Photoacoustic image reconstruction often assumes that the restriction of the acoustic pressure on the detection surface is given. However, commonly used detectors often have a certain directivity and frequency dependence, in which case the measured data are more accurately described as a linear combination of the acoustic pressure and its normal derivative on the detection surface. In this paper, we consider the inverse source problem for data that are a combination of the acoustic pressure and its normal derivative. For the special case of a spherical detection geometry we derive exact frequency domain reconstruction formulas. We present numerical results showing the robustness and validity of the derived formulas. Moreover, we compare several different combinations of the pressure and its normal derivative showing that used measurement model significantly affects the recovered initial pressure.

Deep neural networks have shown very successful performance in a wide range of tasks, but a theory of why they work so well is in the early stage. Recently, the expressive power of neural networks, important for understanding deep learning, has received considerable attention. Classic results, provided by Cybenko, Barron, etc., state that a network with a single hidden layer and suitable activation functions is a universal approximator. A few years ago, one started to study how width affects the expressiveness of neural networks, i.e., a universal approximation theorem for a deep neural network with a Rectified Linear Unit (ReLU) activation function and bounded width. Here, we show how any continuous function on a compact set of ${\mathbb{R}}^{n_{in}},n_{in}\in \mathbb{N}$ can be approximated by a ReLU network having hidden layers with at most $n_{in}+5$ nodes in view of an approximate identity.

Several types of conical Radon transforms have been studied since the introduction of the Compton camera. Several factors of a cone of integration can be considered as variables, for example, a vertex, a central axis, and an opening angle. In this paper, we study the conical Radon transform with a fixed central axis and opening angle. Furthermore, we consider the attenuation effect in the conical Radon transform because it allows us to obtain a high-quality reconstruction image. We construct a nonseparable Hilbert space and its maximal orthonormal set. This maximal orthonormal set comprises the eigenfunctions of the attenuated conical Radon transform, i.e. singular value decomposition (SVD). Finally, the inversion formula of the attenuated conical Radon transform is deduced from the SVD.

A spherical Radon transform that averages a function over all spheres centered on a given sphere is related to not only pure but also applied mathematics topics. Especially, the problem of inverting this spherical Radon transform arising in photoacoustic tomography and sonar has been studied in many articles. We provide two series formulas for inversion: one does not require the compact support of the source function unlike most inversion formulas and the other is singular value decomposition type, i.e., the series formula is based on the complete orthogonal system. Last, we discuss its continuity.