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## Mathematics and Physics of Emerging Biomedical Imaging (1996) Commission on Physical Sciences, Mathematics, and Applications (CPSMA)

### Citation Manager

. "14 A CROSS-CUTTING LOOK AT THE MATHEMATICS OF EMERGING BIOMEDICAL IMAGING." Mathematics and Physics of Emerging Biomedical Imaging. Washington, DC: The National Academies Press, 1996.

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object, then

(compare to section 3.1.2). In the simplest case the ray L may be thought of as a straight line. Modeling L as a strip or cone, possibly with a weight factor to account for detector inhomogeneities, may be more appropriate. Equation 14.1 neglects the dependence of a with the energy (beam hardening effect) and other nonlinear phenomena (e.g., partial volume effect); see section 3.2.3.

The mathematical problem in transmission tomography is to determine a from measurements of I for a large set of rays L. If L is simply the straight line connecting the source x0 with the detector x1, equation 14.1 gives rise to the integral

where dx is the restriction to L of the Lebesgue measure in Rn. The task is to compute a in a domain.W Í Rn.from the values of equation 14.2 where x0 and x1 run through certain subjects of ¶W.

For n = 2, equation 14.2 is simply a reparametrization of the Radon transform R. The operator R is defined to be

where  is a unit vector in Rn ; i.e., q  Î Sn-1 , and s Î R. Thus a is in principle found through Radon's inversion formula for R,

R* is given explicitly by

and the operator K is given by

where H is the Hilbert transform. In fact the numerical implementation of equation 14.4 leads to the filtered backprojection algorithm, which is the standard algorithm in commercial CT scanners.

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 FRONT MATTER (R1-R18) 1 INTRODUCTION AND SUMMARY (1-12) 2 X-RAY PROJECTION IMAGING (13-22) 3 X-RAY COMPUTED TOMOGRAPHY (23-36) 4 MAGNETIC RESONANCE IMAGING (37-88) 5 SINGLE PHOTON EMISSION COMPUTED TOMOGRAPHY (89-104) 6 POSITRON EMISSION TOMOGRAPHY (105-120) 7 ULTRASONICS (121-132) 8 ELECTRICAL SOURCE IMAGING (133-142) 9 ELECTRICAL IMPEDANCE TOMOGRAPHY (143-146) 10 MAGNETIC SOURCE IMAGING (147-156) 11 MEDICAL OPTICAL IMAGING (157-166) 12 IMAGE-GUIDED MINIMALLY INVASIVE DIAGNOSTIC AND THERAPEUTIC INTERVENTIONAL.. (167-186) 13 FRONTIERS OF IMAGE PROCESSING FOR MEDICINE (187-198) 14 A CROSS-CUTTING LOOK AT THE MATHEMATICS OF EMERGING BIOMEDICAL IMAGING (199-230) INDEX (231-238) Color Plates (239-242)