Fast fully 3D image reconstruction using planograms

Presents a method of performing fast and accurate 3D backprojection using only Fourier transform operations based on the geometry of planar detector arrays. This approach, the `planogram method´ is a 3D extension of the 2D-linogram technique of Edholm. By using a special choice of parameters to index a line of response (LOR) for a pair of detectors, rather than the conventional parameters used to index a LOR for a circular tomograph, all the LORs passing through a point in the field of view (FOV) lie on a 2D plane in the 4D data space. Thus backprojection of all the LORs passing through a point in the FOV corresponds to integration of a 2D plane through the 4D `planogram´. The key step is that the integration along a set of parallel 2D planes through the planogram, that is, backprojection of a plane of points, can be replaced by a central section through the origin of the 4D Fourier transform of the data. In other words, the central section theorem is applied to the data space rather than the image space. The planogram method of 3D backprojection can therefore be performed using only Fourier transform operations