split bergman matlab代码

### Split Bregman Algorithm Implementation in MATLAB

The Split Bregman method is an iterative optimization technique used primarily for solving constrained minimization problems that arise frequently in image processing and compressive sensing applications. Below is a basic implementation of the Split Bregman algorithm using MATLAB:

function [u, d] = split_bregman(A, b, lambda, mu, tol, max_iter)
% Inputs:
% A       : Measurement matrix (m x n)
% b       : Observation vector (m x 1)
% lambda  : Regularization parameter
% mu      : Augmented Lagrangian penalty term
% tol     : Tolerance level for convergence criterion
% max_iter: Maximum number of iterations allowed
    
[m,n] = size(A);
d = zeros(n,1); % Initialize dual variable
b_hat = zeros(m,1); % Initialize auxiliary variable
u = zeros(n,1); % Initialize primal variable

for iter = 1:max_iter
        
    % Update u via least squares solution
    u_new = (A' * A + mu * eye(n)) \ (A' * b + mu .* (d - b_hat));
    
    % Soft thresholding operation to update d
    shrinkage_factor = max( abs(u_new + b_hat) - lambda/mu , 0 );
    d = sign(u_new + b_hat).*shrinkage_factor;
        
    % Update multiplier/b_hat
    b_hat = b_hat + u_new - d;

    % Check stopping criteria based on relative change
    if norm(u-u_new,'fro')/norm(u,'fro') < tol
        break;
    end
            
    u = u_new;
end

This function implements the core steps involved in applying the Split Bregman approach as described by Goldstein et al.[^1]. The provided script solves linear inverse problems with L1 regularization through alternating updates between two variables while enforcing constraints implicitly.