bsplines Definition in the Fourier domain [Unser & Blu 2000, Blu & Unser 2003] 10
11 Spline fitting How to find the spline coefficients? 11
12 Spline fitting: (1) spline interpolation Spline interpolation (exact, reversible) discrete input filtering such that Smoothing spline Least square splines (approximation between spline spaces) 12
13 Spline interpolation Discrete Bspline kernels Satisfying interpolation condition: inverse filter! Efficient recursive implementation: cascade of causal and anticausal filters e.g., cubic spline interpolation anticausal causal 13
14 Spline interpolation Generic Ccode 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]); } } 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); } 14
15 Spline interpolation Interpolating or fundamental Bspline 15
16 Spline interpolation The fundamental spline converges to sinc as the degree goes to infinity Shannon s theory appears as a particular case 16
17 Spline fitting: (2) smoothing spline Spline interpolation Smoothing spline discrete and noisy input filtering subject to regularization Least square splines (approximation between spline spaces) 17
18 Smoothing spline The solution (among all functions) of the smoothing spline problem is a cardinal spline of degree 2m1. Its coefficients can be obtained by suitable digital filtering of the input samples: Related to: MMSE (Wiener filtering); splines form optimal space!!! Special case: the draftman s spline Minimum curvature interpolant is obtained for = cubic spline! [Unser & Blu 2005; Ramani et al. 2005] 18
19 Spline fitting: (3) leastsquare spline Spline interpolation Smoothing spline Leastsquare spline (approximation between spline spaces) spline model resampling & filtering error such that 19
20 Leastsquare spline Minimize quadratic error between splines with 1. determine ; e.g., by spline interpolation 2. resample using with 3. obtain samples of new spline representation prefilter resampling postfilter [Unser et al. 1995] 20
21 Leastsquare spline Special case: surface projection firstorder Bsplines on source and target grid weight of sample = overlap between Bsplines support
22 Quantitative approximation quality Best approximation in a space? analog input filtering & sampling analysis function at scale a Orthogonal projection Results for: Fixed scale Asymptotically with error kernel [Blu & Unser 1999] 22
23 Quantitative approximation quality fixed support W=4 [Thévenaz et al. 2000] 23
24 Bspline interpolation in 2D 2D separable model Geometric transformations 2D filtering 2D resampling Applications zooming, rotation, resizing, warping 24
25 Highquality image interpolation performance cost [Thévenaz et al. 2000] 25
26 Interpolation benchmark Cumulative interpolation experiment: the best algorithm wins bilinear windowed sinc cubic spline 26
27 Highquality isosurface rendering 3D Bspline representation of volume data Isosurface analytical knowledge of normal vectors [Thévenaz et al. 2000] 27
28 Multiresolution approximation mscale relation Pyramid or tree algorithms ( ) fast evaluation of binomial filter n=1 for high n ~ Gaussian filter 28
29 Wavelets Admissible scaling function ( father wavelet ) Riesz basis conditions partition of unity twoscale relation Bsplines are perfect candidates Then there exists a wavelet such that forms a Riesz basis of L 2 [MallatMeyer 1989] 29
30 Haar wavelet transform revisited Signal representation basis function: Multiscale signal representation multiscale basis function: 30
31 Haar wavelet transform revisited Wavelet: 31
32 Semiorthogonal wavelets Scaling and wavelet spaces Semiorthogonality conditions
33 Wavelets Wavelets act as differentiators orthogonal cubic Bspline wavelet Effect on transient features: 1) locality 2) sparsity (vanishing moments) 33
34 Wavelets and differentiation Fundamental property: multiscale differentiator Responsible for vanishing moments decorrelation Very successful for coding applications JPEG
35 Hexagonal lattices r 2 Voronoi cell = best tessellation: Six equivalent neighbours Twelvefold symmetry High isotropy r 1 Beesplines? lattice matrix: 35
36 Hexsplines Basis functions for hexagonal grids First order 36
37 Hexsplines Basis functions for hexagonal grids Second order 37
38 Hexsplines Basis functions for hexagonal grids Third order 38
39 Hexsplines Basis functions for hexagonal grids Fourth order 39
40 Hexsplines Bsplinelike construction algorithm: generating functions (~ differentiation operator in 3 directions) localization operators (~ discrete versions of the operators) Bsplinelike properties: convolution property (by construction) positivity, partition of unity, compact support convergence to Gaussian Hexsplines exist for all periodic lattices coincide with separable Bsplines for cartesian lattice Fitting: interpolation, smoothing, leastsquares But no twoscale relation 40
41 Hexsplines versus Bsplines Keep sampling density equal: det(r)=ω Hexsplines on hexagonal grid Bsplines on orthogonal grid r 2 r 2 r 1 r 1 41
42 Hexsplines versus Bsplines Extra samples so approximation quality Bsplines equals that one of hexsplines (asymptotical result) 43
43 Hexsplines versus Bsplines Classical result: isotropic bandlimited signals are better approximated on hexagonal lattices [Mersereau, 1979] Here, result for nonbandlimited signals first order (nearest neighbor) on hexagonal lattices does not pay at least second order (linearlike) hexsplines should be used; secondorder still have easy analytical characterization 44
44 Conclusions Bsplines are a great tool for interpolation and approximation: link continuous and discrete! short support; analytical expression; tunable degree fundamentally linked to differential operators Shiftinvariant spaces due to uniform sampling brings along fast algorithms (filtering, FFTbased, ) powerful theoretical results (error kernel) Multiresolution mscale relation for pyramids and wavelets Multidimensional extensions and variations tensorproduct, hexsplines, boxsplines (see later) 45
45 And finally Many thanks go to Michael Unser Thierry Blu Philippe Thévenaz Papers, demonstrations, source code: The Wavelet Digest: ( subscribers) Annette Unser 46
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