Screened Poisson Hyperfields for Shape Coding

We present a novel perspective on shape characterization using the screened Poisson equation. We discuss that the e↵ect of the screening parameter is a change of measure of the underlying metric space; also indicating a conditioned random walker biased by the choice of measure. A continuum of shape fields is created, by varying the screening parameter or equivalently the bias of the random walker. In addition to creating a regional encoding of the di↵usion with a di↵erent bias, we further break down the influence of boundary interactions by considering a number of independent random walks, each emanating from a certain boundary point, and the superposition of which yields the screened Poisson field. Probing the screened Poisson equation from these two complementary perspectives leads to a high-dimensional hyper-field: a rich characterization of the shape that encodes global, local, interior and boundary interactions. To extract particular shape information as needed in a compact way from the hyper-field, we apply various decompositions either to unveil parts of a shape or parts of boundary or to create consistent mappings. The latter technique involves lower dimensional embeddings, which we call Screened Poisson Encoding Maps (SPEM). The expressive power of the SPEM is demonstrated via illustrative experiments as well as a quantitative shape retrieval experiment over a public benchmark database on which the SPEM method shows a highranking performance among the existing state-of-the-art shape retrieval methods.