This is a MATLAB package for solving discrete linear ill-posed problems with general-form Tikhonov
regularization using the Picard parameter developed by Eitan Levin and Alexander Meltzer. This package
supplements the manuscript titled "Estimation of the regularization parameter in linear discrete
ill-posed problems using the Picard parameter."
This package requires the following external functions to run fully:
- get_l, gravity, heat, phillips from Hansen's Regularization Tools available for
download at: http://www.mathworks.com/matlabcentral/fileexchange/52-regtools
- estimate_Picard_Param from O'Leary's spectral filtering package, available for download
at: https://www.cs.umd.edu/users/oleary/software/
The functions included are:
--Prepared examples:
The codes in this section solve four prepared examples using SS of PMSE, SS of MSE, DFA, GCV, SURE and optimal
MSD minimization. These codes were used to generate data for the manuscript.
-CompareMethods.m: solves three test problems from Hansen's Regularization Tools with 3 different noise levels
and a specified number of noise realizations.
-CompareOnStandardForm.m: solves a large scale image deblurring problem with a seperable blur. The MSD is calculated
for 3 different noise levels and a specified number of noise realizations.
--Solvers:
The codes in this section solve a given ill-posed problem using one of the available parameter
choice methods.
-solveSSproblem.m: uses the SS method with one of three algorithms for estimation of the Picard
parameter.
-solveGCVproblem.m: uses the GCV method.
-solveMSDproblem.m: minimizes MSD to obtain optimal solution. This code requires the true solution
as an additional input.
-solveDFAproblem.m: uses the DFA method.
-solveSUREproblem.m: uses the SURE method.
-solveSS_MSEProblem.m: uses the SS method for the MSE.
--Picard parameter estimators:
The codes in this section estimate the Picard parameter and variance of the noise.
-estPicParP.m: uses the method based on approximating the expected value of the squares of the
Fourier coefficients of the data.
-estPicParML.m: uses the method based on applying the Lilliefors test to subsequences of the Fourier
coefficients.
--Global minima finder:
The codes in this section estimate the global minimum of an anonymous function given
an array of values on which to sample the function. The resulting sample is used as a starting point
for a solver.
-findGlobalMinV1.m: uses the fminbnd solver on intervals centered about every local minima found on
initial sample.
-findGlobalMinV2.m: uses the GlobalSearch solver with initial starting point being the global
minimum of the initial sample.
--Auxilliary functions:
-findLocalMins.m: finds all local minima of an array. Used in findGlobalMinV1 routine.