This paper introduces and demonstrates a computational pipeline for the statistical analysis of shape graph datasets, namely geometric networks embedded in 2D or 3D spaces. Unlike traditional abstract graphs, our purpose is not only to retrieve and distinguish variations in the connectivity structure of the data but also geometric differences of the network branches. Our proposed approach relies on the extraction of a specifically curated and explicit set of topological, geometric and directional features, designed to satisfy key invariance properties. We leverage the resulting feature representation for tasks such as group comparison, clustering and classification on cohorts of shape graphs. The effectiveness of this representation is evaluated on several real-world datasets including urban road/street networks, neuronal traces and astrocyte imaging. These results are benchmarked against several alternative methods, both feature-based and not.
Complex network structures are ubiquitous in both natural systems and engineered environments. From the multiscale ramified branching of neuronal dendrites and other cellular architectures, to the arrangement of blood vessels or lung airways in the human anatomy or root systems in plants, to the sprawling layout of urban road networks and power grids, these structures define the functionality and efficiency of the systems they represent. Unlike standard mathematical graphs that are entirely described by the connectivity information, as for e.g. social or communication networks, many of these examples rather involve graphs embedded in 2D or 3D physical space with the connection between two nodes being itself a geometric curve whose shape may be an equally relevant property of the object. This type of structure is typically referred to as a spatial network [1] in the field of network theory, but has also been an important and longstanding subject of interest in the area of statistical shape analysis, where several recent works [2][3][4] have used the term shape graph to designate such geometric objects. One important specificity of this line of work on shape graphs is the emphasis on the need for a combined analysis of the underlying branching structure (i.e., the graph topology) together with the geometric characteristics of the branches themselves, thus placing shape graphs at the intersection between topology and geometry.
Our primary interest in this work lies in population morphological analysis, that is, in solving problems such as clustering or classification on ensembles of shape graphs. Although being basic standard tasks in statistics and machine learning, these tasks become much more challenging when considering data points that are shape graphs rather than usual Euclidean vectors. Indeed, the unique challenge in this case is to deal with the variable and inconsistent topology from one data point to another, combined with the set of geometric invariances (e.g., to rigid transformation or parameterization) that are required to properly compare such objects.
One of the predominant lines of work towards that goal has been to rely on the definition of a proper metric notion (often a Riemannian metric) on the space of shape graphs from which one can derive a number of statistical analysis tools. This is the case, for instance, of frameworks such as Gromov-Hausdorff, its extension known as the Gromov-Wasserstein metric [5] and other variants which have been applied to problems similar to this paper [6][7][8]. Other related works have alternatively considered extensions of the elastic [2][3][4] or diffeomorphic metric setting [9] from the field of Riemannian shape analysis. One of the known pitfalls of all such approaches, however, is their relatively high computational footprint, as the estimation of just a single distance/geodesic typically involves solving an intricate high-dimensional (often nonconvex) optimization problem. This can be a daunting challenge in problems such as clustering of large datasets, which require computations of pairwise distances between points.
A second main body of work has instead proposed relying on the extraction of features and carrying out statistical analysis tasks in feature space. This approach is usually applied to standard (topological) graphs where features such as average node degree, assortativity, and spectral entropy are well-established to compare the structure of different networks. However, for shape graphs (or spatial networks), the current state-of-the-art tends to be more scattered, with different works having proposed limited numbers of features tailored to a particular data application, such as transport [1], geographic [10], neuronal [11,12] or arterial [13] networks. In contrast to such explicitly engineered features, many recent methods are rather attempting to estimate the feature embedding itself, usually via neural network architectures adapted to the structure of shape graph data. Such methods include, most notably, frameworks derived from graph neural networks [14,15], the PointNet model of [16] (designed for 2D and 3D point clouds) and its variants, or other geometric deep learning architectures [17][18][19]. Nevertheless, there are several outstanding challenges for the applicability of such models to many real-world datasets, such as those we consider in this work. One major challenge is the size of the dataset. In many biomedical datasets, including the datasets we consider in this work, the size of the data ranges from a few hundred to a few thousands at most, which is insufficient to train the learning models computed in [14] or [16] using tens of thousands of samples or more. Another more fundamental obstacle is the difficulty to incorporate all geometric invariances into neural network architectures, in particular invariances to shape graph parametrization, as evidenced empirically in [18]. Last but not least, feature representation o
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