Abstract
We present a method for the recovery of compressively sensed acoustic fields using patterned, instead of point-by-point, detection. From a limited number of such compressed measurements, we propose to reconstruct the field on the sensor plane in each time step independently assuming its sparsity in a Curvelet frame. A modification of the Curvelet frame is proposed to account for the smoothing effects of data acquisition and motivated by a frequency domain model for photoacoustic tomography. An ADMM type algorithm, split augmented Lagrangian shrinkage algorithm, is used to recover the pointwise data in each individual time step from the patterned measurements. For photoacoustic applications, the photoacoustic image of the initial pressure is reconstructed using time reversal in $ {\mathbf k}$ -Wave Toolbox.
Highlights
C OMPRESSED Sensing (CS) is a new measurement paradigm, which allows for the reconstruction of sparse signals sampled at sub-Nyquist rates
We discuss the motivation for using a Curvelet tranform as the sparsifying transformation for the acoustic field at the detector and we propose its modification: a low-frequency Curvelet tranform tailored to the frequency range of the acoustic field on the detector
An appropriate multiscale representation of the time series photoacoustic tomography (PAT) data is considered in Section IV, where we derive the frequency model of sensor data and propose a modified version of Curvelet transform tailored to the range of frequencies of the acoustic field
Summary
C OMPRESSED Sensing (CS) is a new measurement paradigm, which allows for the reconstruction of sparse signals sampled at sub-Nyquist rates. It is common understanding that many digital signals and images admit an adequate representation with far fewer coefficients than their actual length. This phenomena is known as compressibility and it has been a driving force in many image processing applications, most notably the image compression algorithms JPEG and its successor JPEG 2000. As the Curvelets essentially describe the wave front sets their propagation is well approximated through geometrical optics (high frequency asymptotic solution to the wave equation). Of the corresponding Hamilton-Jacobi equation resulting from the high frequency asymptotic Motivated by this result we investigate the Curvelet representation of the cross-section of the wave field by the planar ultrasound sensor. While the arguments in [5] do not directly apply to this situation, the planar cross-section through the acoustic wave front constitutes a singularity along a smooth curve for which Curvelets have been demonstrated to be a nearly optimal representation [6]
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