The author performs image reconstruction for a novel Positron Emission Tomography camera that is optimized for breast cancer imaging. This work addresses for the first time, the problem of fully-3D, tomographic reconstruction using a septa-less, stationary, (i.e. no rotation or linear motion), and rectangular camera whose Field of View (FOV) encompasses the entire volume enclosed by detector modules capable of measuring Depth of Interaction (DOI) information. The camera is rectangular in shape in order to accommodate breasts of varying sizes while allowing for soft compression of the breast during the scan. This non-standard geometry of the camera exacerbates two problems: (a) radial elongation due to crystal penetration and (b) reconstructing images from irregularly sampled data. Packing considerations also give rise to regions in projection space that are not sampled which lead to missing information. The author presents new Fourier Methods based image reconstruction algorithms that incorporate DOI information and accommodate the irregular sampling of the camera in a consistent manner by defining lines of responses (LORs) between the measured interaction points instead of rebinning the events into predefined crystal face LORs which is the only other method to handle DOI information proposed thus far. The new procedures maximize the use of the increased sampling provided by the DOI while minimizing interpolation in the data. The new algorithms use fixed-width evenly spaced radial bins in order to take advantage of the speed of the Fast Fourier Transform (FFT), which necessitates the use of irregular angular sampling in order to minimize the number of unnormalizable Zero-Efficiency Bins (ZEBs). In order to address the persisting ZEBs and the issue of missing information originating from packing considerations, the algorithms (a) perform nearest neighbor smoothing in 2D in the radial bins (b) employ a semi-iterative procedure in order to estimate the unsampled data and (c) mash the in plane projections, i.e. 2D data, with the projection data from the first oblique angles, which are then used to reconstruct the preliminary image in the 3D Reprojection Projection algorithm. The author presents reconstructed images of point sources and extended sources in both 2D and 3D. The images show that the camera is anticipated to eliminate radial elongation and produce artifact free and essentially spatially isotropic images throughout the entire FOV. It has a resolution of 1.50 (+-) 0.75 mm FWHM near the center, 2.25 (+-)0.75 mm FWHM in the bulk of the FOV, and 3.00 (+-) 0.75 mm FWHM near the edge and corners of the FOV.
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