We are currently implementing several spatial relations that are of particular interest such as region adjacency, nearness, overlap and surround. These examples of spatial reasoning can be inferred from the 2-D string [18]. Scale invariance is also provided in the 2-D string, but rotation invariance is not.
The adjacency of regions is resolved from the 2-D string by determining whether or not other regions exist between two candidate regions. The nearness of regions is resolved using adjacency in the 2-D string, but since the 2-D string provides no metric information, it requires an additional computation of the region distance. Since the 2-D string captures the orthogonal relations of the regions in the image, conditions of overlap and surround can also be determined [18]. Overlapping regions require not only the determination of nearness but also require the evaluation of the intersection of the MBRs of the regions. The case of surround is determined by unifying sub-goal queries that evaluate regions to the north, south, east and west of the surrounded-by region.
Under certain conditions the user may desire invariance in scaling and/or rotation. The 2-D string provides scale invariance since a change in scale does not change the order of symbols in the string. However, rotation invariance is not provided in the 2-D string. Therefore, we approximate rotation invariance around the image center point by providing two projections of each image, one normal projection, and another based upon a rotation by 45 degrees.
Figure: 2-D strings rotation invariance (a) 
(A < C < B, ABC), (b) 
(ABC, A < C < B), (c) 
(A < C < B, ABC) with projection onto the rotated axis.
In this case, the rotated 2-D string is extracted by the projection onto the diagonals of the image as illustrated in Figure 11(b). Image rotation by is provided by swapping the x and y projections as illustrated in Figure 12(a) and (b). Rotation by is detected in the rotated 2-D string as illustrated in Figure 12(a) and (c). Other arbitrary rotations are approximated by either the 2-D string or the rotated 2-D string.