Question

# Working to complete parameters for rank approximations in MATLAB.

I would like to know how to generate ranks, root mean square errors between matrices, the value of k, compression ratio and display an image and compute (RMSE) from original image to approximations. CR=2,10,25, and 75.

below is the code I've managed to generate and have had no success in completing an image.

Problem 1
Use the svd() function in MATLAB to compute , the rank-1 approximation of . Clearly state what is, rounded to 4 decimal places. Also, compute the root-mean square error (RMSE) between and .
Solution:
%code
A = [1 2 2; 3 4 5; 6 7 8]
B = double(A);
B = B/225;
[U,S,V] = svd(B)
checkBUSV = U*S*V.'
[QB, RB] = qr(B)
checkBQBRB = QB*RB
Usize = size(U);
Ssize = size(S);
Vsize = size(V);
rankA = rref(A)
rankA1 = B(:,1)
A = max(0,min(1,A))
n = max(size(A))
RMSEAA1 = sqrt(1/n*sum(A-rankA1).^2)

Problem 2
Use the svd() function in MATLAB to compute , the rank-2 approximation of . Clearly state what is, rounded to 4 decimal places. Also, compute the root-mean square error (RMSE) between and . Which approximation is better, or ? Explain.
Solution:
%code
C = zeros(size(A))

rankA2 = B(:,2)
RMSEAA2 = sqrt(1/n*sum(A-rankA2).^2)
Explain:

Problem 3
For the matrix , the singular value decomposition is where . Use MATLAB to compute the dot product .
Also, use MATLAB to compute the cross product and dot product . Clearly state the values for each of these computations. Do these values make sense? Explain.
Solution:
%code
u1 = U(:,1);
u2 = U(:,2);
u3 = U(:,3);
d1 = dot(u1,u2)
c = (cross(u1,u2))
d2 = dot(c,u3)
Explain:

Problem 4
Using the matrix , determine whether or not the columns of span . Explain your approach.
Solution:
%code
temp = rref(U)
[L, U] = lu(U);
Explain: %There are 3 pivots in the temp row reduction of U, which means the row rank is 3. The theorem informs that the row rank = the column rank of a matrix, It is also known that the coloumn rank of A is 3. Therfore there must be 3 linearly independent columns of U that span R.

Problem 5
Use the MATLAB imshow() function to load and display the image stored in the image.mat file, available in the Project Two Supported Materials area in Brightspace. For the loaded image, derive the value of that will result in a compression ratio of . For this value of , construct the rank-k approximation of the image.
Solution:
%code
k1 = rankA1(1,:)
k2 = rankA2(2,:)
rankA3 = B(:,3)
RMSEAA3 = sqrt(1/n*sum(A-rankA3).^2)
k3 = rankA3(3,:)
CR = S./k1(A+rankA1+1)

figure;
imshow(A)
title('My Image Title KG')

Explain:
Problem 6
Display the image and compute the root mean square error (RMSE) between the approximation and the original image. Make sure to include a copy of the approximate image in your report.
Solution:
%code
image(C), axis image

Problem 7
Repeat Problems 5 and 6 for , , and . Explain what trends you observe in the image approximation as increases and provide your recommendation for the best based on your observations. Make sure to include a copy of the approximate images in your report.
Solution:
%code

Explain:

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