A fast algorithm to compute the line through points inside a helix cylinder
Author:
Steven H. Izen
Journal:
Proc. Amer. Math. Soc. 135 (2007), 269276
MSC (2000):
Primary 65H05; Secondary 51N05, 65R10
Published electronically:
July 28, 2006
MathSciNet review:
2280195
Fulltext PDF Free Access
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Abstract: In the context of helical conebeam CT, Danielsson et al. discovered that for each point interior to the cylindrical surface containing a given helix, there is exactly one line segment passing through the point which intersects two points less than one turn apart on the helix. This segment is called a line. A new constructive algebraic proof of this result is presented along with a fast algorithm to compute the endpoints of the line through an arbitrary point in the interior of the helix cylinder. This proof exposes the geometry of the decomposition of a cylinder interior as a disjoint union of lines.
 1.
P.E. Danielsson, P. Edholm, J. Eriksson, and M. Seger Magnusson, Toward exact reconstruction for helical conebeam scanning of long objects. A new detector arrangement and a new completeness condition, Proc. 1997 Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, ed. D. W. Townsend and P. E. Kinahan, 1997, pp. 141144.
 2.
H. Turbell and P.E. Danielsson, An improved PImethod for reconstruction from helical conebeam projections, IEEE Nuclear Science Symposium, Conference Record, 2, 1999, pp. 865868.
 3.
M. Defrise, F. Noo, and H. Kudo, A solution to the longobject problem in helical conebeam tomography, Phys. Med. Biol. 45, 2000, pp. 623643.
 4.
Alexander
Katsevich, An improved exact filtered backprojection algorithm for
spiral computed tomography, Adv. in Appl. Math. 32
(2004), no. 4, 681–697. MR 2053840
(2005b:44002), http://dx.doi.org/10.1016/S01968858(03)00099X
 5.
J.
Stoer and R.
Bulirsch, Introduction to numerical analysis, SpringerVerlag,
New YorkHeidelberg, 1980. Translated from the German by R. Bartels, W.
Gautschi and C. Witzgall. MR 557543
(83d:65002)
 1.
 P.E. Danielsson, P. Edholm, J. Eriksson, and M. Seger Magnusson, Toward exact reconstruction for helical conebeam scanning of long objects. A new detector arrangement and a new completeness condition, Proc. 1997 Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, ed. D. W. Townsend and P. E. Kinahan, 1997, pp. 141144.
 2.
 H. Turbell and P.E. Danielsson, An improved PImethod for reconstruction from helical conebeam projections, IEEE Nuclear Science Symposium, Conference Record, 2, 1999, pp. 865868.
 3.
 M. Defrise, F. Noo, and H. Kudo, A solution to the longobject problem in helical conebeam tomography, Phys. Med. Biol. 45, 2000, pp. 623643.
 4.
 A. Katsevich, Improved Exact FBP Algorithm for Spiral CT, Advances in Applied Mathematics, 32, 2004, pp. 681697. MR 2053840 (2005b:44002)
 5.
 J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, SpringerVerlag, New York, 1980. MR 0557543 (83d:65002)
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Additional Information
Steven H. Izen
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
Email:
shi@cwru.edu
DOI:
http://dx.doi.org/10.1090/S0002993906084498
PII:
S 00029939(06)084498
Received by editor(s):
February 26, 2004
Received by editor(s) in revised form:
July 25, 2005
Published electronically:
July 28, 2006
Communicated by:
M. Gregory Forest
Article copyright:
© Copyright 2006
American Mathematical Society
