In X-ray tomography, the Fourier slice theorem provides a relationship between the\nFourier components of the object being imaged and the measured projection data.\nThe Fourier slice theorem is the basis for X-ray Fourier-based tomographic inversion\ntechniques. A similar relationship, referred to as the ââ?¬Ë?Fourier shell identityââ?¬â?¢ has been\npreviously derived for photoacoustic applications. However, this identity relates the\npressure wavefield data function and its normal derivative measured on an arbitrary\nenclosing aperture to the three-dimensional Fourier transform of the enclosed object\nevaluated on a sphere. Since the normal derivative of pressure is not normally measured,\nthe applicability of the formulation is limited in this form. In this paper, alternative\nderivations of the Fourier shell identity in 1D, 2D polar and 3D spherical polar coordinates\nare presented. The presented formulations do not require the normal derivative\nof pressure, thereby lending the formulas directly adaptable for Fourier based absorber\nreconstructions.
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