Additional Remarks on Approximate Marginalization for Symmetry Detection
R. Lasowski, A. Tevs , H.-P. Seidel, M. Wand: A Probabilistic Framework for Partial Intrinsic Symmetries in Geometric
Data (ICCV 2009).
publication of the paper, we have studied the problem of finding
symmetries by computing marginals in Markov Random Fields more in
detail. We have found some important aspects that we were not aware of
at the time of publication of the original paper but that might be
relevant for applying the technique in practice:
A technical key tool in the paper
is using sum-product loopy belief propagation for computing approximate
marginals. However, it turns out that this algorithm is susceptible to
symmetry breaking. If loopy belief propagation is run for an extended
time, it is likely to happen that the marginal distributions converge
to a uni-modal distribution with only a single peak, discarding the
symmetry information. This problem can often (but not always) be
avoided by using a rather large convergence threshold for stopping the
LBP algorithm. The implementation described in our ICCV 2009 paper does
this implicitly, but we were not yet aware of the importance of doing
You can find a detailed analysis of the problems of loopy belief
propagation in the context of symmetry detection from marginals in Silke Jansen's master thesis available here.
The master thesis examines the problem of detecting symmetries in 2D
images, were the mentioned problems are particularly clearly
observable. The thesis also proposes some solution strategies that
reduce the symmetry breaking issues.
We would like to emphasize that this problem is at the side of the
inference algorithm, not the probabilistic model itself. We have
verified this by using non-loopy graphs as template that are matched to
large quantities of geometry. The resulting marginals are very well
peaked and give very useful cues for extracting symmtries in practice.
No convergence issues are observed in this case, where exact inference
is possible. We are currently preparing a more detailed publication on
Last modified: Michael Wand, September 3rd 2010.