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kaczmarz.m
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kaczmarz.m
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function [X,info] = kaczmarz(A,b,K,x0,options)
%KACZMARZ Kaczmarz's method (often referred to as ART)
%
% [X,info] = kaczmarz(A,b,K)
% [X,info] = kaczmarz(A,b,K,x0)
% [X,info] = kaczmarz(A,b,K,x0,options)
%
% Implements Kaczmarz's method for the system Ax = b:
%
% x^{k+1} = x^k + lambda*(b_i - a^i'*x^k)/(||a^i||_2^2)*a^i
%
% where a_i' is the i-th row of A, and i = (k mod m) + 1.
%
% Input:
% A m times n matrix.
% b m times 1 vector.
% K Number of iterations. If K is a scalar, then K is the maximum
% number of iterations and only the last iterate is saved.
% If K is a vector, then the largest value in K is the maximum
% number of iterations and only iterates corresponding to the
% values in K are saved, together with the last iterate.
% If K is empty then a stopping criterion must be specified.
% x0 n times 1 starting vector. Default: x0 = 0.
% options Struct with the following fields:
% lambda The relaxation parameter. For this method lambda must
% be a scalar; default value is 1.
% stoprule Struct containing the following information about the
% stopping rule:
% type = 'none' : (Default) the only stopping rule
% is the maximum number of iterations.
% 'NCP': Normalized Cumulative Periodogram.
% 'DP' : Discrepancy Principle.
% taudelta = The product of tau and delta, only
% necessary for DP.
% nonneg Logical; if true then nonnegativity in enforced in
% each step.
% box Upper bound L in box constraint [0,L] on pixel values.
% damping A parameter D to avoid division by very small row norms
% by adding D*max_i{||a^i||_2^2} to ||a^i||_2^2.
%
% Output:
% X Matrix containing the saved iterations.
% info Information vector with 2 elements.
% info(1) = 0 : stopped by maximum number of iterations
% 1 : stopped by NCP-rule
% 2 : stopped by DP-rule
% info(2) = no. of iterations.
%
% See also: randkaczmarz, symkaczmarz
% Maria Saxild-Hansen and Per Chr. Hansen, July 5, 2015, DTU Compute.
% Reference: G. T. Herman, Fundamentals of Computerized Tomography,
% Image Reconstruction from Projections, Springer, New York, 2009.
[m,n] = size(A);
A = A'; % Faster to perform sparse column operations.
% Check the number of inputs.
if nargin < 3
error('Too few input arguments')
end
% Default value of starting vector x0.
if nargin < 4 || isempty(x0)
x0 = zeros(n,1);
end
% The sizes of A, b and x must match.
if size(b,1) ~= m || size(b,2) ~= 1
error('The sizes of A and b do not match')
elseif size(x0,1) ~= n || size(x0,2) ~= 1
error('The size of x0 does not match the problem')
end
% Initialization.
if nargin < 5
if isempty(K)
error('No stopping rule specified')
end
stoprule = 'NO';
lambda = 1;
Knew = sort(K);
kmax = Knew(end);
X = zeros(n,length(K));
% Default is no nonnegativity or box constraint or damping.
nonneg = false;
boxcon = false;
damp = 0;
else
% Check the contents of options, if present.
% Nonnegativity.
if isfield(options,'nonneg')
nonneg = options.nonneg;
else
nonneg = false;
end
% Box constraints [0,L].
if isfield(options,'box')
nonneg = true;
boxcon = true;
L = options.box;
else
boxcon = false;
end
% Damping.
if isfield(options,'damping')
damp = options.damping;
if damp<0, error('Damping must be positive'), end
else
damp = 0;
end
if isfield(options,'lambda')
if isnumeric(options.lambda)
lambda = options.lambda;
if lambda <= 0 || lambda >= 2
warning('MATLAB:UnstableRelaxParam',...
'The lambda value is outside the interval (0,2)');
end
else
error('lambda must be numeric')
end
else
lambda = 1;
end
% Stopping rules
if isfield(options,'stoprule') && isfield(options.stoprule,'type')
stoprule = options.stoprule.type;
if ischar(stoprule)
if strncmpi(stoprule,'DP',2)
% DP stopping rule.
if isfield(options.stoprule,'taudelta')
taudelta = options.stoprule.taudelta;
else
error('The factor taudelta must be specified when using DP')
end
% Check that the first iteration should be performed:
nrk = norm(b - A'*x0); % Remember that A is transposed.
if nrk <= taudelta
info = [2 0];
X = x0;
return
end % end the DP-rule.
elseif strncmpi(stoprule,'NC',2)
% NCP stopping rule.
dk = inf;
q = floor(m/2);
c_white = (1:q)'./q;
if ~isempty(K)
K = [K max(K)+1];
end
elseif strncmpi(stoprule,'NO',2)
% No stopping rule.
if isempty(K)
error('No stopping rule specified')
end
else
% Other stopping rules.
error('The chosen stopping rule is not valid')
end % end different stopping rules.
else
error('The stoprule type must be a string')
end % end stoprule is a string.
% Determine the maximum number of iterations and initialize the
% output matrix X.
if isempty(K)
kmax = inf;
X = zeros(n,1);
else
Knew = sort(K);
kmax = Knew(end);
X = zeros(n,length(K));
end % end if isempty K.
else
% Determine the maximum number of iterations and initialize the
% output vector X.
if isempty(K)
error('No stopping rule specified')
else
Knew = sort(K);
kmax = Knew(end);
X = zeros(n,length(K));
stoprule = 'NO';
end
end % end stoprule type specified.
end % end if nargin includes options.
% Initialization before iterations.
xk = x0;
normAi = full(abs(sum(A.*A,1))); % Remember that A is transposed.
I = find(normAi>0);
normAi = normAi + damp*max(normAi);
stop = 0;
k = 0;
l = 0;
klast = 0;
while ~stop
k = k + 1;
if strncmpi(stoprule,'NC',2)
xkm1 = xk;
end
% The Kaczmarz sweep.
for i = I
ai = full(A(:,i)); % Remember that A is transposed.
xk = xk + (lambda*(b(i) - ai'*xk)/normAi(i))*ai;
if nonneg, xk = max(xk,0); end
if boxcon, xk = min(xk,L); end
end
% Stopping rules.
if strncmpi(stoprule,'DP',2)
% DP stopping rule.
nrk = norm(b - A'*xk); % Remember that A is transposed.
if nrk <= taudelta || k >= kmax
stop = 1;
if k >= kmax
info = [0 k];
else
info = [2 k];
end
end % end the DP-rule.
elseif strncmpi(stoprule,'NC',2)
% NCP stopping rule.
rkh = fft(b - A'*xk); % Remember that A is transposed.
pk = abs(rkh(1:q+1)).^2;
c = zeros(q,1);
for index = 1:q
c(index) = sum(pk(2:index+1))/sum(pk(2:end));
end
if dk < norm(c-c_white) || k >= kmax
stop = 1;
if k >= kmax
info = [0 k-1];
else
info = [1 k-1];
end
else
dk = norm(c-c_white);
end % end NCP-rule.
elseif strncmpi(stoprule,'NO',2)
% No stopping rule.
if k >= kmax
stop = 1;
info = [0 k];
end
end % end stoprule type.
% If the current iteration is requested saved.
if (~isempty(K) && k == Knew(l+1)) || stop
l = l + 1;
% Save the correct iteration.
if strncmpi(stoprule,'NC',2)
if ~(stop && klast == k-1)
X(:,l) = xkm1;
else
l = l - 1;
end
else
X(:,l) = xk;
end
klast = k;
end
end
X = X(:,1:l);