Singular values of the attenuated photoacoustic imaging operator

Abstract

We analyse the ill-posedness of the photoacoustic imaging problem in the case of an attenuating medium. To this end, we introduce an attenuated photoacoustic operator and determine the asymptotic behaviour of its singular values. Dividing the known attenuation models into strong and weak attenuation classes, we show that for strong attenuation, the singular values of the attenuated photoacoustic operator decay exponentially, and in the weak attenuation case the singular values of the attenuated photoacoustic operator decay with the same rate as the singular values of the non-attenuated photoacoustic operator. arXiv:1611.05807v1 [math.FA] 17 Nov 2016 c 1 +χ(ω) in the electrodynamic model).We will interpret the equation (1.2) as an equation in the space of tempered distributions S (R × R 3 ) so that the Fourier transform and the δ-distribution are both well-defined. To make sense of A κ as an operator on S (R × R 3 ) and to be able to find a solution of (1.2), we impose the following conditions on the function κ. Definition 2.1 We call a non-zero function κ ∈ C ∞ (R; H), where H = {z ∈ C | m z > 0} denotes the upper half complex plane and H its closure in C, an attenuation coefficient if (i) all the derivatives of κ are polynomially bounded. That is, for every ∈ N 0 there exist constants κ 1 > 0 and N ∈ N such that |κ ( ) (ω)| ≤ κ 1 (1 + |ω|) N , ((2.2)) (ii) there exists a holomorphic continuationκ : H → H of κ on the upper half plane, that is,κ ∈ C(H; H) withκ| R = κ andκ : H → H is holomorphic; with |κ(z)| ≤κ 1 (1 + |z|)Ñ for all z ∈ H for some constantsκ 1 > 0 andÑ ∈ N.(iii) we have the symmetry κ(−ω) = −κ(ω) for all ω ∈ R.

Reconstruction formulas for photoacoustic imaging in attenuating media

“…Proof: The first item has been proven in [4]. The second item is an immediate consequence of the definition of a tempered distribution.…”

“…For such attenuation coefficients we proved in [4] well-posedness of the attenuated wave equation: Moreover, q a := I [p a ] is a tempered distribution and satisfies the equation…”

“…Remark: The four conditions in Definition 2.1 on κ encode the following physical properties of the attenuated wave equation (see [4]):…”

“…The literature on Photoacoustics in attenuating media concentrates on time-reversal and attenuation compensation based on power laws: We mention the k-wave toolbox implementation and the according papers [3,24], [2,7]. In [4] several attenuation laws from the literature have been cataloged into two classes, namely weak and strong attenuation laws. Power laws lead, in general, to severely ill-posed photoacoustic imaging problem, while mathematically sophisticatedly derived models, like the Nachman-Smith-Waag model [15], lead to moderately ill-posed problems.…”

Fractional time stepping and adjoint based gradient computation in an inverse problem for a fractionally damped wave equation

“…( 21)], see also [2].) To numerically solve the semidiscrete system (7), in Section 2 we derive and analyze a time stepping scheme, that is based on the Newmark method, equipped with a quadrature formula for the fractional derivative.…”