An algorithm for total variation regularized photoacoustic imaging
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An algorithm for total variation regularized photoacoustic imaging Yiqiu Dong · Torsten G¨orner · Stefan Kunis
Received: 20 February 2013 / Accepted: 12 May 2014 © Springer Science+Business Media New York 2014
Abstract Recovery of image data from photoacoustic measurements asks for the inversion of the spherical mean value operator. In contrast to direct inversion methods for specific geometries, we consider a semismooth Newton scheme to solve a total variation regularized least squares problem. During the iteration, each matrix vector multiplication is realized in an efficient way using a recently proposed spectral discretization of the spherical mean value operator. All theoretical results are illustrated by numerical experiments. Keywords Spherical mean operator · Fast Fourier transform · Total variation regularization · Photoacoustic imaging Mathematics Subject Classifications (2010) 65T50 · 44A12 · 92C55
Communicated by: Leslie Greengard Y. Dong Department of Applied Mathematics and Computer Science, Technical University of Denmark, Copenhagen, Denmark e-mail: [email protected] T. G¨orner · S. Kunis () Institute of Mathematics, University Osnabr¨uck, Osnabr¨uck, Germany e-mail: [email protected] T. G¨orner e-mail: [email protected] S. Kunis Institute of Computational Biology, Helmholtz Zentrum M¨unchen, M¨unchen, Germany
Y. Dong et al.
1 Introduction Analogously to the inversion of the Radon transform in computerized tomography, recovering a function from its mean values over a family of spheres is the crucial ingredient in photoacoustic imaging [6, 25, 40]. The recovery of a function from such spherical means has been studied recently in [1, 3, 11, 25, 36] and references therein. In all practical applications the so called center points are located on a fixed measurement curve or surface and for specific geometries, direct reconstruction algorithms are discussed in [2, 4, 10, 13, 17, 18, 27–29]. Generalizations to integrating detectors and to variable speed of sound, having no direct relation to the spherical mean value operator, are studied in [7, 16, 19, 22, 33, 35, 39, 42, 43], respectively. In this paper we consider constant speed of sound and center points located on an arbitrary curve or surface. For this situation, the reconstruction problem is known to be ill-posed, see e.g. [25, 32] for a detailed discussion, and it has been pointed out recently that direct reconstruction formulae are out of reach, cf. [14, 15, 30]. We regularize the original problem by a total variation (TV) term with the aim of preserving significant edges in the reconstructed images [37]. We set up an iterative method to solve the total variation regularized least squares problem based on the Fenchel-Rockafellar-duality and inexact semismooth Newton techniques following the approach in [21]. In each iteration, we apply the recently proposed algorithm [12] for the fast and accurate computation of spherical means. The structure of our paper is as follows. Section 2 reviews the Cauchy problem for the wave equation and it
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