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Tomographic algorithms have been used to generate cross-sectional images of sound velocity in the human body from time-of-flight measurements of ultrasonic pulses. Similar algorithms have been employed to reconstruct optical refractive-index fields, in which optical path lengths are measured interferometrically rather than transit times. Using a ray propagation model, an ultrasonic transit time or optical path length measurement is proportional to the line integral of the ultrasonic (or optical) refractive index over the ray path. Under the assumption that the propagation paths are straight, conventional computerized tomography (CT) algorithms have been used to perform the reconstructions. In real media, ray refraction introduces a time-of-arrival (or optical path length) error in the measurement, leading to image degradation. To date, only iterative techniques based on numerical ray tracing have been proposed to correct for the effects of refraction. In this paper, we present a perturbation approach to this problem which, for relatively small refractive-index fluctuations, requires neither iteration nor ray tracing. Assuming that the average deviation of the refractive index from its mean is on the order of the small quantity e, an expression is derived for the ray trajectory whose departure from a straight line is first order in E. Using this first-order ray path, we obtain a perturbation expansion of the path integral of the refractive index along the refracted ray and derive a path-length correction of order e2 arising from the deviation of the refracted ray from a straight line. This second order correction can then be applied to refraction-degraded time-of-flight or optical path length measurements before submission to a conventional CT algorithm. The result is an improvement in image quality after correction.
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