Mathematical Aspects of Computerized Tomography pp 100-111 | Cite as

# Stability and Consistency for the Divergent Beam X-Ray Transform

Conference paper

## Abstract

Let Ω be a bounded open convex subset of the plane and f be a square integrable function that vanishes outside of Ω , i.e. f ∈ L We assume throughout the paper that a is not in Ω , the closure of Ω.

^{2}(Ω). The divergent beam x-ray transform of f from the source point a in the direction θ is defined by$${D_{\rm{a}}}{\rm{f}}({\rm{\theta }}) = {\rm{ \backslash smallint}}_o^\infty {\rm{f}}({\rm{a + t\theta )dt}}.$$

(1.1)

## Keywords

Compact Operator Consistency Condition Line Passing Finite Union Closed Graph Theorem
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## Bibliography

- 1.K. J. Falconer: Consistency conditions for a finite set of projections of a function. Math. Proc, Cambridge Phil. Soc. 85, (1979), pp. 61–68.CrossRefMATHMathSciNetGoogle Scholar
- 2.E. Gagliardo: On integral transformations with positive kernel. Proc. A.M.S. Vol. 16, No. 3, (1965), pp. 429–434.MathSciNetGoogle Scholar
- 3.J. V. Leahy, K. T. Smith, and D. C. Solmon: Uniqueness, nonuniqueness and inversion in the x-ray and Radon problems, to appear Proc. of the International Symposium on Ill-Posed Problems, Newark, Del. Oct. 1979.Google Scholar
- 4.B. E. Petersen, K. T. Smith, and D. C. Solmon: Sums of plane waves, and the range of the Radon transform. Math. Ann. 243, (1979), pp. 153–161.CrossRefMATHMathSciNetGoogle Scholar
- 5.K. T. Smith: Formulas to represent functions by their derivatives. Math. Ann. 188, (1970), pp. 53–77.CrossRefMathSciNetGoogle Scholar
- 6.L. Svensson. When is the sum of closed subspaces closed? An example arising in computed tomography. Research report, Royal Institute of Technology, Stockholm 1980.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1981