Robust elastic 2D/3D geometric graph matching

Abstract

We present an algorithm for geometric matching of graphs embedded in 2D or 3D space. It is applicable for registering any graph-like structures appearing in biomedical images, such as blood vessels, pulmonary bronchi, nerve fibers, or dendritic arbors. Our approach does not rely on the similarity of local appearance features, so it is suitable for multimodal registration with a large difference in appearance. Unlike earlier methods, the algorithm uses edge shape, does not require an initial pose estimate, can handle partial matches, and can cope with nonlinear deformations and topological differences.The matching consists of two steps. First, we find an affine transform that roughly aligns the graphs by exploring the set of all consistent correspondences between the nodes. This can be done at an acceptably low computational expense by using parameter uncertainties for pruning, backtracking as needed. Parameter uncertainties are updated in a Kalman-like scheme with each match.In the second step we allow for a nonlinear part of the deformation, modeled as a Gaussian Process. Short sequences of edges are grouped into superedges, which are then matched between graphs. This allows for topological differences. A maximum consistent set of superedge matches is found using a dedicated branch-and-bound solver, which is over 100 times faster than a standard linear programming approach. Geometrical and topological consistency of candidate matches is determined in a fast hierarchical manner.We demonstrate the effectiveness of our technique at registering angiography and retinal fundus images, as well as neural image stacks.

Non-Rigid Graph Registration Using Active Testing Search

“…Global nodal matches are then estimated using multidimensional optimization schemes such as graduated assignment [16], spectral techniques [18], [19], [38] or considering the graphs as an absorbing Markov chain [7]. Considering compatibilities as binary tests, the largest consistent set of matches corresponds to the maximum weighted independent set or equivalently, to the maximum weighted clique [12], [31]. Due to its high computational cost, the method is only applicable to small graphs.…”

Robust non-rigid registration of 2D and 3D graphs

“…We also clearly outperform [21], mainly for large levels of deformation. While our non-linear algorithm is able to warp the graph while searching for matching nodes, the affine search of [21] only yields reasonable results for low levels of deformation.…”

“…For the synthetic experiments we compare our approach (denoted Non-Linear GP) to the Coherent Point Drift (CPD) [16], which is a representative example of the state-of-the-art in non-rigid point matching and shape recovery. We also compare it against [21], which uses a Kalman filter based approach to learn an initial affine transform and refines the output with a local non-linear warping. We refer this approach as Affine Kalman.…”

“…While our non-linear algorithm is able to warp the graph while searching for matching nodes, the affine search of [21] only yields reasonable results for low levels of deformation. In addition, observe that the magnitudes of error obtained by our approach are in fact very good approximations.…”

“…The final class of methods involves simultaneously searching for correspondences and estimating the transformation parameters using a Kalman filter approach [14,22,20,23,21]. As soon as a few initial correspondences have been established, the set of potential correspondences is rapidly reduced, making the search complexity manageable.…”