Abstract
In X-ray tomography, the Fourier slice theorem provides a relationship between the Fourier components of the object being imaged and the measured projection data. The Fourier slice theorem is the basis for X-ray Fourier-based tomographic inversion techniques. A similar relationship, referred to as the ‘Fourier shell identity’ has been previously derived for photoacoustic applications. However, this identity relates the pressure wavefield data function and its normal derivative measured on an arbitrary enclosing aperture to the three-dimensional Fourier transform of the enclosed object evaluated on a sphere. Since the normal derivative of pressure is not normally measured, the applicability of the formulation is limited in this form. In this paper, alternative derivations of the Fourier shell identity in 1D, 2D polar and 3D spherical polar coordinates are presented. The presented formulations do not require the normal derivative of pressure, thereby lending the formulas directly adaptable for Fourier based absorber reconstructions.
Highlights
Photoacoustics is the generation of acoustic waves as a consequence of the absorption of light energy by an absorbing material and the subsequent thermo-elastic expansion of the material
Anastasio et al (2007) derived the “Fourier shell identity”, a mathematical relationship between the pressure wavefield data function, its normal derivative measured on an arbitrary aperture that encloses the object and the three-dimensional (3D) Fourier transform of the optical absorption distribution evaluated on concentric spheres
Fourier shell identity Anastasio et al (2007) demonstrated a mathematical relationship between the pressure wavefield data function and its normal derivative measured on an arbitrary aperture that encloses the object and the three-dimensional (3D) Fourier transform of the optical absorption distribution evaluated on concentric spheres
Summary
Photoacoustics is the generation of acoustic waves as a consequence of the absorption of light energy by an absorbing material and the subsequent thermo-elastic expansion of the material. Anastasio et al (2007) derived the “Fourier shell identity”, a mathematical relationship between the pressure wavefield data function, its normal derivative measured on an arbitrary aperture that encloses the object and the three-dimensional (3D) Fourier transform of the optical absorption distribution evaluated on concentric spheres. This relationship can be regarded as the PAT analog of the classic Fourier slice theorem from X-ray tomography. Fourier shell identity Anastasio et al (2007) demonstrated a mathematical relationship between the pressure wavefield data function and its normal derivative measured on an arbitrary aperture that encloses the object and the three-dimensional (3D) Fourier transform of the optical.
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