Biomedical Engineering Reference
In-Depth Information
3.5.1
A Consideration of Sampling and Interpolation Theory
The origins of the sampling theorem remain a matter of debate. According to
conventional wisdom, Shannon showed in that a band-limited signal
sampled with an infinite periodic sampling function can be perfectly interpo-
lated using the sinc function interpolant previously proposed by the mathe-
matician Whittaker.
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According to Butzer and Stens,
50
Shannon had first
written down this theorem in 1940, but the Russian Kotel'nikov had discov-
ered it independently in 1933. Furthermore, it appears that Ogura had formu-
lated the theorem even earlier, in 1920 in a Japanese publication, and
erroneously attributed his result to Whittaker.
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Many medical images are,
however, not band limited. For example, multislice datasets are not band lim-
ited in the through-slice direction, as the field of view is truncated with a top
hat function. Even in MR image volumes reconstructed using a 3D Fourier
transform, the condition is not usually satisfied because the image data pro-
vided by the scanner are often truncated to remove slices at the periphery of
the field of view. Also, data provided are modulus, and taking the modulus
is a nonlinear operation that can increase the spatial frequency content.*
Even if the images being transformed were band limited, it would not be
possible to carry out perfect interpolation using a sinc function, because a
sinc function is infinite in extent.
For many purposes, this problem is entirely ignored during medical
image analysis. The most widely used image interpolation function is prob-
ably trilinear interpolation, in which a voxel value in the transformed coor-
dinates is estimated by taking a weighted average of the nearest eight
neighbors in the original dataset. The weightings, which add up to one, are
inversely proportional to the distance of each neighbor from the new sample
point. For accurate comparison of registered images, for example by sub-
tracting one image from another, the errors introduced by trilinear interpo-
lation become important. It can be shown that trilinear interpolation applies
a low-pass filter to the image and introduces aliasing.
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For transformations
that contain rotations, the amount of low-pass filtering varies with position
in the image. If subtracting one image from another to detect small change,
for example in serial MR imaging, the low-pass filtering in this process can
lead to substantial artifacts. Subtracting a low-pass filtered version of an
image from the original is a well known edge enhancement method, so even
in the case of identical images differing only by a rigid-body transformation,
using linear interpolation followed by subtraction does not result in the
expected null result, but instead results in an edge-enhanced version of the
original.
Hajnal recently brought this issue to the attention of the MR image analysis
community
31
and proposed that the solution is to interpolate using a sinc func-
tion truncated with a suitable window function such as a Hamming window.
1949
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* According to Butzer
50
and Unser,
51
more general versions of the sampling theory for functions
that are not necessarily band limited were published even earlier than the theorem for the classic
band-limited case; in 1927 by Whittaker, and even in 1908 by de la
Vallee
Poussin.
