TY - JOUR
T1 - Convolution backprojection formulas for 3-D vector tomography with application to MRI
AU - Prince, Jerry L.
N1 - Funding Information:
Manuscript received September 11, 1994; revised December 29, 1995. This work was supported by the National Institutes of Health under Grant R01-HL45090 and the National Science Foundation under Grant MIP93-50336. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ken D. Sauer.
PY - 1996
Y1 - 1996
N2 - Vector tomography is the reconstruction of vector fields from measurements of their projections. In previous work, it has been shown that 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 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, 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 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.
AB - Vector tomography is the reconstruction of vector fields from measurements of their projections. In previous work, it has been shown that 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 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, 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 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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U2 - 10.1109/83.536894
DO - 10.1109/83.536894
M3 - Article
C2 - 18290063
AN - SCOPUS:0030269648
SN - 1057-7149
VL - 5
SP - 1462
EP - 1472
JO - IEEE Transactions on Image Processing
JF - IEEE Transactions on Image Processing
IS - 10
ER -