Least Squares Image Restoration Using Spline Basis Functions

This paper presents a theoretical analysis and computational technique for constrained least squares image restoration using spline basis functions. A realistic continuous–discrete physical imaging model has been adopted throughout the formulation. The optical system is assumed to be incoherent, and the general problem of image restoration with space-variant or space-invariant point-spread function degradations has been studied.

[1]  D. Goldfarb,et al.  Conjugate Gradient Method for Nonlinear Programming Problems with Linear Constraints , 1968 .

[2]  On the convergence of a family of regularizing algorithms , 1972 .

[3]  A Tikhonov,et al.  Solution of Incorrectly Formulated Problems and the Regularization Method , 1963 .

[4]  Åke Björck,et al.  Numerical Methods , 1995, Handbook of Marine Craft Hydrodynamics and Motion Control.

[5]  Approximation by polygonal paths in a quadratic metric in the solution of integral equations of the first kind , 1971 .

[6]  William C. Davidon,et al.  Variance Algorithm for Minimization , 1968, Comput. J..

[7]  D. Goldfarb.  Extension of Davidon’s Variable Metric Method to Maximization Under Linear Inequality and Equality Constraints , 1969 .

[8]  Magnus R. Hestenes,et al.  Pseudoinversus and conjugate gradients , 1975, CACM.

[9]  Alexander A. Sawchuk,et al.  Restoration of astigmatism and curvature of field , 1975 .

[10]  I. J. Schoenberg,et al.  On Pólya frequency functions IV: The fundamental spline functions and their limits , 1966 .

[11]  Carl de Boor,et al.  On uniform approximation by splines , 1968 .

[12]  C. E. Lemke,et al.  The Constrained Gradient Method of Linear Programming , 1961 .

[13]  G. Golub,et al.  Linear least squares and quadratic programming , 1969 .

[14]  H. Schwarz.  Numerical analysis of symmetric matrices , 1974 .

[15]  G. Zoutendijk,et al.  Methods of Feasible Directions , 1962, The Mathematical Gazette.

[16]  L. Schumaker,et al.  Computation of Smoothing and Interpolating Natural Splines via Local Bases , 1973 .

[17]  R. Fletcher,et al.  A New Approach to Variable Metric Algorithms , 1970, Comput. J..

[18]  A. N. Tikhonov,et al.  REGULARIZATION OF INCORRECTLY POSED PROBLEMS , 1963 .

[19]  B. R. Hunt,et al.  The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer , 1973, IEEE Transactions on Computers.

[20]  G. Golub.  MATRIX DECOMPOSITIONS AND STATISTICAL CALCULATIONS , 1969 .

[21]  C. Reinsch.  Smoothing by spline functions , 1967 .

[22]  W. Pratt,et al.  Digital image restoration under a regression model , 1975 .

[23]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[24]  J. B. Rosen.  The Gradient Projection Method for Nonlinear Programming. Part I. Linear Constraints , 1960 .

[25]  J. Philip.  Reconstruction from measurements of positive quantities by the maximum-likelihood method , 1963 .

[26]  P. Wolfe.  THE SIMPLEX METHOD FOR QUADRATIC PROGRAMMING , 1959 .

[27]  Thomas S. Huang,et al.  Inverse filtering for linear shift-variant imaging systems , 1972 .

[28]  I. J. Schoenberg,et al.  On Pólya frequency functions IV: The fundamental spline functions and their limits , 1966 .

[29]  Y. Khudak.  The convergence of regularizing algorithms , 1971 .

[30]  A. Sawchuk.  Space-variant image motion degradation and restoration , 1972 .

[31]  Michael P. Ekstrom,et al.  A Numerical Algorithm for Identifying Spread Functions of Shift-Invariant Imaging Systems , 1973, IEEE Transactions on Computers.

[32]  Ragnar Frisch.  The multiplex method for linear programming , 1958 .

[33]  J. Horner.  Optical Spatial Filtering with the Least Mean-Square-Error Filter* , 1969 .