In three experiments a simple Euclidean transformation (reflection, translation, rotation) was applied to collections of twelve dots in such a way that they contained equal lower-order structure, defined on the pairwise grouping of elements with their partner following transformation (e.g. parallel virtual lines), but differed in the presence vs absence of higher-order structure, defined on pairs of pairwise groupings (e.g. virtual quadrangles with correlated angles). Based on the much better performance levels (d') in the case of additional higher-order structure, we conclude that global regularities are easier to detect when the local correspondences are supported by higher-order ones formed between them. These enable the lower-order groupings to spread out across the whole pattern very rapidly (called bootstrapping). As a preliminary attempt to specify these principles, we proposed a working model with two basic components: first, a function expressing the cost of a perceptual grouping or the lack of regularity, and, secondly, an algorithm based on simulated annealing to minimize the cost function. The simulation results obtained with our current implementation of these principles showed satisfactory qualitative agreement with human regularity detection performance. Finally, the theory was shown to capture the essence of a large number of grouping phenomena taken from diverse domains such as detection of symmetry in dot patterns, global structure in Glass and vector patterns, correspondence in stereoscopic transparency and apparent motion. Therefore, we are convinced that, in principle, the mechanism used by the human visual system to detect regularity incorporates something like bootstrapping based on higher-order structure. We regard this as a promising step towards unraveling the intriguing mechanisms of classic Gestalt phenomena.