A Method for Taking Cross Sections
211
A Method for Taking Cross Sections
of ThreeDimensional Gridded Data
Kelly Slater Cline
Kacee Jay Giger
Timothy O&Conner
Eastern Oregon University
LaGrande, OR 97850
Advisor: Norris Preyer
Summary
Effective threedimensional magnetic resonance imaging (MRI) requires an
accurate method for taking planar cross sections. However, if an oblique cross
section is taken, the plane may not intersect any known data points. Thus, a
method is needed to interpolate water density between data points.
Interpolation assumes continuity of density, but there are discontinuities in
the human body at the borders of different types of tissue. Most interpolation
methods try to smooth these sharp borders, blurring the data and possibly
destroying useful information.
To capture qualitatively the key dif±culties of this problem, we created a
sequence of simulated biological data sets, such as a brain and an arm, each
with some speci±c defect. Our data sets are cubic arrays with 100 elements on
each side, for a total of one million elements, specifying water density at each
point with an integer in the range
[0
,
255]
. In each data set, we use differentiable
functions to describe several tissue types with discontinuities between them.
To analyze these data, we created a group of algorithms, implemented in
C++, and compared their effectiveness in generating accurate cross sections.
We used local interpolation techniques, because the data are not continuous
on a global level. Our ±nal algorithm searches for discontinuities between
tissues. If it ±nds one at a point, it preserves sharp edges by assigning to that
point the water density of the nearest data point. If there is no discontinuity,
the algorithm does a polynomial ±t in three dimensions to the nearest 64 data
points and interpolates the water density.
The UMAP Journal
19 (3) (1998) 211—221. c
°
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The UMAP Journal 19.3
We measured the accuracy of the algorithms by &nding the mean absolute
difference between the interpolated water density and the actual water density
at each point in the cross sections. Our &nal algorithm has an error 16% lower
than a simple closestpoint technique, 17% lower than a continuous linear inter
polation, and 22% lower than a continuous polynomial interpolation without
discontinuity detection.
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 Spring '11
 Li
 Numerical Analysis, Harshad number, Continuous function, Taking Cross Sections

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