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A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation
Momentum ray transforms
1. | TIFR Centre for Applicable Mathematics, Sharada Nagar, Chikkabommasandra, Yelahanka New Town, Bangalore, India |
2. | Sobolev Institute of Mathematics; 4 Koptyug Avenue, Novosibirsk, 630090, Russia |
3. | Novosibirsk State University, 2 Pirogov street, 630090, Russia |
The momentum ray transform $ I^k $ integrates a rank $ m $ symmetric tensor field $ f $ over lines in $ \mathbb{R}^n $ with the weight $ t^k $: $ (I^k\!f)(x,\xi) = \int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\, \mathrm{d} t. $ In particular, the ray transform $ I = I^0 $ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $ f $ from the data $ (I^0\!f,I^1\!f,\dots, I^m\!f) $. In the cases of $ m = 1 $ and $ m = 2 $, we derive the Reshetnyak formula that expresses $ \|f\|_{H^s_t({\mathbb R}^n)} $ through some norm of $ (I^0\!f,I^1\!f,\dots, I^m\!f) $. The $ H^{s}_{t} $-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.
References:
[1] |
A. Abhishek and R. K. Mishra, Support theorems and an injectivity result for integral moments of a symmetric m-tensor field, https://arXiv.org/abs/1704.02010, Journal of Fourier Analysis and Applications, 2018, 1–26.
doi: 10.1007/s00041-018-09649-7. |
[2] |
P. Kuchment and F. Terizoglu,
Inversion of weighted divergent beam and cone transforms, Inverse Problems and Imaging, 11 (2017), 1071-1090.
doi: 10.3934/ipi.2017049. |
[3] |
W. Lionheart and V. A. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, Ed. Habib Ammari and Hyeonbae Kang, Contemporary Mathematics, 494 (2009), 137–159.
doi: 10.1090/conm/494/09648. |
[4] |
V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series. VSP, Utrecht, 1994.
doi: 10.1515/9783110900095. |
[5] |
V. A. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002, 20pp.
doi: 10.1088/1361-6420/33/2/025002. |
[6] |
V. A. Sharafutdinov and J.-N. Wang, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp.
doi: 10.1088/0266-5611/28/6/065017. |
show all references
References:
[1] |
A. Abhishek and R. K. Mishra, Support theorems and an injectivity result for integral moments of a symmetric m-tensor field, https://arXiv.org/abs/1704.02010, Journal of Fourier Analysis and Applications, 2018, 1–26.
doi: 10.1007/s00041-018-09649-7. |
[2] |
P. Kuchment and F. Terizoglu,
Inversion of weighted divergent beam and cone transforms, Inverse Problems and Imaging, 11 (2017), 1071-1090.
doi: 10.3934/ipi.2017049. |
[3] |
W. Lionheart and V. A. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, Ed. Habib Ammari and Hyeonbae Kang, Contemporary Mathematics, 494 (2009), 137–159.
doi: 10.1090/conm/494/09648. |
[4] |
V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series. VSP, Utrecht, 1994.
doi: 10.1515/9783110900095. |
[5] |
V. A. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002, 20pp.
doi: 10.1088/1361-6420/33/2/025002. |
[6] |
V. A. Sharafutdinov and J.-N. Wang, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp.
doi: 10.1088/0266-5611/28/6/065017. |
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