Many-to-Many Feature Matching Using Spherical Coding of Directed Graphs

Abstract

Abstract. In recent work, we presented a framework for many-to-many matching of multi-scale feature hierarchies, in which features and their relations 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 Matousek [13]. However, the method was limited by the fact that two graphs to be matched were typically embedded into vector spaces with different 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 efficient embedding procedure based on a spherical coding of directed 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 matching algorithms exist, such as the Earth Mover's Distance. We apply the approach to the problem of multi-scale, view-based object recognition, in which an image is decomposed into a set of blobs and ridges with automatic scale selection.

Learning Hierarchical Shape Models from Examples

“…Our technique is based on a recent approach to this problem, proposed by Demirci et al [16], which transforms the many-to-many graph matching problem to a many-to-many weighted point matching problem, for which an efficient algorithm exists. Given a shortest-path distance matrix encoding node-to-node distances, the algorithm employs a spherical coding technique to yield a low-distortion embedding of the nodes in a low-dimensional Euclidean space (we adopt a simpler, spectral embedding technique).…”

“…A number of techniques are available for embedding the distance matrix into Euclidean space; examples include metric tree embedding [17], spherical codes [16], and ISOMAP [19]. We adopt a spectral embedding of a distance matrix computed in terms of shortest paths between nodes in a blob graph, similar to [19].…”

Video-Based Face Recognition Using Earth Mover’s Distance

“…In order to define the similarity function f (A, B) between two videos, we introduce the notion of Earth Mover's Distance (EMD). EMD is a general distance measure with application to image retrieval [12][13] and graph matching [16] [17]. It is proved much better than other well-known metrics (e.g., Euclidean distance between two vectors).…”

Object Categorization and the Need for Many-to-Many Matching

“…Moreover, the computed flows between piles and holes can be translated back to the original problem, yielding a many-to-many node correspondence between the original graphs. Details can be found in [4,2].…”