Articulated Shape Matching Using Laplacian Eigenfunctions and Unsupervised Point Registration (2012.07340v1)

Abstract: Matching articulated shapes represented by voxel-sets reduces to maximal sub-graph isomorphism when each set is described by a weighted graph. Spectral graph theory can be used to map these graphs onto lower dimensional spaces and match shapes by aligning their embeddings in virtue of their invariance to change of pose. Classical graph isomorphism schemes relying on the ordering of the eigenvalues to align the eigenspaces fail when handling large data-sets or noisy data. We derive a new formulation that finds the best alignment between two congruent $K$-dimensional sets of points by selecting the best subset of eigenfunctions of the Laplacian matrix. The selection is done by matching eigenfunction signatures built with histograms, and the retained set provides a smart initialization for the alignment problem with a considerable impact on the overall performance. Dense shape matching casted into graph matching reduces then, to point registration of embeddings under orthogonal transformations; the registration is solved using the framework of unsupervised clustering and the EM algorithm. Maximal subset matching of non identical shapes is handled by defining an appropriate outlier class. Experimental results on challenging examples show how the algorithm naturally treats changes of topology, shape variations and different sampling densities.

• The paper's main contribution is the introduction of an eigenfunction-based method that initializes shape matching using histogram comparisons.
• It transforms dense shape matching into a graph matching task by employing unsupervised point registration and iterative EM clustering.
• Empirical results validate its robustness by accurately matching large voxel-sets with minimal unmatched points despite high noise and diverse sampling.

Articulated Shape Matching Using Laplacian Eigenfunctions and Unsupervised Point Registration

The paper presented introduces a method for matching articulated shapes using spectral graph theory and unsupervised point registration. This approach builds upon the foundations of maximal sub-graph isomorphism and spectral graph matching, providing a means to handle articulated shapes represented by voxel-sets. The proposed method focuses on aligning embeddings derived from Laplacian eigenfunctions, offering robustness against issues typically associated with large datasets and noisy data.

Key Highlights of the Paper

The principal innovation in this work is the introduction of a new formulation that initializes the shape matching process using eigenfunction signatures. The use of histograms to match these eigenfunction signatures offers a sophisticated means to start the alignment problem, enhancing the overall performance. The method translates dense shape matching into a graph matching task, which is further simplified through point registration of embeddings subjected to orthogonal transformations. This translation allows the problem to be approached within the framework of unsupervised clustering and the EM algorithm. Furthermore, the consideration of outlier classes helps in managing non-identical shapes, thereby improving the versatility of the approach.

Experimental Results and Numerical Strengths

Empirical evidence is provided through challenging examples, where the algorithm demonstrates its capability to address changes in topology, accommodate shape variations, and handle different sampling densities. For instance, in one experiment, two voxel-sets comprising 12,577 and 12,346 voxels were matched in a 7-dimensional space using the proposed method. Eigenfunction alignment effectively initialized the orthogonal transformation required, and EM converged after nine iterations, leaving only 90 voxels unmatched—illustrating the robustness and efficiency of the method.

Implications and Future Developments

From a theoretical perspective, the use of Laplacian eigenfunctions effectively captures the local geometric and topological properties of articulated shapes, making it well-suited for dimensional reduction and mapping shapes to a lower-dimensional space. The paper's approach transcends the limitations of classical graph isomorphism schemes by addressing the inherent difficulties in ordering eigenvalues, especially in large and noisy datasets. The introduction of eigenfunction histograms presents a novel way to match graphs, providing a potentially influential method for the community working on graph-based shape matching.

This work also paves the way for future research in AI and computational vision. The novel approach to spectral graph matching could be further explored in various applications, such as 3-D object recognition, motion analysis, and beyond. The techniques discussed offer a promising alternative for researchers interested in the problem of shape matching under complex transformations, including projects involving more complex datasets and higher-dimensional embeddings.

The paper sets a robust foundation for the future exploration of articulated shape matching using spectral methods and offers valuable insights into both the theoretical and practical aspects of graph-based matching algorithms. As AI continues to evolve, the implications of such work are likely to expand into broader applications across diverse fields dealing with spatial and geometric data.