Abstract

In recent work, we presented a framework for many-to-many matching of multi-scale feature hierarchies, in which features and their re- lations were captured in a vertex-labeled, edge-weighted directed graph. The algorithm was based on a metric-tree representation of labeled graphs and their metric embedding into normed vector spaces, using the embedding algorithm of Matoufisek [13]. However, the method was limited by the fact that two graphs to be matched were typically embedded into vector spaces with difierent dimensionality. Before the embeddings could be matched, a dimensionality reduction technique (PCA) was required, which was both costly and prone to error. In this paper, we introduce a more eficient embedding procedure based on a spherical coding of di- rected graphs. The advantage of this novel embedding technique is that it prescribes a single vector space into which both graphs are embedded. This reduces the problem of directed graph matching to the problem of geometric point matching, for which efficient many-to-many match- ing algorithms exist, such as the Earth Mover’s Distance. We apply the approach to the problem of multi-scale, view-based ob ject recognition, in which an image is decomposed into a set of blobs and ridges with automatic scale selection.