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.

In this article, we provide a method to improve the depth resolution of wide-field depth-resolved wavenumber-scanning interferometry (DRWSI), because its depth resolution is limited by the range of the wavenumber scanning and mode hopping of the light source. An optical wedge is put into the optical path to measure the series of the wavenumber on time using a 2D spatial Fourier transform (FT) of the interferograms. Those uncorrelated multiple bands of the wavenumbers due to mode hopping of the diode laser can be synthesized into one band, to enlarge the range of the wavenumber scanning. A random-sampling FT is put forward to evaluate the distribution of frequencies and phases of the multiple surfaces measured. The benefit is that the depth resolution of the DRWSI is enhanced significantly with a higher signal-to-noise ratio. Because of its simplicity and practicability, this method broadens the way to employing multiple different lasers or lasers with mode hopping as the light sources in the DRWSI.

This paper aims at imaging the dynamics of metabolic activity of cells. Using dynamic optical coherence tomography, we introduce a new multiparticle dynamical model to simulate the movements of the collagen and the cell metabolic activity and develop an efficient signal separation technique for sub-cellular imaging. We perform a singular-value decomposition of the dynamic optical images to isolate the intensity of the metabolic activity. We prove that the largest eigenvalue of the associated Casorati matrix corresponds to the collagen. We present several numerical simulations to illustrate and validate our approach.Mathematics Subject Classification (MSC2000): 92C55, 78A46, 65Z05

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In this paper we study the problem of photoacoustic inversion in a weakly attenuating medium. We present explicit reconstruction formulas in such media and show that the inversion based on such formulas is moderately ill-posed. Moreover, we present a numerical algorithm for imaging and demonstrate in numerical experiments the feasibility of this approach. arXiv:1705.07466v1 [math.NA] 21 May 2017 R, appearing as a source term in the wave equationfrom some measurements over time of the pressure p on a two-dimensional manifold Γ outside of the specimen, that is outside of the support of the absorption density function. This problem has been studied extensively in the literature (see e.g. [14,25,13], to mention just a few survey articles). Biological tissue has a non-vanishing viscosity, thus there is thermal consumption of energy. These effects can be described mathematically by attenuation. Common models of such are the thermo-viscous model [9], its modification [11], Szabo's power law [22, 21] and a causal modification [10], Hanyga & Seredy'nska [5], Sushilov & Cobbold [20], and the Nachman-Smith-Waag model [15]. Photoacoustic imaging in attenuating medium then consists in computing the absorption density function h from measurements of the attenuated pressure p a on a surface containing the object of interest. The attenuated pressure equation reads as follows