On affine rigidity

We define the notion of affine rigidity of a hypergraph and prove a variety of fundamental results for this notion. First, we show that affine rigidity can be determined by the rank of a specific matrix which implies that affine rigidity is a generic property of the hypergraph.Then we prove that if a graph is is $(d+1)$-vertex-connected, then it must be "generically neighborhood affinely rigid" in $d$-dimensional space. This implies that if a graph is $(d+1)$-vertex-connected then any generic framework of its squared graph must be universally rigid. Our results, and affine rigidity more generally, have natural applications in point registration and localization, as well as connections to manifold learning.Comment: Updated abstrac

Large-Scale Sensor Network Localization via Rigid Subnetwork Registration

In this paper, we describe an algorithm for sensor network localization (SNL) that proceeds by dividing the whole network into smaller subnetworks, then localizes them in parallel using some fast and accurate algorithm, and finally registers the localized subnetworks in a global coordinate system. We demonstrate that this divide-and-conquer algorithm can be used to leverage existing high-precision SNL algorithms to large-scale networks, which could otherwise only be applied to small-to-medium sized networks. The main contribution of this paper concerns the final registration phase. In particular, we consider a least-squares formulation of the registration problem (both with and without anchor constraints) and demonstrate how this otherwise non-convex problem can be relaxed into a tractable convex program. We provide some preliminary simulation results for large-scale SNL demonstrating that the proposed registration algorithm (together with an accurate localization scheme) offers a good tradeoff between run time and accuracy.Comment: 5 pages, 8 figures, 1 table. To appear in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, April 19-24, 201

Nonrigid Registration of Brain Tumor Resection MR Images Based on Joint Saliency Map and Keypoint Clustering

This paper proposes a novel global-to-local nonrigid brain MR image registration to compensate for the brain shift and the unmatchable outliers caused by the tumor resection. The mutual information between the corresponding salient structures, which are enhanced by the joint saliency map (JSM), is maximized to achieve a global rigid registration of the two images. Being detected and clustered at the paired contiguous matching areas in the globally registered images, the paired pools of DoG keypoints in combination with the JSM provide a useful cluster-to-cluster correspondence to guide the local control-point correspondence detection and the outlier keypoint rejection. Lastly, a quasi-inverse consistent deformation is smoothly approximated to locally register brain images through the mapping the clustered control points by compact support radial basis functions. The 2D implementation of the method can model the brain shift in brain tumor resection MR images, though the theory holds for the 3D case

A Combinatorial Solution to Non-Rigid 3D Shape-to-Image Matching

We propose a combinatorial solution for the problem of non-rigidly matching a 3D shape to 3D image data. To this end, we model the shape as a triangular mesh and allow each triangle of this mesh to be rigidly transformed to achieve a suitable matching to the image. By penalising the distance and the relative rotation between neighbouring triangles our matching compromises between image and shape information. In this paper, we resolve two major challenges: Firstly, we address the resulting large and NP-hard combinatorial problem with a suitable graph-theoretic approach. Secondly, we propose an efficient discretisation of the unbounded 6-dimensional Lie group SE(3). To our knowledge this is the first combinatorial formulation for non-rigid 3D shape-to-image matching. In contrast to existing local (gradient descent) optimisation methods, we obtain solutions that do not require a good initialisation and that are within a bound of the optimal solution. We evaluate the proposed method on the two problems of non-rigid 3D shape-to-shape and non-rigid 3D shape-to-image registration and demonstrate that it provides promising results.Comment: 10 pages, 7 figure