Phase-contrast x-ray computed tomography (CT) is an emerging imaging technique that can be implemented at third-generation synchrotron radiation sources or by using a microfocus x-ray source. Promising results have recently been obtained in materials science and medicine. At the same time, the lack of a mathematical theory comparable with that of conventional CT limits the progress in this field. Such a theory is now suggested, establishing a fundamental relation between the three-dimensional Radon transform of the object function and the two-dimensional Radon transform of the phase-contrast projection. A reconstruction algorithm is derived in the form of a filtered backprojection. The filter function is given in the space and spatial-frequency domains. The theory suggested enables one to quantitatively determine the refractive index of a weakly absorbing medium from x-ray intensity data measured in the near-field region. The results of computer simulations are discussed.

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Abstract-Methods of quantitative emission computed tomography require compensation for linear photon attenuation. A current trend in single-photon emission computed tomography (SPECT) and positron emission tomography (PET) is to employ transmission scanning to reconstruct the attenuation map. Such an approach, however, considerably complicates both the scanner design and the data acquisition protocol. A dramatic simplification could be made if the attenuation map could be obtained directly from the emission projections, without the use of a transmission scan. This can be done by applying the consistency conditions that enable us to identify the operator of the problem and, thus, to reconstruct the attenuation map. In this paper, we propose a new approach based on the discrete consistency conditions. One of the main advantages of the suggested method over previously used continuous conditions is that it can easily be applied in various scanning configurations, including fully three-dimensional (3-D) data acquisition protocols. Also, it provides a stable numerical implementation, allowing us to avoid the crosstalk between the attenuation map and the source function. A computationally efficient algorithm is implemented by using the QR and Cholesky decompositions. Application of the algorithm to computer-generated and experimentally measured SPECT data is considered.Index Terms-Attenuation correction, positron emission tomography (PET), single-photon emission computed tomography (SPECT).

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The attenuated Radon transform serves as a mathematical tool for single-photon emission computerized tomography (SPECT). The identification problem for the attenuated Radon transform is to find the attenuation coefficient, which is a parameter of the transform, from the values of the transform alone. Previous attempts to solve this problem used range theorems for the continuous attenuated/exponential Radon transform. We consider a matrix representation of the transform and formulate the corresponding discrete consistency conditions in the form of the orthogonal projection of the data vector onto the orthogonal complement of the column space of the matrix. The singular value decomposition is applied to compute the orthogonal projector and its Fréchet derivative. The numerical algorithm suggested is based on the Newton method with the Tikhonov regularization. Results of numerical experiments and inversion of the measured SPECT data are considered.