box spline reconstruction
filters on BCC lattice
Alireza Entezari
Ramsay 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
(image courtesy of [1])
• multiplication and convolution are dual
operations: F(fg)=F(f)F(g) F(fg)=F(f)F(g)
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])
reconstruction
• 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
(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.
defects due to filters
• post-aliasing - “sample frequency ripple”
(image courtesy of [5])
• ringing (“overshoot”)
(image courtesy of [5])
sampling theory
in 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 pattern
in 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
FCC lattice
BCC 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
linear box spline filter
• Fourier transform of a linear box spline
filter can be obtained by projection-slice
theorem.
4D hypercube T(x,y,z,w)
F
projection
linear box spline
on BBC lattice LRD(x,y,z)
F(T)
slicing
F
F(LRD)
• zero-crossings at all the frequencies of
replicas ([7])  no “sampling frequency
ripple” ([5])
cubic box spline filter
4D hypercube
tensor product of
self-convolution four 1D triangle functions
projection
linear box spline
on BBC lattice self-convolution
projection
cubic box spline
on BBC lattice
cubic box spline filter (cont’d)
• 1D-2D analogy
self-convolution
projection
self-convolution
(image courtesy of [7],[8])
projection
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