The extraction of symmetries as quench points of propagating orientation from edge maps of grey scale images is faced with fundamental theoretical and computational problems. Theoretical difficulties arise because low-level processes do not yield perfect information and symmetries are drastically altered with small changes such as (i) missing edges (gaps) (ii) occlusion and parts, (iii) spurious edges. alter the underlying symmetry set. While the full symmetry set retains much of the original figure's symmetries it does so at the expense of bringing to bear many unintuitive branches, thus requiring further selection for object recognition. In this paper, we view the full symmetry set as the superposition of shocks arising from {\em multiple generations of waves}: the quenching points of the waves from the initial edge map constitute the first generation of shocks. A second generation of waves initiated at these points, simulate {\em interpenetrating waves} and generate a second generation of shocks, and so on until no further shocks can be formed. This view of the full symmetry set supports a {\em selective} continuation of waves, e.g., at shock loops to remove spurious edges, and at shock-hypothesized limbs to partition shape and close boundary gaps. This selective continuation of waves brings out relevant symmetries, but avoids the ambiguity of the full symmetry set. The second point of the paper is to propose the use of a computational framework based on the Contour-based Euclidean Distance Transform (CEDT) for shock detection, classification, labeling, as well as for simulating interpenetrating waves and multiple generation shocks described above. The key feature of CEDT that makes this possible is the explicit simultaneous propagation of orientation and distance, as well as additional features, e.g. shock labels,which are necessary to deal with the theoretical issues raised above. In addition, CEDT is exact and of very low numerical complexity. The results for a number of illustrative examples indicate the suitability of this framework for the recovery of object symmetries from real imagery.