 # Consider the one dimensional image restoration problem of finding f...

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Consider the one dimensional image restoration problem of finding f = (fi /n) € R" such that is minimized, where d € R" is the obtained data (blurred image), K is the given blurring matrix, and Ja is the approximate Tikhonov Total Variation function (see Lecture Note for more details). This problem can be solved by the Picard iteration Set m = 0. A.z. = 1, f = zeros(in), A/ = ones(n). while > € m=m+1 evaluate L(fm). Solve [KTK + - KTd - Update end where at each step L(f) D'diag(©2 -1 1 D and -1 1 Q (Notice that we have simplified the computations by setting Ar = 1). Problem: Write a program in MATLAB/Julia that does the following tasks: 1. Set n = 100 and set forg = (fi fr) by 7 [1,10] [11,25] 26,45 46,55 56, 70] 71,80 [81,90] [91,100] f; 0.2 0.7 0.4 0.6 0.8 0.2 0.5 0.4 2. Set m = 2 and find the (sparse) matrix K = by Rig - my/2w exp ( (1-1) = 0, otherwise. 3. Generate the blurred image d from the original image f by the following command (in Julia) d = K*f+0.01*rand(n) (in MATLAB) d = K*f+0.01*rand(n, 1); 4. Use E = 10-4, 3 = 10-5 and select two values a of your choice in the interval [10-3,1] to find the (reconstruction) image / with the algorithm above. 5. For each value of a, plot the original image forg: the blurred image d. and the reconstruction f in the same figure. 1 Instructions: Please submit your assignments with hard copy or a SINGLE PDF that includes: the program and all figures. Help & Tips: Sparse matrices work just like regular ones, i.e., you can add, subtract, multiply, etc. See examples below for how to generate sparse matrices. # Julia # MATLAB n 100; n = 100; A spzeros(n,n); A sparse(n,n); for i=1:n for i=1:n A[i,i]=-1; A(i, i)=-1; end end 2

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clear; clc; close all;
global beta epsil
%Original image
f_org=[0.2*ones(1,10), 0.7*ones(1,20),0.4*ones(1,15),0.6*ones(1,10),0.8*ones(1,15),0.2*ones(1,10),0.5*ones(1,10),0.4*ones(1,10)]';%Original 1D image
n=length(f_org);
m=2;
%Generate the blurring sparse kernel
K=zeros(n,n);
for i=1:n
for j=1:n
if abs(i-j)<=2*m
K(i,j)=(1/(m*sqrt(2*pi)))*exp(-(i-j)^2/(2*m^2));
end
end
end
%Generate blurred image
d=K*f_org+0.01*rand(n,1);
%Reconstruct image
epsil=1e-4; beta=1e-5;
alpha1=0.1; alpha2=0.002;
f_reconstructed_alph1=picardSolve(d,K,alpha1);
plotResults(f_org,d,f_reconstructed_alph1,alpha1);
f_reconstructed_alph1=picardSolve(d,K,alpha2);
plotResults(f_org,d,f_reconstructed_alph1,alpha2);...

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