Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A fast algorithm to compute the $ \pi$-line through points inside a helix cylinder


Author: Steven H. Izen
Journal: Proc. Amer. Math. Soc. 135 (2007), 269-276
MSC (2000): Primary 65H05; Secondary 51N05, 65R10
Published electronically: July 28, 2006
MathSciNet review: 2280195
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the context of helical cone-beam 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 $ \pi$-line. A new constructive algebraic proof of this result is presented along with a fast algorithm to compute the endpoints of the $ \pi$-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 $ \pi$-lines.


References [Enhancements On Off] (What's this?)

  • 1. P.E. Danielsson, P. Edholm, J. Eriksson, and M. Seger Magnusson, Toward exact reconstruction for helical cone-beam 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. 141-144.
  • 2. H. Turbell and P.E. Danielsson, An improved PI-method for reconstruction from helical cone-beam projections, IEEE Nuclear Science Symposium, Conference Record, 2, 1999, pp. 865-868.
  • 3. M. Defrise, F. Noo, and H. Kudo, A solution to the long-object problem in helical cone-beam tomography, Phys. Med. Biol. 45, 2000, pp. 623-643.
  • 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/S0196-8858(03)00099-X
  • 5. J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR 557543 (83d:65002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 65H05, 51N05, 65R10

Retrieve articles in all journals with MSC (2000): 65H05, 51N05, 65R10


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/S0002-9939-06-08449-8
PII: S 0002-9939(06)08449-8
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