Any pair of 2D curves is consistent with a 3D symmetric interpretation

Symmetry has been shown to be a very effective a priori constraint in solving a 3D shape recovery problem. Symmetry is useful in 3D recovery because it is a form of redundancy. The effectiveness of symmetry constraint is illustrated in Demo 1. There are, however, some fundamental limits to the effectiveness of symmetry. Specifically, given two arbitrary curves in a single 2D image, one can always find a 3D mirror-symmetric interpretation of these curves under quite general assumptions. This fact is illustrated in Demos 2-8. These demos show a number of 2D curves that do not look like 2D projections of symmetric pairs of 3D curves. However, each of the 2D pairs of curves do have 3D symmetric interpretations. These interpretations are shown as well. Note that the 3D symmetric interpretations always correspond to degenerate views.



Demo 1. A symmetric pair of 3D curves produced from the pair of 2D curves in Figure 1.

Demo 2. A symmetric pair of 3D curves produced from the pair of 2D curves in Figure 2.

Demo 3. A symmetric pair of 3D curves produced from the pair of 2D curves in Figure 3.

Demo 4. A symmetric pair of 3D curves produced from the pair of 2D curves in Figure 5.

Demo 5. A symmetric pair of 3D curves produced from the pair of 2D curves in Figure 7.

Demo 6. A symmetric pair of 3D curves produced from the pair of 2D curves in Figure 8.

Demo 7. A symmetric pair of 3D curves produced from the pair of 2D curves in Figure 12.

Demo 8. Two different 3D symmetric curves produced from the same 2D curve in Figure 13.