Photoacoustic tomography with a single detector in a reverberant cavity
A rectangular reverberant cavity with a single detector at point and a point source.
Image sources and detectors. The shaded region is the actual rectangular cavity; the other rectangles are images. (a) Image sources. Those within a circle of radius have contributed to the signal measured at the detector by time , where is the sound speed. Rays are shown connecting the image sources to the omnidirectional detector, which records the times of arrivals of the pulses but not their directions. (b) Image detectors. These can be useful when reconstructing the position of a point source. The circles illustrate the backprojection of three impulses received at the detector for three of the image sources, showing that they coincide at the source point.
The first 0.25 ms of the train of pulses received by the detector as calculated using image sources. The amplitude shows cylindrical spreading dependence (sound speed ). On three occasions two pulses arrive at the detector simultaneously, hence the three pulses with larger amplitude.
Backprojecting the train of pulses from the nearest 500 image detectors shows that the point source can be located. The detector is located at the corner (0 mm, 20 mm) and the source is at (8 mm, 7 mm).
The first part of a photoacoustic time series simulated from the initial pressure distribution in Fig. 6(A) for the point (0.03 mm, 19.96 mm).
(A) Circular initial pressure distribution used to generate the time series in Fig. 5. (B) Image formed by backprojecting the reverberant time series from the 500 image detector positions nearest to the actual detector. The circular initial pressure distribution has been recovered, but the image suffers from artifacts and noise.
(A) Initial pressure distribution, , consisting of nine small circles with Gaussian profiles. (B) Image of obtained by backprojection using Eq. (2). (C) Image of obtained by a modal reconstruction in a rectangular cavity. (D) Image of obtained by a modal reconstruction in a chaotic cavity. (E) Profiles at through (A), (B), (C), and (D), corresponding to the exact (solid line), backprojection (dashed), modal reconstruction with rectangular cavity (dot-dashed), and chaotic cavity (dotted). (F) Mean squared error as a function of truncation order for the images from rectangular and chaotic (○) cavities.
Singular values of matrix for the rectangular cavity.
Histograms showing the distributions of the spaces between the lowest 4000 modal frequencies as a function of normalized modal spacing, (modal spacing/number of modes included). (a) For a rectangular cavity, the distribution agrees closely with Poisson’s distribution (solid line). (b) For a chaotic cavity (a quarter Sinai billiard), the distribution agrees closely with Wigner’s distribution (solid line).
Singular values of matrix for rectangular and chaotic cavities.
(A) Example initial pressure distribution, . (B) Image of obtained using a rectangular cavity. (C) Image of obtained using a chaotic cavity. (D) Profiles at through (A), (B), and (C), corresponding to the exact (solid line), reconstructed using modal reconstruction with rectangular cavity (dashed), and reconstruction with chaotic cavity (dotted).
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