CONVOLUTION BACKPROJECTION FORMULAS FOR 3-D VECTOR TOMOGRAPHY WITH APPLICATION TO MRI

Vector tomography is the reconstruction of vector fields from measurem ents of their projections, In previous work, it has been shown that re construction of a general three-dimensional (3-D) vector field is poss ible from the so-called inner product measurements, It has also been s hown how reconstruction of either the irrotational or solenoidal compo nent of a vector field can be accomplished with fewer measurements tha n that required for the full field, The present paper makes three cont ributions. First, in analogy to the two-dimensional (2-D) approach of Norton, several 3-D projection theorems are developed. These lead dire ctly to new vector field reconstruction formulas that are convolution backprojection formulas, It is shown how the local reconstruction prop erty of these 3-D reconstruction formulas permits reconstruction of po int flow or of regional flow from a limited data set, Second, simulati ons demonstrating 3-D reconstructions, both local and nonlocal, are pr esented, Using the formulas derived herein and those derived in previo us work, these results demonstrate reconstruction of the irrotational and solenoidal components, their potential functions, and the field it self from simulated inner product measurement data, Finally, it is sho wn how 3-D inner product measurements can be acquired using a magnetic resonance scanner.