We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In $mathbb{R}^3$ it maps a function to its surface integrals over circular cones, and in $mathbb{R}^2$ it maps a function to its integrals along tw
o rays with a common vertex. Such transforms appear in various mathematical models arising in medical imaging, nuclear industry and homeland security. This paper contains new results about inversion of conical Radon transform with fixed opening angle and vertical central axis in $mathbb{R}^2$ and $mathbb{R}^3$. New simple explicit inversion formulae are presented in these cases. Numerical simulations were performed to demonstrate the efficiency of the suggested algorithm in 2D.
Exterior inverse problem for the circular means transform (CMT) arises in the intravascular photoacoustic imaging (IVPA), in the intravascular ultrasound imaging (IVUS), as well as in radar and sonar. The reduction of the IPVA to the CMT is quite str
aightforward. As shown in the paper, in IVUS the circular means can be recovered from measurements by solving a certain Volterra integral equation. Thus, a tomography reconstruction in both modalities requires solving the exterior problem for the CMT. Numerical solution of this problem usually is not attempted due to the presence of invisible wavefronts, which results in severe instability of the reconstruction. The novel inversion algorithm proposed in this paper yields a stable partial reconstruction: it reproduces the visible part of the image and blurs the invisible part. If the image contains little or no invisible wavefronts (as frequently happens in the IVPA and IVUS) the reconstruction is quantitatively accurate. The presented numerical simulations demonstrate the feasibility of tomography-like reconstruction in these modalities.
In this article, we consider a generalized Radon transform that comes up in ultrasound reflection tomography. In our model, the ultrasound emitter and receiver move at a constant distance apart along a circle. We analyze the microlocal properties of
the transform $R$ that arises from this model. As a consequence, we show that for distributions with support sufficiently inside the circle, $R^*R$ is an elliptic pseudodifferential operator of order $-1$ and hence all the singularities of such distributions can be recovered.
