Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields

Svetov, Ivan E; Polyakova, Anna P

doi:10.1088/1361-6420/ad0fac

Cited by 2 publications

(1 citation statement)

References 44 publications

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“…Next we consider the null space of the transversal transform, see [22] and for the proof [35], theorem 5.…”

Section: Null Spaces and Inversion Formulae For The Two Radon Transformsmentioning

confidence: 99%

A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality

Louis

2024

Inverse Problems

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Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 The Mathematics of Computerized Tomography (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 Inverse Problems 38 065008) containing the results from Louis (2022 Inverse Problems 38 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.

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“…Next we consider the null space of the transversal transform, see [22] and for the proof [35], theorem 5.…”

Section: Null Spaces and Inversion Formulae For The Two Radon Transformsmentioning

confidence: 99%

A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality

Louis

2024

Inverse Problems

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A non-axisymmetric temperature field reconstruction method based on the compressive sensing background oriented Schlieren method

Wu,

Pan,

Zhang

et al. 2024

Physics of Fluids

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Measurement of the combustion temperature field is an extremely important issue in industrial production. Temperature is one of the key parameters in combustion studies. With the temperature field distribution of the combustion field obtained, heat transfer, heat convection, and heat radiation can be calculated directly and efficiently. Traditional background oriented Schlieren (BOS) is an effective method for non-axisymmetric temperature field measurements, but it requires simultaneous Schlieren imaging at multiple angles for tomographic reconstruction, which will greatly limit its application. In this paper, the compressive sensing algorithm is introduced into the temperature field reconstruction, which establishes the system of equations between the deflection angle and the refractive index gradient. Then, the reconstruction of the non-axisymmetric temperature field is realized by solving the underdetermined system of equations by the method of solving the sparse solution through the compressive sensing. First, light offsets across the non-axisymmetric temperature field are measured by the under-angled BOS system and image processing method. Second, the spatial refractive index field is reconstructed by the compressive sensing BOS method proposed in this paper. Finally, the spatial temperature field is obtained. The experimental results show that by comparing the iRadon reconstruction algorithm and the compressive sensing reconstruction algorithm, the temperature field reconstructed by the compressive sensing under the condition of the under-angled sampling of projection data had a higher accuracy than that reconstructed by the tomographic reconstruction algorithm under the same condition. The average error of the temperature field was reduced from 34.6 to 29.7 K under the same measurement conditions; however, the accuracy is better maintained by using the compressive sensing algorithm under the condition of undersampling projection.

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Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields

Cited by 2 publications

References 44 publications

A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality

A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality

A non-axisymmetric temperature field reconstruction method based on the compressive sensing background oriented Schlieren method

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