Three-dimensional reconstruction from planar projections

Abstract

The technique of direct three-dimensional reconstruction from planar projections is analyzed from a linear system viewpoint. It is found that unfiltered back projection and summation of the one-dimensional planar projections gives a point-spread function that behaves like 1/r in three-dimensional space. Thus an analogy between this reconstruction problem and the familiar electrostatic problem is set up. To correct the 1/r blurring, a Laplacian operation on the unfiltered summation image is required. Another method for reconstruction is to perform a second derivative operation on the one-dimensional planar projection set before the back projection. For spherically symmetric objects, this algorithm reduces to the Vest—Steel formula. The advantages of this reconstruction scheme as compard with reconstruction from line projections are also discussed.