Let's say you have a big box (0.999, 0.999) and a small box (0.001, 0.001).
Euclidean distance = sqrt((0.999 - 0.001)^2 + (0.999 - 0.001)^2) = 1.41.
Now, we look at a small box (0.001, 0.001) and an even smaller box (0.00000001, 0.000000001).
Euclidean distance = sqrt((0.001 - 0.00000001)^2 + (0.001 - 0.000000001)^2) = 0.00141
If you look at the number of the 2nd example, the Euclidean distance
thinks that the second box is pretty similar to the first box. But
this is not true.
IOU solves the problem because it measures the boxes relative to each
other. The result is a percentage, so it would give both boxes a
percentage near 0 which is correct.
As for the Jaccard Index, why when the denominator equals the numerator, J(b1,b2)=1 ?
I suppose we still have terms (w1*h1+w2*h2) in the denominator, so how can denominator equals numerator ?
J(b1,b2) = 1 iff intersection/(w1h1 + w2h2 - intersection) = 1 where
intersection = min_of_width * min_of_height
Assume w1h1 = w2h2, then intersection/(2 * w1h1 - intersection). But
intersection is just w1h1, then w1h1/(2 * w1h1 - w1h1) = w1h1/wh1h1 = 1