We address the problem of reconstructing functions from their nonuniform data and zero/threshold crossings. We introduce a deterministic process via the Gram-Schmidt orthonormalization procedure to reconstruct functions from their nonuniform data and zero/threshold crossings. This is achieved by first introducing the nonorthogonal basis functions in a chosen 2-D domain (e.g., for a band-limited signal, a possible choice is the 2-D Fourier domain of the image) that span the signal subspace of the nonuniform data. We then use the Gram-Schmidt procedure to construct a set of orthogonal basis functions that span the linear signal subspace defined by the nonorthogonal basis functions. Next, we project the N-dimensional measurement vector (N is the number of nonuniform data or threshold crossings) onto the newly constructed orthogonal basis functions. Finally, the function at any point can be reconstructed by projecting the representation with respect to the newly constructed orthonormal basis functions onto the reconstruction basis functions that span the signal subspace of the evenly spaced sampled data. The reconstructed signal gives the minimum mean square error estimate of the original signal. This procedure gives error-free reconstruction provided that the nonorthogonal basis functions that span the signal subspace of the nonuniform data form a complete set in the signal subspace of the original band-limited signal. We apply this algorithm to reconstruct functions from their unevenly spaced sampled data and zero crossings and also apply it to solve the problem of synthesis of a 2-D band-limited function with the prescribed level crossings.
"Image reconstruction/synthesis from nonuniform data and zero/threshold crossings," Optical Engineering 33(10), (1 October 1994). https://doi.org/10.1117/12.182339