IPI
A support theorem for the geodesic ray transform of symmetric tensor fields
Venkateswaran P. Krishnan Plamen Stefanov
Inverse Problems & Imaging 2009, 3(3): 453-464 doi: 10.3934/ipi.2009.3.453
Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of such a field $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown that there exists a vector field $v$ satisfying $f=dv$ on the set of points lying on these geodesics and $v=0$ on the intersection of this set with the boundary ∂$ M$ of the manifold $M$. Using this result, a Helgason's type of a support theorem for the geodesic ray transform is proven. The approach is based on analytic microlocal techniques.
keywords: integral geometry support theorem. X ray transform tensors
IPI
The weighted Doppler transform
Sean Holman Plamen Stefanov
Inverse Problems & Imaging 2010, 4(1): 111-130 doi: 10.3934/ipi.2010.4.111
We consider the tomography problem of recovering a covector field on a simple Riemannian manifold based on its weighted Doppler transformation over a family of curves $\Gamma$. This is a generalization of the attenuated Doppler transform. Uniqueness is proven for a generic set of weights and families of curves under a condition on the weight function. This condition is satisfied in particular if the weight function is never zero, and its derivatives along the curves in $\Gamma$ are never zero.
keywords: Integral geometry Pseudodifferential operators. Inverse problems
IPI
Instability of the linearized problem in multiwave tomography of recovery both the source and the speed
Plamen Stefanov Gunther Uhlmann
Inverse Problems & Imaging 2013, 7(4): 1367-1377 doi: 10.3934/ipi.2013.7.1367
In this paper we consider the linearized problem of recovering both the sound speed and the thermal absorption arising in thermoacoustic and photoacoustic tomography. We show that the problem is unstable in any scale of Sobolev spaces.
keywords: source and speed. instability multiwave tomography inverse problem Thermoacoustic tomography
IPI
Weyl asymptotics of the transmission eigenvalues for a constant index of refraction
Ha Pham Plamen Stefanov
Inverse Problems & Imaging 2014, 8(3): 795-810 doi: 10.3934/ipi.2014.8.795
We prove Weyl-type asymptotic formulas for the real and the complex internal transmission eigenvalues when the domain is a ball and the index of refraction is constant.
keywords: Interior transmission problem spectral theory interior transmission eigenvalue scattering. asymptotic distribution of eigenvalues
IPI
Multiwave tomography with reflectors: Landweber's iteration
Plamen Stefanov Yang Yang
Inverse Problems & Imaging 2017, 11(2): 373-401 doi: 10.3934/ipi.2017018

We use the Landweber method for numerical simulations for the multiwave tomography problem with a reflecting boundary and compare it with the averaged time reversal method. We also analyze the rate of convergence and the dependence on the step size for the Landweber iterations on a Hilbert space.

keywords: Thermoacoustic tomography microlocal analysis Landweber iterations medical imaging
IPI
Modulated luminescence tomography
Plamen Stefanov Wenxiang Cong Ge Wang
Inverse Problems & Imaging 2015, 9(2): 579-589 doi: 10.3934/ipi.2015.9.579
We propose and analyze a mathematical model of Modulated Luminescence Tomography. We show that when single X-rays or focused X-rays are used as an excitation, the problem is similar to the inversion of weighted X-ray transforms. In particular, we give an explicit inversion in the case of Dual Cone X-ray excitation.
keywords: medical imaging Xray microlocal analysis modulated tomography.
IPI
Gauge equivalence in stationary radiative transport through media with varying index of refraction
Stephen McDowall Plamen Stefanov Alexandru Tamasan
Inverse Problems & Imaging 2010, 4(1): 151-167 doi: 10.3934/ipi.2010.4.151
Three dimensional anisotropic attenuating and scattering media sharing the same albedo operator have been shown to be related via a gauge transformation. Such transformations define an equivalence relation. We show that the gauge equivalence is also valid in media with non-constant index of refraction, modeled by a Riemannian metric. The two dimensional model is also investigated.
keywords: Inverse transport gauge equivalence.
IPI
On the stable recovery of a metric from the hyperbolic DN map with incomplete data
Plamen Stefanov Gunther Uhlmann Andras Vasy
Inverse Problems & Imaging 2016, 10(4): 1141-1147 doi: 10.3934/ipi.2016035
We show that given two hyperbolic Dirichlet to Neumann maps associated to two Riemannian metrics of a Riemannian manifold with boundary which coincide near the boundary are close then the lens data of the two metrics is the same. As a consequence, we prove uniqueness of recovery a conformal factor (sound speed) locally under some conditions on the latter.
keywords: metric. DN map Inverse problem

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