Basic Indeterminacy of Reconstruction

The history of determining an arbitrary function from its projections is at least a half century old. During that time, mathematicians have been able to provide some encouraging results when they assumed (the unrealistic situation) than an infinite amount of projection data is available. However, it was not until 1973 that fundamental work addressed the more practical question which follows from the fact that only a finite number of projections can ever become available. This recent result, of Kennan T. Smith, demonstrates that even with very strong restrictions on the objective function, a finite number of radiographs seem to say nothing at all about determining this function. This chapter shows that the basic indeterminacy demonstrated by Kennan Smith does not depend on his choice of restrictions, i. e., the non-uniqueness is independent of almost any choice of the space from which the objective function comes (as long as it is infinite dimensional). This chapter concludes that some change in the mathematical model is needed.