Concepts in projection-reconstruction.

The Achilles heel of conventional multidimensional NMR spectroscopy is the long duration of the measurements, set by the Nyquist sampling condition and the resolution requirements in the evolution dimensions. Projection-reconstruction solves this problem by radial sampling of the evolution-domain signals, relying on Bracewell's Fourier transform slice/projection theorem to generate a set of projections at different inclinations. Reconstruction is implemented by one of three possible deterministic back-projection schemes (additive, lowest-value, or algebraic), or by a statistical model-fitting program. For simplicity the treatment focuses principally on the three-dimensional case, and then extends the analysis to four dimensions. The concept of hyperdimensional spectroscopy is described for dealing with even higher dimensions.