Abstract

The discrete formulation of spectral extrapolation requires the solution of a set of linear equations that are generally inconsistent and redundant. A recently proposed method for spectral extrapolation [ R. J. Mammone and G. Eichmann, Appl. Opt. 21, 496– 501( 1982)] addresses these two properties in the following way. The inconsistencies are first removed from the measured signal by elimination of the noise components that fall outside the passband of the degradation operator. The resulting set of consistent but redundant equations possesses many solutions. This ambiguity is resolved by selecting the solution that contains the least number of nonnegative elements. The technique used to find this solution employs linear programming (LP) to minimize the L1 norm of the residual error. The removal of the inconsistencies guarantees an exact solution. Therefore the minimum L1 norm of the error should be zero. The deviation from zero provides a check on the accuracy of the estimate. A technique is presented that reduces the computational burden of finding the minimum L1 norm. This is accomplished by the elimination of the first phase of LP and the introduction of a new pivot strategy for LP. The elimination of the first phase of LP reduces the computer-memory requirements and the number of iterations necessary to find the minimum L1 solution. This more efficient LP formulation of the minimum L1 norm estimation problem was developed previously [ I. Barrodale and F. Roberts, Commun. ACM 17, 319 ( 1973)] in a slightly different way. It is the new pivot strategy that differentiates the method presented here from numerous L1 methods developed previously. The pivot strategy presented in this paper, which accelerates the convergence of the L1 norm method, is sufficiently general to apply to any LP problem. Computer simulations demonstrate the advantages of the new pivot strategy.

© 1983 Optical Society of America

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