Filtered-Backprojection Reconstruction Formula for 2D Tomography with Bilateral Truncation

We present a reconstruction formula for classical two-dimensional image reconstruction that is impervious to the effects of lateral truncation. The formula applies to objects lying inside an elliptical region whose area is less than that of a scanner's circular field-of-view. This situation would occur for example for a patient whose arms are truncated in anterior and posterior projection views. This formula differs fundamentally from existing algorithms in that it does not require a two-step processing of a DBP (differentiated backprojection) followed by subsequent Hilbert inversion. The formula represents a true filtered backprojection (FBP) algorithm, although the filtering step sometimes requires certain neighboring projections to also contribute to the current filtered projection. The derivation is based on a result published in 2004 on multiple Radon inversion formulas for functions with circular support. We have changed the support to an ellipse and identified a formula that does not access the truncated part of the sinogram. We implemented this reconstruction approach and tested it using simulated truncated data of a modified Shepp-Logan phantom. We suggest that potential advantages of this method would be the reconstruction of a small region-of-interest, which could be performed at high pixel resolution with only local backprojection and localized filtering required.