paper1 - A Method for Taking Cross Sections 211 A Method...

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A Method for Taking Cross Sections 211 A Method for Taking Cross Sections of Three-Dimensional Gridded Data Kelly Slater Cline Kacee Jay Giger Timothy O&Conner Eastern Oregon University LaGrande, OR 97850 Advisor: Norris Preyer Summary Effective three-dimensional 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 ° Copyright 1998 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro±t or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.
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212 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 closest-point technique, 17% lower than a continuous linear inter- polation, and 22% lower than a continuous polynomial interpolation without discontinuity detection.
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This note was uploaded on 01/16/2012 for the course MAD 4103 taught by Professor Li during the Spring '11 term at University of Central Florida.

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paper1 - A Method for Taking Cross Sections 211 A Method...

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