IMAGE-RECONSTRUCTION SYNTHESIS FROM NONUNIFORM DATA AND ZERO THRESHOLD CROSSINGS

We address the problem of reconstructing functions from their nonunifo rm data and zero/threshold crossings. We introduce a deterministic pro cess via the Gram-Schmidt orthonormalization procedure to reconstruct functions from their nonuniform data and zero/threshold crossings. Thi s is achieved by first introducing the nonorthogonal basis functions i n a chosen 2-D domain (e.g., for a band-limited signal, a possible cho ice is the 2-D Fourier domain of the image) that span the signal subsp ace of the nonuniform data. We then use the Gram-Schmidt procedure to construct a set of orthogonal basis functions that span the linear sig nal subspace defined by the nonorthogonal basis functions. Next, we pr oject the N-dimensional measurement vector (N is the number of nonunif orm data or threshold crossings) onto the newly constructed orthogonal basis functions. Finally, the function at any point can be reconstruc ted by projecting the representation with respect to the newly constru cted orthonormal basis functions onto the reconstruction basis functio ns that span the signal subspace of the evenly spaced sampled data. Th e reconstructed signal gives the minimum mean square error estimate of the original signal. This procedure gives error-free reconstruction p rovided that the nonorthogonal basis functions that span the signal su bspace of the nonuniform data form a complete set in the signal subspa ce of the original band-limited signal. We apply this algorithm to rec onstruct functions from their unevenly spaced sampled data and zero cr ossings and also apply it to solve the problem of synthesis of a 2-D b and-limited function with the prescribed level crossings.