Article

# A Reconstruction Algorithm for Photoacoustic Imaging Based on the Nonuniform FFT

Department of Mathematics, University Innsbruck, 6020 Innsbruck, Austria.
(Impact Factor: 3.54). 11/2009; 28(11):1727-35. DOI: 10.1109/TMI.2009.2022623
Source: arXiv

ABSTRACT

Fourier reconstruction algorithms significantly outperform conventional backprojection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction.

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Available from: Markus Haltmeier, Apr 25, 2014
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• "Using such algorithms for reconstructing a 3D phantom at a reasonable resolution would require several hours, which is significantly slower than our implementations. In combination with existing Fourier algorithms for the spherical mean Radon transform with linear and circular center sets (see [30] [36] [44] [46]) our decomposition approach would even yield O(N log N) algorithms for inverting the spherical mean Radon transform with centers on a circular cylinder in R 3 . "
##### Article: The spherical mean Radon transform with centers on cylindrical surfaces
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ABSTRACT: The spherical mean Radon transform maps a function in an $n$-dimensional space to its integrals (or averages) over spherical surfaces. Recovering a function from its spherical mean Radon transform with centers of spheres of integration restricted to some hypersurface is at the heart of several modern imaging technologies, including SONAR, ultrasound imaging, and photo- and thermoacoustic tomography. In this paper we study the inversion of the spherical mean Radon transform with centers restricted to a cylindrical domain of the form $B \times \mathbb{R}^{n_2}$, where $B$ is any bounded domain in $\mathbb{R}^{n_1}$. As our main results we derive explicit inversion formulas of the backprojection type for that transform. For that purpose we show that the spherical mean Radon transform with centers on a cylindrical surface can be decomposed into two lower dimensional spherical mean Radon transforms. For a cylindrical center set in $\mathbb{R}^3$, we demonstrate that the derived inversion formulas can be implemented by filtered-backprojection type algorithms only requiring $\mathcal O(N^{4/3})$ floating point operations, where $N$ is the total number of unknowns to be recovered. We finally present some results of our numerical simulations performed in Matlab, showing that our implementations accurately recover a 3D image of millions of unknowns in a few minutes on an ordinary notebook.
Preview · Article · Jan 2015
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• "Unfortunately, this kind of interpolation seems to be sub-optimal for Fourier-interpolation with respect to both accuracy and computational costs [7] [25] A regularized inverse k-space interpolation has already been shown to yield better reconstruction results [10]. The superiority of applying the NUFFT, compared to linear interpolation, has been shown theoretically and computationally by [9] "
##### Article: Computational Realization of a Non-Equidistant Grid Sampling in Photoacoustics with a Non-Uniform FFT
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ABSTRACT: To obtain the initial pressure from the collected data on a planar sensor arrangement in Photoacoustic tomography, there exists an exact analytic frequency domain reconstruction formula. An efficient realization of this formula needs to cope with the evaluation of the datas Fourier transform on a non-equispaced mesh. In this paper, we use the non-uniform fast Fourier transform to handle this issue and show its feasibility in 3D experiments. This is done in comparison to the standard approach that uses polynomial interpolation. Moreover, we investigate the effect and the utility of flexible sensor location on the quality of photoacoustic image reconstruction. The computational realization is accomplished by the use of a multi-dimensional non-uniform fast Fourier algorithm, where non-uniform data sampling is performed both in frequency and spatial domain. We show that with appropriate sampling the imaging quality can be significantly improved. Reconstructions with synthetic and real data show the superiority of this method.
Full-text · Article · Jan 2015
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• "In the recent years, many reconstruction techniques for the wave inversion and the inversion from circular means (or spherical means in higher dimensions) have been developed. These techniques can be classified in iterative reconstruction methods (see [4] [7] [8] [9] [10]), model based time reversal (see [11] [12] [13] [14]), Fourier domain algorithms (see [6] [15] [16] [17] [18] [19]), and algorithms based on explicit reconstruction formulas of the back-projection type (see [5] [12] [20] [21] [22] [23] [24] [25] [26] [27]). The back-projection approach is particularly appealing since it is theoretically exact, stable with respect to data and modeling imperfections, mathematically elegant, and quite straightforward to implement numerically. "
##### Article: Inversion of circular means and the wave equation on convex planardomains
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ABSTRACT: We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary $\partial \Om$ of a smooth convex bounded domain $\Om \subset \R^2$. As a main result we establish back-projection type inversion formulas that recover any initial data with support in $\Om$ modulo an explicitly computed smoothing integral operator $\K_\Om$. For circular and elliptical domains the operator $\K_\Om$ is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on $\partial \Om$. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography.
Full-text · Article · Apr 2013 · Computers & Mathematics with Applications