scholarly journals Inversion of the Attenuated X-Ray Transforms: Method of Riesz Potentials

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yu Yufeng
Keyword(s):  
Single Photon ◽  
Photon Emission ◽  
Emission Computed Tomography ◽  
Riesz Potentials ◽  
X Ray ◽  
Ray Transform ◽  
Straight Line ◽  
Single Photon Emission Computed ◽  
Ray Transforms ◽  
Photon Emission Computed Tomography

The attenuated X-ray transform arises from the image reconstruction in single-photon emission computed tomography. The theory of attenuated X-ray transforms is so far incomplete, and many questions remain open. This paper is devoted to the inversion of the attenuated X-ray transforms with nonnegative varying attenuation functions μ, integrable on any straight line of the plane. By constructing the symmetric attenuated X-ray transform Aμ on the plane and using the method of Riesz potentials, we obtain the inversion formula of the attenuated X-ray transforms on Lpℝ21≤p<2 space, with nonnegative attenuation functions μ, integrable on any straight line in ℝ2. These results are succinct and may be used in the type of computerized tomography with attenuation.

scholarly journals A geometric based preprocessing for weighted ray transforms with applications in SPECT

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Fedor Goncharov
Keyword(s):  
Single Photon ◽  
Photon Emission ◽  
Reconstruction Algorithm ◽  
Emission Computed Tomography ◽  
Radon Transforms ◽  
Single Photon Emission Computed ◽  
Ray Transforms ◽  
Straight Lines ◽  
The Impact ◽  
Photon Emission Computed Tomography

AbstractIn this work we investigate numerically the reconstruction approach proposed in [F. O. Goncharov and R. G. Novikov, An analog of Chang inversion formula for weighted Radon transforms in multidimensions, Eurasian J. Math. Comput. Appl. 4 2016, 2, 23–32] for weighted ray transforms (weighted Radon transforms along oriented straight lines) in 3D. In particular, the approach is based on a geometric reduction of the data modeled by weighted ray transforms to new data modeled by weighted Radon transforms along two-dimensional planes in 3D. Such reduction could be seen as a preprocessing procedure which could be further completed by any preferred reconstruction algorithm. In a series of numerical tests on modelized and real SPECT (single photon emission computed tomography) data we demonstrate that such procedure can significantly reduce the impact of noise on reconstructions.

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