Donald C. Solmon

Practical and mathematical aspects of the problem of reconstructing objects from radiographs

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A note on k-plane integral transforms

Abstract Let Π be a k -dimensional subspace of R n , n ⩾ 2, and write x = ( x ′, x ″) with x ′ in Π and x ″ in the orthogonal complement Π ⊥ . The k -plane transform of a measurable function ƒ in the direction Π at the point x ″ is defined by Lƒ(Π, x″) = ∝Πƒ(x′, x″) dx′ . In this article certain a priori inequalities are established which show in particular that if ƒ ϵ L p (R n ) , 1 ⩽ p

Filtered-backprojection and the exponential Radon transform

n k , then ƒ is integrable over almost every translate of almost every k -space. Mapping properties of the k -plane transform between the spaces L p ( R n ), p ⩽ 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k -spaces in R n are also obtained.

A Characterization of the Range of the Divergent Beam x-Ray Transform

On etudie des conditions sur la mesure μ qui assurent la validite de #7B-R t (#7B-Rf*K)=f*E, ou E=#7B-R t K pour une transformee de Radon generalisee #7B-Rμ et une grande classe de filtres K correspondant a des fonctions delta approchees E

Addendum to “Practical and mathematical aspects of the problem of reconstructing objects from radiographs”

In this paper we give a characterization of the range of the divergent beam x-ray transform when the source set is a sphere. The result is analogous to the theorem of Helgason [Acta. Math., 113 (1965), pp. 153–180] and Ludwig [Comm. Pure Appl. Math., 69 (1966), pp. 49–81] on the range of the Radon transform.

The Diagnosis of Breast Cancer in Mammograms by the Evaluation of Density Patterns

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Sums of homogeneous functions and the range of the divergent beam x-ray transform

Standard mammograms from 33 patients with surgically proved adenocarcinoma or fibrocystic disease were analyzed with a scanning microdensitometer and computer. A quickly computable number called the linear mass ratio is introduced. This simple ratio discriminated correctly between the 16 adenocarcinomas and 17 fibrocystic lesions of the study, all cases in which diagnosis had required biopsy.

Stability and Consistency for the Divergent Beam X-Ray Transform

In this paper we analyze the divergent beam x-ray transform (and generalizations) with finite source set in as an operator between Lρspaces. The main results give conditions for this operator to have closed range when n = 2, 3 and give a characterization of the range. The dual result asserts closure in Lρ of sums of spaces of functions homogeneous of given degree from the several sources.

Iterated Products of Projections in Hilbert Space

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 ∈ L2 (Ω). The divergent beam x-ray transform of f from the source point a in the direction θ is defined by