Direct reconstruction method for wavelet transform extrema representation

In contrast to the iterative reconstruction algorithm of projections onto convex sets (Mallat and coworkers, 1992; Liew and Nguyen, 1995; Cvetkovic and Vetterli, 1995) a noniterative method that completely solves the problem of reconstructing from the wavelet transform extrema representation is presented for the first time. The solution obtained by the proposed method is mathematically consistent and is indistinguishable from the true solution, i. e. both give the same representation. The proposed method consists of first finding a leastsquares solution in the space spanned by the wavelet sampling bases. An orthogonal component that is to be added to the leastsquares solution to form a consistent solution is then found by solving a set of linear inequalities specified by the a priori information in the representation using the linear programming technique. Numerical results presented show that the reconstructions are of good quality.