Add LS analytical solution, and find_BFS function

LP/find_BFS.m

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+function [optimal_sol,optimal_cost,feasible_solutions] = find_BFS(A,b, c)
+%find_BFS Finds basic feasible soltions of standard form LP problem The function receives constraints of the form Ax=b, 
+%and a column vector of representing the cost function.
+%   - A - a matrix of equality constraints
+%   - b - a column vetor of constants for the equation
+%   - c - a column vetor of costs
+
+%choose columns subsets
+[subset_size, col_num] = size(A);
+subsets = nchoosek(1:col_num,subset_size);
+%find basic solutions
+basic_solutions = [];
+for ii = 1:size(subsets,1) %iterate over all subsets
+    A_beta = A(:,subsets(ii,:));
+    if rank(A_beta) == subset_size %% if A_beta is invertible
+        tmp = zeros(col_num,1);
+        tmp(subsets(ii,:)) = A_beta\b;
+        basic_solutions = [basic_solutions tmp];
+    end
+end
+if isempty(basic_solutions)
+    error('no basic solutions found, check for errors in your constraints equation')
+end
+%feasability of basic solutions
+%verify no typos in A, by verfying equality constraints
+if max(max(abs(A*basic_solutions - repmat(b,1,size(basic_solutions,2))))) > 1e-5
+    warning('equality constraints violated')
+else
+    disp('equality constraints satisfied')
+end
+%feasible solutions require only positive entries
+feasible_solutions = [];
+for ii = 1:size(basic_solutions,2) %iterate over all columns
+    if min(basic_solutions(:,ii))>=0
+        feasible_solutions = [feasible_solutions basic_solutions(:,ii)];
+    end
+end
+if isempty(feasible_solutions)
+    error('no feasible solutions found')
+end
+costs = c'*feasible_solutions;
+[optimal_cost, optimal_idx] = min(costs);
+optimal_sol = feasible_solutions(:,optimal_idx);
+end
+

LS/basic_LS.m

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+%% reference and blurring
+clear
+close all
+
+pic = imread('tree.jpg');
+[A,y,X] = blurring(pic);
+
+[n,m] = size(X);
+blurred_pic = reshape(y, n, m);
+
+figure(1)
+imshow(X)
+title('Reference picture')
+%saveas(gcf,'Reference.eps','epsc')
+%% blurred picture
+
+figure(2)
+imshow(blurred_pic)
+title('Blurred picture')
+%saveas(gcf,'Blurred.eps','epsc')
+%% LS
+x_LS = (A'* A)\(A'*y);
+
+figure(3)
+imshow(reshape(x_LS, n, m))
+title('LS reconstructed picture')
+%saveas(gcf,'LS.eps','epsc')
+%% RLS 
+lambda = 5*1e-4; %required some trial and error
+x_RLS = (A'* A + lambda*speye(size(A'* A,1)) )\(A'*y);
+
+figure(4)
+imshow(reshape(x_RLS, n, m))
+title('RLS reconstructed picture, \lambda = 5*1e-4')
+%saveas(gcf,'RLS.eps','epsc')
+%% MSE 
+LS_MSE = norm(x_LS - X(:))^2 / length(x_LS)
+RLS_MSE = norm(x_RLS - X(:))^2 / length(x_RLS)