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The alignment of a set of objects by means of transformations plays an important role in computer vision. Whilst the case for only two objects can be solved globally, when multiple objects are considered usually iterative methods are used. In practice the iterative methods perform well if the relative transformations between any pair of objects are free of noise. However, if only noisy relative transformations are available (e.g. due to missing data or wrong correspondences) the iterative methods may fail.Based on the observation that the underlying noise-free transformations can be retrieved from the null space of a matrix that can directly be obtained from pairwise alignments, this paper presents a novel method for the synchronisation of pairwise transformations such that they are transitively consistent.Simulations demonstrate that for noisy transformations, a large proportion of missing data and even for wrong correspondence assignments the method delivers encouraging results.

The Kuramoto model of a system of coupled phase oscillators describe synchronization phenomena in nature. We propose a generalization of the Kuramoto model where each oscillator state lives on the compact, real Stiefel manifold St(p, n). Previous work on high-dimensional Kuramoto models have largely been influenced by results and techniques that pertain to the original model. This paper uses optimization and control theory to prove that the generalized Kuramoto model on St(p, n) converges to a completely synchronized state for any connected graph from almost all initial conditions provided (p, n) satisfies p ≤ 2 3 n − 1 and all oscillator frequencies are equal. This result could not have been predicted based on knowledge of the Kuramoto model in complex networks on the circle with homogeneous oscillator frequencies. In that case, almost global synchronization is graph dependent; it applies if the network is acyclic or sufficiently dense. The problem of characterizing all such graphs is still open. This paper hence identifies a property that distinguishes many highdimensional generalizations of the Kuramoto model from the original model. It should therefore have important implications for modeling of synchronization phenomena in physics and control of multi-agent systems in engineering applications.

α = +[ PCA (global support) Our (local support) PCA (global support) Our (local support) PCA (global support)Our (local support)Global support factors of PCA lead to implausible body shapes, whereas the local support factors of our method give more realistic results. See our accompanying video for animated results. AbstractRepresenting 3D shape deformations by highdimensional linear models has many applications in computer vision and medical imaging. Commonly, using Principal Components Analysis a low-dimensional subspace of the high-dimensional shape space is determined. However, the resulting factors (the most dominant eigenvectors of the covariance matrix) have global support, i.e. changing the coefficient of a single factor deforms the entire shape. Based on matrix factorisation with sparsity and graph-based regularisation terms, we present a method to obtain deformation factors with local support. The benefits include better flexibility and interpretability as well as the possibility of interactively deforming shapes locally. We demonstrate that for brain shapes our method outperforms the state of the art in local support models with respect to generalisation and sparse reconstruction, whereas for body shapes our method gives more realistic deformations. 0 c 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

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