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

State-space and singular-value decomposition-based approximation methods for the harmonic retrieval problem

S. Y. Kung, K. S. Arun, and D. V. Bhaskar Rao
J. Opt. Soc. Am. 73(12) 1799-1811 (1983)

Unified Hilbert space approach to iterative least-squares linear signal restoration

Jorge L. C. Sanz and Thomas S. Huang
J. Opt. Soc. Am. 73(11) 1455-1465 (1983)

Signal restoration from phase by projections onto convex sets

Aharon Levi and Henry Stark
J. Opt. Soc. Am. 73(6) 810-822 (1983)

Direct fast method for time-limited signal reconstruction

Yanfei Wang, Zaiwen Wen, Zuhair Nashed, and Qiyu Sun
Appl. Opt. 45(13) 3111-3126 (2006)

Image restoration and resolution enhancement

Charles L. Byrne, Raymond M. Fitzgerald, Michael A. Fiddy, Trevor J. Hall, and Angela M. Darling
J. Opt. Soc. Am. 73(11) 1481-1487 (1983)