A fast algorithm to compute the -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

DOI:
https://doi.org/10.1090/S0002-9939-06-08449-8

Published electronically:
July 28, 2006

MathSciNet review:
2280195

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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 -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 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.**A. Katsevich,*Improved Exact FBP Algorithm for Spiral CT*, Advances in Applied Mathematics,**32**, 2004, pp. 681-697. MR**2053840 (2005b:44002)****5.**J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, 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:
https://doi.org/10.1090/S0002-9939-06-08449-8

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