PET image reconstruction from finite linogram via Direct Fourier and Logarithmic Barrier Method

Dual-head PET (Positron Emission Tomography) and PEM (Positron Emission Mammography) are encountered in many biological imaging and medical diagnosis applications. Since the data produced by flat-panel detectors are more closed to the linogram, and rebinning to the sinogram causes errors of the interpolation, image reconstruction based on the linogram is superior to keep the feature of the raw data for dual-head. Length of the linogram produced by actual PET or PEM systems is finite in practice. The projection views of the data are incomplete, and then the finite linogram reconstruction is a limited-view problem in mathematics. Conventional algorithms cannot achieve the exact reconstruction of the limited-view problem. In this work, we propose a half-analytic and half-iterative method named DF/LBM (Direct Fourier and Logarithmic Barrier Method) to solve the limited-view problem. The least square solution is obtained via DFM (Direct Fourier Method), and then a convex optimization method named logarithmic barrier method is employed to correct the least square solution. The substance of the convex optimization is defining the components in the null-space to satisfy some prior information, as the least square solution has no components in the null-space. Applying this method to Hoffman brain phantom, the artifact generated by the least square solution is eliminated, and PSNR is 90.2112, 94.8699 and 99.0507 when the tangent of the half allowable angle is 1.0, 1.5 and 2.0 respectively.