Search for: All records

Abstract The forward problem arising in several hybrid imaging modalities can be modeled by the Cauchy problem for the free space wave equation. Solution to this problems describes propagation of a pressure wave, generated by a source supported inside unit sphereS. The datagrepresent the time-dependent values of the pressure on the observation surfaceS. Finding initial pressureffrom the known values ofgconsitutes the inverse problem. The latter is also frequently formulated in terms of the spherical means offwith centers onS. Here we consider a problem of range description of the wave operator mappingfintog. Such a problem was considered before, with datagknown on time interval at least $[0,2]$(assuming the unit speed of sound). Range conditions were also found in terms of spherical means, with radii of integration spheres lying in the range $[0,2]$. However, such data are redundant. We present necessary and sufficient conditions for functiongto be in the range of the wave operator, forggiven on a half-time interval $[0,1]$. This also implies range conditions on spherical means measured for the radii in the range $[0,1]$. 

Abstract Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon transforms of vector fields. As usual, we use the standard Helmholtz decomposition of smooth and fast decreasing vector fields over the whole space. We show that such a decomposition produces potential and solenoidal components decreasing at infinity fast enough to guarantee the existence of the unweighted longitudinal and transversal Radon transforms of these components. It is known that reconstruction of an arbitrary vector field from only longitudinal or only transversal transforms is impossible. However, for the cases when both linearly weighted and unweighted transforms of either one of the types are known, we derive explicit inversion formulas for the full reconstruction of the field. Our interest in the inversion of such transforms stems from a certain inverse problem arising in magnetoacoustoelectric tomography (MAET). The connection between the weighted Radon transforms and MAET is exhibited in the paper. Finally, we demonstrate performance and noise sensitivity of the new inversion formulas in numerical simulations.