box spline reconstruction filters on BCC lattice

Alireza EntezariRamsay Dyer

Torsten Möller

Minho Kim

problem

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

sampling theory in 1D

Fourier transform

• one-to-one mapping between spatial and Fourier domains

• multiplication and convolution are dual operations: F(fg)=F(f)F(g) F(fg)=F(f)F(g)

(image courtesy of [1])

Dirac comb function (cω)

• infinite series of equidistant Dirac impulses

• Fourier transform has the same shape

(image courtesy of [1])

sampling

• F(fcω)=F(f)F(cω)

(image courtesy of [1])

• to remove all the “replicated spectra” except the “primary spectrum”

• requires ω>2B (B: highest frequency of f)

• requires “low-pass filter” bπ/ω

• F-1(bπ/ω)(t) = sinc(t) = sin(t)/t

reconstruction

(image courtesy of [1])

recontruction (cont’d)

• F-1(bπ/ωF(fg))=F-1(bπ/ω)(fg)

– weighted sum of basis functions, sinc

(image courtesy of [1])

reconstruction (cont’d)

(image courtesy of [2])

aliasing

• happens when the condition “ω>2B”is not met

• cannot reconstruct the original signal

(image courtesy of [5])

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.

• post-aliasing - “sample frequency ripple”

• ringing (“overshoot”)

defects due to filters

(image courtesy of [5])

(image courtesy of [5])

sampling theoryin higher dimensions

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

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

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’1 T’2 T’3], T’1=[1 0 -1], T’2=[0 1 -1], T’3=[-1 -1 2], which is the sampling matrix of FCC lattice

BCC and FCC lattices

BCC lattice FCC lattice

(image courtesy of Wikipedia)

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.

rhombic dodecahedron

• the first neighbor cell of BCC lattice (image courtesy of [7])

• animated version (from MathWorld): http://mathworld.wolfram.com/RhombicDodecahedron.html

• 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])

linear box spline filter

4D hypercube T(x,y,z,w)

linear box splineon BBC lattice LRD(x,y,z)

F(T)

F(LRD)

projection

F

Fslicing

cubic box spline filter

4D hypercube

linear box splineon BBC lattice

cubic box splineon BBC lattice

tensor product offour 1D triangle functionsself-convolution

self-convolution

projection projection

cubic box spline filter (cont’d)

• 1D-2D analogy

self-convolution

self-convolution

projection projection

(image courtesy of [7],[8])

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