Convolution backprojection formulas for 3-D vector tomography withapplication to MRI
Prince, J.L.
Image Processing, IEEE Transactions on
Volume 5, Issue 10, Oct 1996 Page(s):1462 - 1472
Digital Object Identifier 10.1109/83.536894
Summary:Vector tomography is the reconstruction of vector fields from
measurements of their projections. In previous work, it has been shown
that the reconstruction of a general three-dimensional (3-D) vector
field is possible from the so-called inner product measurements. It has
also been shown how the reconstruction of either the irrotational or
solenoidal component of a vector field can be accomplished with fewer
measurements than that required for the full field. The present paper
makes three contributions. First, in analogy to the two-dimensional
(2-D) approach of Norton (1988), several 3-D projection theorems are
developed. These lead directly to new vector field reconstruction
formulas that are convolution backprojection formulas. It is shown how
the local reconstruction property of these 3-D reconstruction formulas
permits reconstruction of point flow or of regional flow from a limited
data set. Second, simulations demonstrating 3-D reconstructions, both
local and nonlocal, are presented. Using the formulas derived herein and
those derived in previous work, these results demonstrate the
reconstruction of the irrotational and solenoidal components, their
potential functions, and the field itself from simulated inner product
measurement data. Finally, it is shown how 3-D inner product
measurements can be acquired using a magnetic resonance scanner
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