  A recurring task in image processing, approximation theory, and the numerical
solution of partial differential equations is to reconstruct a piecewise-smooth
real-valued function f(x) in multiple dimensions from its truncated Fourier
transform (its truncated spectrum). An essential step is edge detection for
which a variety of one-dimensional schemes have been developed over the last
few decades. Most higher-dimensional edge detection algorithms consist of
applying one-dimensional detectors in each component direction in order to
recover the locations in R^N where f(x) is singular (the singular support).
  In this paper, we present a multidimensional algorithm which identifies the
wavefront of a function from spectral data. The wavefront of f(x) is the set of
points $(x,k) \in R^N \times (S^{N-1} / \{\pm 1\})$ which encode both the
location of the singular points of a function and the orientation of the
singularities. (Here $S^{N-1}$ denotes the unit sphere in N dimensions.) More
precisely, k is the direction of the normal line to the curve or surface of
discontinuity at x. Note that the singular support is simply the projection of
the wavefront onto its x-component. In one dimension, the wavefront is a subset
of R^1 and coincides with the singular support. In higher dimensions, geometry
comes into play and they are distinct. We discuss the advantages of wavefront
reconstruction and indicate how it can be used for segmentation in magnetic
resonance imaging (MRI).
