Shape from Phase: An Integrated Level Set and Probability Density Shape Representation

The past twenty years has seen the explosion of the "shape zoo": myriad shape representations, each with pros and cons. Of the varied denizens, distance transforms and density function shape representations have proven to be the most utile. Distance transforms inherit the numerous geometric advantages of implicit curve representations while density functions are unmatched in their approach to the modeling of uncertainty and noise in shape features. We have not seen much rapprochement between these two representations in general. In this work, we introduce a complex wave representation (CWR) of shape which has the ability to simultaneously carry probabilistic information via its magnitude and geometric information via its phase, achieving an integration of distance transforms and density function shape representations. The CWR is a parametric representation with cluster centers akin to a mixture model and curve normal information akin to signed distance functions. We demonstrate the perceptual gains of the CWR, highlight the advantages of the probabilistic aspect for noisy shape alignment by a likelihood approach, and fusing both aspects we show that the CWR leads to a feature space in which kernel PCA yields approximate closed curves and probability density functions.