Notes on Symmetry

Many structures of interest are approximately symmetric, meaning that a reflected version of the object could also be an example of the class. For instance, though an individual face is rarely strictly symmetric, a reflection of a face is also an example of a face. When building a model from such objects the points should be placed symmetrically (i.e. for every point on the left there should be an equivalent point on the right). In that case each annotated example can be used twice, once in its original form, once in its reflected state.

However, when reflecting sets of points, the index numbers must be permuted.

For instance, in the case above when reflected the left eye (Pt 1) becomes the right eye (Pt 0).
Many of the model builders can take advantage of this to get two examples for every set of points.
The parameter files include a line defining the permutation of points under symmetry, for instance for the 28 point face markup in the examples:
reflection_symmetry: { 1 0 3 2 7 6 5 4 13 12 11 10 9 8 14 16 15 17 18 19 21 20 23 22 26 27 24 25 } The i-th element in this list indicates the new index of point i after the reflection. Points on the axis of symmetry, such as the nose tip (Pt 14) in the example above, are unchanged.

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