Problem #5

We use the gray image cameraman.tif, which is coincidentally 256x256, so it decomposes into 16x16 pixel blocks perfectly. The program hw5.m returns the eigenvalues and eigenvectors for the KL Transform. The first and last 10 eignevectors are given in links below.

Note: I used the MATLAB command "eig" to solve the eigenvalue problem for R. I realized later that eig returns the eigenvectors in the order from smallest to largest eigenvector. Since our most important eigenvector is the one with the largest eigenvalue, I switched the order here. Hence, the first eigenvector shown is the image "hw5_eig256" which is the plot of E(:,256). As we move on, the eigenvectors appear to be more and more varied.

Original image

First 10 eigenvalues
1e6*
0.0077 0.0078 0.0093 0.0105 0.0125 0.0173 0.0277 0.0397 0.0564 4.3271

First 10 eigenvectors.
The ith column represents the (11-i)th eigenvector.

Eigenvector 1	Eigenvector 2	Eigenvector 3	Eigenvector 4	Eigenvector 5
Eigenvector 6	Eigenvector 7	Eigenvector 8	Eigenvector 9	Eigenvector 10

Last 10 eigenvalues
0.0000 0.0003 0.0007 0.0016 0.0027 0.0053 0.0064 0.0096 0.0115 0.0172

Last 10 eigenvectors
The ith column represents the (257-i)th eigenvector.

Eigenvector 247	Eigenvector 248	Eigenvector 249	Eigenvector 250	Eigenvector 251
Eigenvector 252	Eigenvector 253	Eigenvector 254	Eigenvector 255	Eigenvector 256

Source Code: hw5.m

Back to HW 1 Page.