Higher Dimensional Affine Registration and Vision Applications

Abstract

Abstract-Affine registration has a long and venerable history in computer vision literature, and in particular, extensive work has been done for affine registration in IR 2 and IR 3 . This paper studies affine registration in IR m with m typically ranging from 4 to 12. To justify breaking of this dimension barrier, the first part of the paper describes three novel matching problems that can be formulated and solved as affine point-set registration problems in dimensions greater than three: stereo correspondence under motion, image set matching, and covariant point-set matching, problems that are not only interesting in their own right but also have potential for important vision applications. Unfortunately, most of the existing affine registration algorithms do not generalize easily to higher dimensions due to their inefficiency. Therefore, the second part of this paper develops a novel algorithm for estimating the affine transform between two point sets in IR m . Specifically, the algorithm follows the common approach of iteratively solving the correspondences and transform. The initial correspondences are determined using the novel notion of local spectral features, features constructed from local distance matrices. Unlike many correspondence-based methods, the proposed algorithm is capable of registering point sets of different size, and the use of local features provides some degree of robustness against noise and outliers. The proposed algorithm is validated on a variety of synthetic point sets in different dimensions with varying degrees of deformation and noise, and the paper also shows experimentally that several instances of the aforementioned three matching problems can indeed be solved satisfactorily using the proposed affine registration algorithm.

Unsupervised Word Mapping Using Structural Similarities in Monolingual Embeddings

“…The unsupervised mapping problem arises in other contexts where an optimal alignment between two isomorphic point sets is sought. In image registration and shape recognition, various efficient methods can be used to find an optimal alignment between two sets of low-dimensional points that correspond to images with various degrees of deformation (Myronenko and Song, 2010;Chi et al, 2008). In manifold learning, two sets of related high-dimensional points are projected into a shared lower dimensional space where the points can be compared and mapped to one other, such as the alignment of isomorphic protein structures (Wang and Mahadevan, 2009) and cross-lingual document alignment with unsupervised topic models (Diaz and Metzler, 2007;Wang and Mahadevan, 2008).…”

“…To analyze local structures in monolingual vector spaces, we treat each word embedding as a point in a high-dimensional space and further embed each point into a local invariant feature space, as proposed in (Chi et al, 2008) for affine registration of image point sets. The local invariant features are produced through eigendecomposition of the k-nearestneighbor (knn) graph for each point in the vector space as described below.…”

A 3D point matching algorithm for affine registration

“…Interestingly enough, they took a different route in addressing 3D point matching problem [25], in which an algebraic approach for registrating point sets in R 2 and R 3 using higher-order moments and sophisticated polynomial optimization was proposed. In another paper [26], they addressed affine registration in R m for m > 3 using spectral algorithms. We investigated the generalization of their original method for 2D point matching to 3D point matching using quaternion representation of 3D points, which turns out to preserve all the positive properties of the algorithm in 2D case.…”