## Transcribed Text

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);...