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Register a new userMicrolocal inversion of a 3-dimensional restricted transverse ray transform of symmetric $m$-tensor fields
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We study the problem of inverting a restricted transverse ray transform to recover a symmetric $m$-tensor field in $mathbb{R}^3$ using microlocal analysis techniques. More precisely, we prove that a symmetric $m$-tensor field can be recovered up to a known singular term and a smoothing term if its transverse ray transform is known along all lines intersecting a fixed smooth curve satisfying the Kirillov-Tuy condition.
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For an integer $rge0$, we prove the $r$th order Reshetnyak formula for the ray transform of rank $m$ symmetric tensor fields on $mathbb{R}^n$. Certain differential operators $A^{(m,r,l)} (0le lle r)$ on the sphere $mathbb{S}^{n-1}$ are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any $r$ although the volume of calculations grows fast with $r$. The algorithm is realized for small values of $r$ and Reshetnyak formulas of orders $0,1,2$ are presented in an explicit form.
An analysis of the stability of the spindle transform, introduced in (Three dimensional Compton scattering tomography arXiv:1704.03378 [math.FA]), is presented. We do this via a microlocal approach and show that the normal operator for the spindle transform is a type of paired Lagrangian operator with blowdown--blowdown singularities analogous to that of a limited data synthetic aperture radar (SAR) problem studied by Felea et. al. (Microlocal analysis of SAR imaging of a dynamic reflectivity function SIAM 2013). We find that the normal operator for the spindle transform belongs to a class of distibutions $I^{p,l}(Deltacupwidetilde{Delta},Lambda)$ studied by Felea and Marhuenda (Microlocal analysis of SAR imaging of a dynamic reflectivity function SIAM 2013 and Microlocal analysis of some isospectral deformations Trans. Amer. Math.), where $widetilde{Delta}$ is reflection through the origin, and $Lambda$ is associated to a rotation artefact. Later, we derive a filter to reduce the strength of the image artefact and show that it is of convolution type. We also provide simulated reconstructions to show the artefacts produced by $Lambda$ and show how the filter we derived can be applied to reduce the strength of the artefact.
In this paper we study the attenuated $X$-ray transform of 2-tensors supported in strictly convex bounded subsets in the Euclidean plane. We characterize its range and reconstruct all possible 2-tensors yielding identical $X$-ray data. The characterization is in terms of a Hilbert-transform associated with $A$-analytic maps in the sense of Bukhgeim.
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.
In two dimensions, we consider the problem of inversion of the attenuated $X$-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the convex hull of this arc. The attenuation is assumed known. The method of proof uses the Hilbert transform associated with $A$-analytic functions in the sense of Bukhgeim.
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