Figure 2 - uploaded by Kamran Sadiq
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An even order m-tensor field f is determined by the odd negative angular modes on or above the diagonal k = −n (green region), and the odd negative angular modes (marked red) on the m 2 red lines n + 2k = −(m + 1) for k ≥ 0. All the odd non-positive angular modes on and below the line n + 2k = −(m + 1), and left of the line n = −(m + 1) vanish.
Contexts in source publication
Context 1
... following result is a direct consequence of the algebraic interaction of the range conditions in (2.14), (2.15), (2.16), and (2.17). To illustrate the result of these interactions, let us consider the partition of Z − × Z as in Figure 2: ...
Context 3
... following result is a direct consequence of the algebraic interaction of the range conditions in (2.14), (2.15), (2.16), and (2.17). To illustrate the result of these interactions, let us consider the partition of Z − × Z as in Figure 2: ...
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Citations
... Different from the works above, in here we present a reconstruction method, which recovers f from momenta ray transform in fan-beam coordinates. Specific to the Euclidean plane, our method is based on Bukhgeim's theory of A-analyticity [6], developed in [27,30,29] and extended here to a system of inhomogeneous Bukhgeim-Beltrami equations. In addition, we also consider the case of the attenuated momenta ray transform. ...
We present a reconstruction method which stably recovers some sufficiently smooth, real valued, symmetric tensor fields compactly supported in the Euclidean plane, from knowledge of their non/attenuated momenta ray transform. The reconstruction method extends Bukhgeim's A-analytic theory from an equation to a system.
... Different from the works above, in here we present a reconstruction method, which recovers f from momenta ray transform in fan-beam coordinates. Specific to the Euclidean plane, our method is based on Bukhgeim's theory of A-analyticity [6], developed in [27,30,29] and extended here to a system of inhomogeneous Bukhgeim-Beltrami equations. In addition, we also consider the case of the attenuated momenta ray transform. ...
We present a reconstruction method which stably recovers some sufficiently smooth, real valued, symmetric tensor fields compactly supported in the Euclidean plane, from knowledge of their non/attenuated momenta ray transform. The reconstruction method extends Bukhgeim's $A$-analytic theory from an equation to a system.
... The latter work also includes a numerical implementation. Different from the above referenced works, in here we introduce a new reconstruction method of the full vector field (see the proof of Theorem 1.1), which is based on Bukhgeim's theory of A-analyticity [3] and its extension in [18,21,20]. Numerical results from its implementation are also presented. ...
... An application of Lemma A.2 to (20) estimates the normal derivative of solutions of (9) in terms of their tangential derivative, ...
We introduce an analytic method which stably reconstructs both components of a (sufficiently) smooth, real valued, vector field compactly supported in the plane from knowledge of its Doppler transform and its first moment Doppler transform. The method of proof is constructive. Numerical inversion results indicate robustness of the method.
In this article we characterize the range of the attenuated and non-attenuated X -ray transform of compactly supported symmetric tensor fields in the Euclidean plane. The characterization is in terms of a Hilbert-transform associated with A -analytic maps in the sense of Bukhgeim.
We define the algebraic range AR(X) of the X-ray transform of symmetric tensors via the algebraic constraints of the Fourier coefficients on the lattice \(\mathbb {Z}\times \mathbb {Z}\), recently introduced by the authors. The algebraic range is the \(L^2\)-closure of the range of the X-ray transform of smooth tensors of compact support. Orthogonal projection of the data on AR(X) reduces the noise by annihilating its orthogonal component. In numerical experiments for 0-order tensors, we illustrate the effect of inverting the X-ray transform from such projections.KeywordsX-ray transformRadon transformFan-beam coordinatesSymmetric tensorsGeneralized Gelfand–Graev–Helgason–Ludwig moment conditionsNoise reduction
