Reconstruction Algorithms For Single-photon Emission Computed Tomography

Single photon emission computed tomography (SPECT), which is widely used in clin-ical applications, is one of the important nuclear imaging techniques. The variety ofexisting SPECT reconstruction algorithms can be split into a family of analytical meth-ods and a wide class of iterative techniques. The analytical methods are based on theinversion of the exponential or attenuated Radon transform. The main advantage ofthem is low computational cost, and the main disadvantage of them is that the recon-structed image is undesired for the noised projection data. The iterative methods takea lot of degraded factors of projection data into system matrix, and the qualities of re-constructed images are higher. However, the computational cost of them is very high.In the third and forth chapters of this thesis, we studied the analytical methods for180projection data. It had been pointed out that the weight-differentiated backprojec-tion (WDBP) of projection data of the exponential Radon transform could reduce theSPECT reconstruction to inverting a one-dimensional cosh-Hilbert transform (CHT).Although the uniqueness of the inversion of CHT had been proved under some condi-tions, there is no analytically and accurately inverse formula of it. Based on the Tricomiinversion formula for Hilbert transform and the characteristics of the Taylor expansionof hyperbolic cosine function, we proposed a moment-based method for the inversionof CHT numerically in the third chapter. Furthermore, Based on the CHT, we proposeda semi-analytical method for SPECT image reconstruction in the forth chapter. Dueto the lack of noise immunity, the reconstructed images by the moment-based methodhave a lot of distortions. In order to suppress the distortions, we proposed a regulariza-tion model based on the CHT, in which the balance parameter varies according to theweight function of WDBP. We validated the performances of the proposed methods bynumerical simulations.In the fifth chapter of the thesis, we investigated the algebraic methods of SPECTimage reconstruction. We studied the convergence of the superiorized EM algorithm,and applied to SPECT image reconstruction. Because the projection data of SPECTobey the Poisson distribution, EM iteration is widely used in the field of SPECT im- age reconstruction. However, the reconstructed images have a lot of distortions dueto the noise of projection data and uncertainty of the system matrix. Regularizationmethods are the common techniques to improve the qualities of reconstructed images,which implies that we have to solve optimal problems. Due to the large scale of imag-ing problems, there is no efficient algorithm at present. The superiorization of iterativealgorithms is a new idea to handle the optimal problems. Firstly, we proved the con-vergence of perturbed EM iteration under some conditions. Secondly, we designed thesuperiorized EM algorithm based on the convergent conditions, and discussed the de-tails of implementations for total variation and l1-norm minimization problems. Lastly,the numerical experiments were conducted to validate the efficiency of the proposedalgorithms.