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**Additional resources for Efficient Visual Recognition Using the Hausdorff Distance**

**Example text**

In general, I use this second interpretation of the Hausdorffdistance in terms of disks; in Chap. 3, I show how this interpretation can be very useful. The undirected Hausdorff distance, H(A, B), is the maximum of h(A, B) and h(B, A). It measures the degree of mismatch between two sets by finding the point of A that is farthest from any point of B and vice versa. Intuitively, if the Hausdorff distance is d, then every point of A must be within a distance d of some point of B and vice versa; also, at least one point of A or B must be exactly d away from its closest neighbour in the other set.

1. 1)2, []~(/(P1 ) -- ~(1(1)2)[] = IIp - z, zll. t h e , H G = Pn)of Let A and B be any two point sets, and 9a. g2 E G any two transformations. qI ( A ). q~ ( B ) ) = max( h,( 91( A ). g2 ( B ) ) . h,( 92 (13 ). ql ( A ) ) ) --- max ( sup inf I [ . ql ( a ' ) - . ))- I'll. ~"l, inf ] ] . q 2 ' ( . q~(A)). g2(B)) inf HLq~ I ( g l ( A ) ) . G gl,gzEG = inf H ( 9 ( A } . B) so H c ( A. B) = I2fc( A. [1)" Ha is therefore synametric. [] When G is the group of translations and [1" II is any norm, [1(t'1 + t) - (I'2 + t)ll = It holds also when G is the group of rigid motions and I1,,1-p2 II.

If the transformation is incorrect, though, the function should be much more irregular, and have a larger overall amplitude. On the other hand, if the transformation is correct, but the image of the object is incomplete, due to occlusion, then there will be some large local peaks in this function, corresponding to the parts of the model which are not visible in the image, but apart from these peaks, it should also be small. These last two cases might have a similar overall chamfer distance, but inspection of this function might disambiguate them; the large local peaks in the second case might then be discounted, yielding a lower corrected distance value.