How to quantify shape similarity between two vectors.

I need to quanitfy how similarity between two spectral profile. I have looked at procrustes distance but it dosent work when spectrum is shifted on x axis. Is there a way to quanitfy this, ideally between 0 (no similairty) and 1 (perfect similarity) ?

x1 = 0:0.1:10;
Y1 = gaussmf(x1,[0.8 5 ]);
Y2 = gaussmf(x1,[0.8 3 ]);
Y3 = gaussmf(x1,[0.4 2 ]);

figure(1)
plot(x1,Y1, 'k','Linewidth',2)
hold on
plot(x1,Y2, 'b','Linewidth',2)
hold on
plot(x1,Y3, 'r','Linewidth',2)
legend(["A","B","C"])

figure(1)

In the toy example, A and B are very similar so that should have a high similairty index but A-C and B-C should have lower similarity.

Answer 1

[Torsten]

Shift the curves by their means first so that the means of all objects lie in the origin (0). This will remove translational differences between the objects.

Answer 2

[Matt J]

Perhaps you could use correlation-based similarity?

x1 = 0:0.1:10;
Y1 = gaussmf(x1,[0.8 5 ]);
Y2 = gaussmf(x1,[0.8 3 ]);
Y3 = gaussmf(x1,[0.4 2 ]);

corrcoeff(Y2,Y1)

ans = 1.0000

corrcoeff(Y2,Y3)

ans = 0.8944

corrcoeff(Y1,Y3)

ans = 0.8944

function coeff=corrcoeff(u,v)

 coeff=max(xcorr(u,v))/norm(u)/norm(v);

end

Answer 3

[Image Analyst]

See the Mathworks page on spectral signature matching:

https://www.mathworks.com/help/images/target-detection-and-spectral-matching-techniques.html