Analytic signals for multivariate data

Hilbert transformation is a standard tool in univariate signal-processing. It leaves the information content unaffected for, apart from a change of sign, the iterated transform reproduces the original data. As we know from Fourier transforms, such alternative representations of the same information, notwithstanding their theoretical equivalence with the data, can assist powerfully with extracting and interpreting that information. Although the extension to multivariate data is not so obvious for Hilbert as for Fourier transforms, Nabighian gave a treatment of the bivariate situation in 1984. Fueter, some 50 years earlier, had worked on an analogous extension problem, seeking to generalize complex function theory. On comparing these two developments we learn that, although Nabighian's transforms fit naturally into Fueter's theory, they are only one among many alternative possibilities. This paper presents a general theory, of higher dimensional Hilbert transforms and analytic signals, applicable to data of all dimensions less than eight. The change-of-field-direction filters used in geophysical data processing are shown to arise as special situations.