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16 Apr 1998 (Updated )

Analysis and Solution of Discrete Ill-Posed Problems.

shaw(n)
function [A,b,x] = shaw(n)
%SHAW Test problem: one-dimensional image restoration model.
%
% [A,b,x] = shaw(n)
%
% Discretization of a first kind Fredholm integral equation with
% [-pi/2,pi/2] as both integration intervals.  The kernel K and
% the solution f, which are given by
%    K(s,t) = (cos(s) + cos(t))*(sin(u)/u)^2
%    u = pi*(sin(s) + sin(t))
%    f(t) = a1*exp(-c1*(t - t1)^2) + a2*exp(-c2*(t - t2)^2) ,
% are discretized by simple quadrature to produce A and x.
% Then the discrete right-hand b side is produced as b = A*x.
%
% The order n must be even.

% Reference: C. B. Shaw, Jr., "Improvements of the resolution of
% an instrument by numerical solution of an integral equation",
% J. Math. Anal. Appl. 37 (1972), 83-112.

% Per Christian Hansen, IMM, 08/20/91.

% Check input.
if (rem(n,2)~=0), error('The order n must be even'), end

% Initialization.
h = pi/n; A = zeros(n,n);

% Compute the matrix A.
co = cos(-pi/2 + (.5:n-.5)*h);
psi = pi*sin(-pi/2 + (.5:n-.5)*h);
for i=1:n/2
  for j=i:n-i
    ss = psi(i) + psi(j);
    A(i,j) = ((co(i) + co(j))*sin(ss)/ss)^2;
    A(n-j+1,n-i+1) = A(i,j);
  end
  A(i,n-i+1) = (2*co(i))^2;
end
A = A + triu(A,1)'; A = A*h;

% Compute the vectors x and b.
a1 = 2; c1 = 6; t1 =  .8;
a2 = 1; c2 = 2; t2 = -.5;
if (nargout>1)
  x =   a1*exp(-c1*(-pi/2 + (.5:n-.5)'*h - t1).^2) ...
      + a2*exp(-c2*(-pi/2 + (.5:n-.5)'*h - t2).^2);
  b = A*x;
end

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