Calculating the eigenvalues of simple shapes

Hi there,

I've used the strel function to create a range of shapes and I would like to now calculate the eigenvalues of each shape although I am struggling to do this and would really appreciate any help regarding this.

Thank you in advance,

[the cyclist]

Do you have a reference for what the eigenvalue of a binary shape is? I did some googling of keywords, but didn't find something definite. (Maybe this is well known in image processing, but that is not my specialty.)

Are you stuck on the math of it, or the MATLAB coding? Have you written any code?

Answer 1

[Walter Roberson]

M = double(strel('disk',5).Neighborhood)

M = 9×9

0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0

E = simplify(eig(sym(M)))

E = 

vpa(E)

ans = 

(the imaginary component is due to round-off error)

M2 = double(strel('octagon',12).Neighborhood)

M2 = 25×25

0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

E2 = simplify(eig(sym(M2)))

E2 = 

vpa(imag(E2))

ans = 

vpa(E2)

ans = 

[Bjorn Gustavsson]

The/One benefit of using svd instead of eig is that one get real singular values - which is not a guarantee with eig. Appart from that the soutions should be comparable/similar/identical.

Answer 2

[Bjorn Gustavsson]

If you have a binary image then why not just run through the svd and see what you get:

I = zeros(256);
I(64:(64+128),64:(64+128)) = 1;
[U,S,V] = svd(I);
figure
subplot(1,2,1)
plot(diag(S))
subplot(1,2,2)
imagesc(U(:,1)*S(1,1)*V(:,1)')
% Or for a funnier example:
I = numgrid('B',258);
I = I(2:end-1,2:end-1);
[U,S,V] = svd(I);
subplot(2,2,1)
plot(diag(S))
subplot(2,2,2)
imagesc(U(:,1)*S(1,1)*V(:,1)')
subplot(2,2,2)
imagesc(U(:,1:4)*S(1:4,1:4)*V(:,1:4)')
subplot(2,2,2)
imagesc(U(:,1:16)*S(1:16,1:16)*V(:,1:16)')

You can also look at the individual eigen-images by something like:

imagesc(U(:,7)*V(:,7)')

HTH