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Subsections
Baseband Optimal Sampling
Figure 2:
2D regular rectangular and hexagonal sampling in the spatial and frequency domains.

spatial domain 
frequency domain 



rectangular sampling 





hexagonal sampling 



Usually, no a priori knowledge is available about the (continuous)
underlying functions we are sampling. Therefore, we assume that these
functions are isotropic, i.e., that they do not have a preferred
direction. Another common assumption is that they are bandlimited.
Both assumptions together result in the property that the frequency
responses of such functions are hyperspheres (circles in 2D and
spheres in 3D, respectively).
Sampling such spherically bandlimited functions results in
replicating the primary spectrum in the frequency domain
[15]. In order to reconstruct the underlying continuous
signal perfectly, we need to ensure that the samples in spatial domain
are close enough, so that the aliased spectra in the frequency domain
do not overlap. Optimal sampling is achieved when the number of
samples that fulfill this condition is minimal. In 1D this is also
known as the Nyquist sampling rate. In order to optimally
sample in higher dimensions (i.e., to use as few samples as possible),
aliased spectra in the frequency domain have to be packed as closely
as possible. This problem is known as the sphere packing
problem [14], which has been extensively studied and
solutions for regular packing structures in 2D and 3D exist.
As a motivation and for the sake of simplicity we will first describe
our method to find an optimal sampling pattern in 2D before we delve
into the mysterious structure of 3D Euclidian space.
We will describe sampling as a mapping from indices to actual sample
positions [4]:

(1) 
Here the integers are the indices of the sample and is
the corresponding sampling position. The matrix , which is called
sampling matrix, describes the mapping itself, e.g.,

(2) 
is the matrix for rectangular sampling which simplifies to the
commonly used regular (Cartesian) sampling for . Hexagonal
sampling is most conveniently described by the
matrix

(3) 
which virtually just performs a shear of the rectangular samples
followed by a forshortening.
When describes the sampling in spatial domain, the matrix , satisfying

(4) 
with being the transpose of and being the identity
matrix, describes the positions of the replicas in frequency domain.
is therefore called the periodicity
matrix [4]. Applying
Eq. 4 to Eq. 2 we
obtain the periodicity matrix for rectangular sampling

(5) 
with
and
. The
periodicity matrix for hexagonal sampling is

(6) 
where again
.
The 2D Fourier Transform of a circularly bandlimited
signal has the property

(7) 
where is the maximum frequency in the data set. This baseband can
be inscribed, for example, in a square with length
(corresponding to rectangular sampling). In other words, and
in Eq. 5 have to be equal to . On the other
hand, the baseband can also be inscribed in a hexagon with side length
(corresponding to hexagonal sampling). This
means that in Eq. 6 must be equal to (see
Fig. 2). Calculating the sampling matrices from
these periodicity matrices, we end up with

(8) 
where

(9) 
and

(10) 
where

(11) 
The sampling density is proportional to
[4]. By taking the
ratio

(12) 
we see that hexagonal sampling requires 13.4% fewer samples than
rectangular sampling.
Analogous to the 2D case, we can describe the mapping from indices to coordinates for rectilinear grids using the following matrix:

(13) 
which is regular (rectangular) when
.
As expected, regular rectangular sampling in 3D is (as in 2D) not
optimal. It is important to note that an optimal sphere packing for
arbitrary packing structures in 3D is not known. However, several
optimal packing structures, all with equal sampling density, are known
for the case of regular sampling, i.e., structures that can be
described by three base vectors. Fortunately, this is exactly what we
need, since we do not want to sacrifice the implicit indexing of the
grid points that makes regular grid representations so attractive.
Among the optimal regular packing structures are the hexagonal close
packed (hcp) structure and the face centered cubic (fcc)
structure [3]. In order to achieve a close packing in the
frequency domain using an fcc lattice (the reason why not using an hcp
lattice is explained at the end of this section), we use the following
matrix:

(14) 
An fcc lattice consists of simple cubic cells with additional
sampling points in the center of each cell face. One cell of an fcc
lattice is depicted in Fig. 3 with the base vectors from
Eq. 14.
By plugging Eq. 14 in Eq. 4
we end up with a sampling matrix in spatial domain which describes a
body centered cubic (bcc) lattice:

(15) 
with
.
A bcc lattice also consists of a simple cubic cell but with only one
additional sampling point, which is right in the center of the cell.
Fig. 4 depicts one cell of a bcc lattice. Note, that the
base vectors of Fig. 4 do not correspond to
Eq. 15, as these are rather unintuitive. We chose
another set of base vectors, which is more convenient for our purpose
of indexing the data points (see Section 3.1).
They are described by the sampling matrix

(16) 
It is easily verified that the sampling matrices in
Eqs. 15 and 16 describe the same set of
sample points.
To guarantee that the replicas in frequency domain do not overlap,
must be equal to for rectangular sampling. Since the periodicity
matrix is analogous to the 2D periodicity matrix we end up with

(17) 
For the fcc lattice, must be equal to , which yields

(18) 
By again taking the ratio

(19) 
we see that we need 29.3% fewer samples than with rectangular
sampling. This means that if we sample a function on a regular
rectangular grid with sampling distance , we can increase the
sampling distance for a bcc grid to
.
Figure 3:
One cell of an fcc lattice with base vectors . The black
dots mark additional sample points (in the center of the faces)
as compared to a simple cubic cell.

Figure 4:
One cell of a bcc structure with base vectors . The only
difference to a simple cubic cell is one additional sample
point right in the center of the cell, marked with a black dot.

In the above example we started with an fcc lattice in the frequency
domain which resulted in a bcc lattice in the spatial domain. We could
also choose an hcp lattice, since it has the same packing density as
an fcc lattice [3]. However, an hcp lattice in the
frequency domain is also an hcp lattice in the spatial domain and hcp
lattices are rather difficult to index [7]. Therefore, we
prefer to use a bcc grid.
Next: Implementation Details
Up: Optimal Regular Volume Sampling
Previous: Introduction
Thomas Theußl
20010805