The Arccos and Lommel Diffraction an Approximate Fourier Transform Pair Leading to a Unified Theory of Spatially Averaged Diffraction Corrections for Ultrasonic Piston
نویسنده
چکیده
This work has three goals. The first is to derive closed-form spatially averaged two-way diffraction corrections for focused piston transducers operating in pulsed mode. The goal is attained by (i) establishing that the arccos and Lommel diffraction formulations form an approximate Fourier transform pair and (ii) exploiting this newly established Fourier equivalence in rigorous and original derivations of spatially averaged diffraction corrections for one-way and two-way diffraction. In addition to the first goal, this research has two larger goals. The first is to develop a unified theory of spatially averaged diffraction corrections for piston transducers. The second is to advance the scientific community's understanding of diffraction from a circular aperture, a ubiquitous and fundamental physical problem. When viewed in its entirety, this research attains these larger goals by presenting rigorous derivations and original analyses that unify and extend existing theory with generalized results and fresh insights. Finally, an aspect of the unified theory is tested in a practical application. Specifically, two-way diffraction corrections are applied to ultrasonic data obtained from laboratory experiments. The diffraction corrections are implemented as time-varying filters using a short-time Fourier technique known as the weighted overlap-add method. Raw and diffraction-corrected RF data are quantitatively compared via spectral centroids, and B-mode images are reconstructed from diffraction-corrected data.
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