box spline reconstruction filters on BCC lattice Alireza Entezari Ramsay Dyer Torsten Möller

1. box spline reconstruction filters on BCC latticeAlireza EntezariRamsay DyerTorsten Möller Minho Kim

2. problem • What’s the optimal sampling pattern in 3D and which reconstruction filter can we use for it?

3. sampling theory in 1D

4. Fourier transform • one-to-one mapping between spatial and Fourier domains • multiplication and convolution are dual operations: F(fg)=F(f)F(g) F(fg)=F(f)F(g) (image courtesy of [1])

5. Dirac comb function (cω) • infinite series of equidistant Dirac impulses • Fourier transform has the same shape (image courtesy of [1])

6. sampling • F(fcω)=F(f)F(cω) (image courtesy of [1])

7. reconstruction • to remove all the “replicated spectra” except the “primaryspectrum” • requires ω>2B (B: highest frequency of f) • requires “low-pass filter” bπ/ω • F-1(bπ/ω)(t) = sinc(t) = sin(t)/t (image courtesy of [1])

8. recontruction (cont’d) • F-1(bπ/ωF(fg))=F-1(bπ/ω)(fg) • weighted sum of basis functions, sinc (image courtesy of [1])

9. reconstruction (cont’d) (image courtesy of [2])

10. aliasing • happens when the condition “ω>2B”is not met • cannot reconstruct the original signal (image courtesy of [5])

11. reconstruction filters • Ideal low-pass filter (sinc function) is impractical since it has infinite support in spatial domain. • We need alternative filters but they may have defects such as post-aliasing, smoothing (“blur”), ringing (“overshoot”), anisotropy. • examples: Barlett filter (linear filter), cubic filter, truncated sinc filter, etc.

12. defects due to filters • post-aliasing - “sample frequency ripple” • ringing (“overshoot”) (image courtesy of [5]) (image courtesy of [5])

13. sampling theoryin higher dimensions

14. reconstruction filters • two ways of extending filters • separable (tensor-product) extension • for Cartesian lattice only • spherical extension • doesn’t guarantee zero-crossings of frequency responses at all replicas of the spectrum

15. optimal sampling patternin 3D • sparsest pattern in spatial domain  tightest arrangement of the replicas of the spectrum in Fourier domain • densest sphere packing lattice  FCC (Face Centered Cubic) lattice • dual of FCC lattice  BCC (Body Centered Cubic) lattice

16. dual lattice • Fourier transform of a sampling lattice with sampling matrix T has sampling matrix T-T ([6], Theorem 1.) • example: • for BCC lattice, T=[T1,T2,T3], T1=[2 0 0]T, T2=[0 2 0]T, T3=[1 1 1]T • T-T=1/2[T’1T’2T’3], T’1=[1 0 -1], T’2=[0 1 -1], T’3=[-1 -1 2], which is the sampling matrix of FCC lattice

17. BCC and FCC lattices FCC lattice BCC lattice (image courtesy of Wikipedia)

18. reconstruction filters • Ideally, the reconstruction filter is the inverse Fourier transform of the characteristic function of the Voronoi cell of FCC lattice, which is impractical. • Alternatively, we use linear or cubic box spline filters of which support is rhombic dodecahedron, (3D shadow of a 4D hypercube) the first neighbor cell of BCC lattice.

19. rhombic dodecahedron • the first neighbor cell of BCC lattice (image courtesy of [7]) • animated version (from MathWorld): http://mathworld.wolfram.com/RhombicDodecahedron.html

20. linear box spline filter • Fourier transform of a linear box spline filter can be obtained by projection-slice theorem. • zero-crossings at all the frequencies of replicas ([7])  no “sampling frequency ripple” ([5]) 4D hypercube T(x,y,z,w) F(T) F projection slicing linear box spline on BBC lattice LRD(x,y,z) F(LRD) F

21. cubic box spline filter 4D hypercube tensor product of four 1D triangle functions self-convolution projection projection linear box spline on BBC lattice cubic box spline on BBC lattice self-convolution

22. cubic box spline filter (cont’d) • 1D-2D analogy self-convolution projection projection self-convolution (image courtesy of [7],[8])

23. references [1] Oliver Kreylos, “Sampling Theory 101,” http://graphics.cs.ucdavis.edu/~okreylos/PhDStudies/Winter2000/SamplingTheory.html, 2000 [2] Rebecca Willett, “Sampling Theory and Spline Interpolation,” http://cnx.org/content/m11126/latest [3] “truncated octahedron,” http://mathworld.wolfram.com/TruncatedOctahedron.html, MathWorld [4] “rhombic dodecahedron,” http://mathworld.wolfram.com/RhombicDodecahedron.html, MathWorld [5] Stephen R. Marschner and Richard J. Lobb, “An Evaluation of Reconstruction Filters for Volume Rendering,” Proceedings of Visualization '94 [6] Alireza Entezari, Ramsay Dyer, and Torsten Möller, “From Sphere Packing to the Theory of Optimal Lattice Sampling,” PIMS/BIRS Workshop, May 22-27, 2004 [7] Alireza Entezari, Ramsay Dyer, Torsten Möller, “Linear and Cubic Box Splines for the Body Centered Cubic Lattice,”, Proceedings of IEEE Visualization 2004 [8] Hartmut Prautzsch and Wolfgang Boehm, “Box Splines,” 2002