Image reconstruction based on zero-crossing representations of wavelet transform

A reconstruction method of the image signal from the zero-crossing representation is discussed. First, the one- and two-dimensional multiscale wavelet transforms are described with completeness and translation invariance. It is shown that the original signal can be reconstructed with a stable and fast convergence from the zero-crossing representation of the signal through the iterated projections to the convex set. The effect of the wavelet basis filter on the convergence in the reconstruction is investigated, and a necessary condition for the optimal basis filter is derived considering the convergence. Finally, a new basis filter is designed using the B-spline function and the usefulness of the proposed method is demonstrated for the one-dimensional signal and the image signal. It is also shown that the signal reconstruction procedure converges with stability, even if there exists an error at the extracted zero-crossing point.