A 1 a 2 a 3 a 3 a 2 a 1 * * * * a 1 a 2 a 3 a 3 a 2 a 1

Figure 12: Example of interactions between equality constraints in a horizontal window. the window. The rst row of Figure 12 gives the values of n. The second row shows the equality constraints due to the symmetry of the corner and to the periodicity. The third row, nally, gives the equality constraints due to the symmetry of the image corner. It is clear form the second row that, for example, w h 0] = w h 5], but, from the third row we can see that w h 5] = w h 4] and this imply also w h 0] = w h 4] and from the second row it follows w h 4] = w h 1] that implies w h 0] = w h 1] = w h 4] = w h 5] and so on. It is clear that it is very diicult to give the equality constraints in a closed form that depends on N x , x and x. However, there exists a simple algorithm to compute them. Indeed, we want to know, for each couple of integer n 1 , n 2 between 0 and N x , if there is a chain of integers m 0 ; m 1 ; : : :; m ` , such that w h n 1 ] = w h m 0 ] = = w h m ` ] = w h n 1 ], where each equality descends directly from the symmetry constraints of the corner or of the image corner. This is the problem of nding the transitive closure of the relation \a and b are such that w h a] = w h b] because of the corner or the image corner symmetry" 9]. The easiest way to compute the transitive closure of a relation between M elements is to construct an M M matrix R, such that R ij = 1 if and only if the element i and the element j are in such relation and zero otherwise. (74) where the product and the sum used in (74) are, respectively, the boolean AND and the boolean OR. Expression (74) can be slow to compute (although it is not too slow for windows of normal size and current computers) and a more eecient algorithm exists, see 10].