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Distributed methods for synchronization of orthogonal matrices over graphs Thunberg, Johan ; Bernard, Florian ; Goncalves, Jorge in Automatica (2017), 80 This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We ... [more ▼] This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We formulate the synchronization problem as a least-squares optimization problem with nonlinear constraints. The synchronization problem appears as one of the key components in applications ranging from 3D-localization to image registration. The main contributions of this work can be summarized as the introduction of two novel algorithms; one for symmetric graphs and one for graphs that are possibly asymmetric. Under general conditions, the former has guaranteed convergence to the solution of a spectral relaxation to the synchronization problem. The latter is stable for small step sizes when the graph is quasi-strongly connected. The proposed methods are verified in numerical simulations. [less ▲] Detailed reference viewed: 15 (0 UL)Shape-aware surface reconstruction from sparse 3D point-clouds Bernard, Florian ; Salamanca Mino, Luis ; Thunberg, Johan et al in Medical Image Analysis (2017), 38 The reconstruction of an object’s shape or surface from a set of 3D points plays an important role in medical image analysis, e.g. in anatomy reconstruction from tomographic measurements or in the process ... [more ▼] The reconstruction of an object’s shape or surface from a set of 3D points plays an important role in medical image analysis, e.g. in anatomy reconstruction from tomographic measurements or in the process of aligning intra-operative navigation and preoperative planning data. In such scenarios, one usually has to deal with sparse data, which significantly aggravates the problem of reconstruction. However, medical applications often provide contextual information about the 3D point data that allow to incorporate prior knowledge about the shape that is to be reconstructed. To this end, we propose the use of a statistical shape model (SSM) as a prior for surface reconstruction. The SSM is represented by a point distribution model (PDM), which is associated with a surface mesh. Using the shape distribution that is modelled by the PDM, we formulate the problem of surface reconstruction from a probabilistic perspective based on a Gaussian Mixture Model (GMM). In order to do so, the given points are interpreted as samples of the GMM. By using mixture components with anisotropic covariances that are “oriented” according to the surface normals at the PDM points, a surface-based fitting is accomplished. Estimating the parameters of the GMM in a maximum a posteriori manner yields the reconstruction of the surface from the given data points. We compare our method to the extensively used Iterative Closest Points method on several different anatomical datasets/SSMs (brain, femur, tibia, hip, liver) and demonstrate superior accuracy and robustness on sparse data. [less ▲] Detailed reference viewed: 29 (9 UL)Integration of sparse electrophysiological measurements with preoperative MRI using 3D surface estimation in deep brain stimulation surgery Husch, Andreas Dominik ; ; Thunberg, Johan et al in Webster, Robert; Fei, Baowei (Eds.) Medical Imaging 2017: Image-Guided Procedures, Robotic Interventions, and Modeling (2017, February 14) Intraoperative microelectrode recordings (MER) have been used for several decades to guide neurosurgeons during the implantation of Deep Brain Stimulation (DBS) electrodes, especially when targeting the ... [more ▼] Intraoperative microelectrode recordings (MER) have been used for several decades to guide neurosurgeons during the implantation of Deep Brain Stimulation (DBS) electrodes, especially when targeting the subthalamic nucleus (STN) to suppress the symptoms of Parkinson’s Disease. The standard approach is to use an array of up to five MER electrodes in a fixed configuration. Interpretation of the recorded signals yields a spatiallyvery sparse set of information about the morphology of the respective brain structures in the targeted area. However, no aid is currently available for surgeons to intraoperatively integrate this information with other data available on the patient’s individual morphology (e.g. MR imaging data used for surgical planning). This integration might allow surgeons to better determine the most probable position of the electrodes within the target structure during surgery. This paper suggests a method for reconstructing a surface patch from the sparse MER dataset utilizing additional a-priori knowledge about the geometrical configuration of the measurement electrodes. The conventional representation of MER measurements as intervals of target region/non-target region is therefore transformed into an equivalent boundary set representation, allowing efficient point-based calculations. Subsequently, the problem is to integrate the resulting patch with a preoperative model of the target structure, which can be formulated as registration problem minimizing a distance measure between the two surfaces. When restricting this registration procedure to translations, which is reasonable given certain geometric considerations, the problem can be solved globally by employing an exhaustive search with arbitrary precision in polynomial time. The proposed method is demonstrated using bilateral STN/Substantia Nigra segmentation data from preoperative MRIs of 17 Patients with simulated MER electrode placement. When using simulated data of heavily perturbed electrodes and subsequent MER measuremen [less ▲] Detailed reference viewed: 10 (0 UL)Local Lyapunov Functions for Consensus in Switching Nonlinear Systems Thunberg, Johan ; Goncalves, Jorge ; in IEEE Transactions on Automatic Control (2017) This note presents two theorems on asymptotic state consensus of continuous time nonlinear multi-agent systems. The agents reside in Rm and have switching interconnection topologies. Both the first ... [more ▼] This note presents two theorems on asymptotic state consensus of continuous time nonlinear multi-agent systems. The agents reside in Rm and have switching interconnection topologies. Both the first theorem, formulated in terms of the states of individual agents, and the second theorem, formulated in terms of the pairwise states for pairs of agents, can be interpreted as variants of Lyapunov’s second method. The two theorems complement each other; the second provides stronger convergence results under weaker graph topology assumptions, whereas the first often can be applied in a wider context in terms of the structure of the right-hand sides of the systems. The second theorem also sheds some new light on well-known results for consensus of nonlinear systems where the right-hand sides of the agents’ dynamics are convex combinations of directions to neighboring agents. For such systems, instead of proving consensus by using the theory of contracting convex sets, a local quadratic Lyapunov function can be used. [less ▲] Detailed reference viewed: 27 (1 UL)A Combinatorial Solution to Non-Rigid 3D Shape-to-Image Matching Bernard, Florian ; ; Thunberg, Johan et al in The proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2017) 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 ... [more ▼] 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 the image and the 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 combinatorial 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. [less ▲] Detailed reference viewed: 13 (4 UL)Fast Correspondences for Statistical Shape Models of Brain Structures Bernard, Florian ; Vlassis, Nikos ; et al in SPIE Medical Imaging (2016, March) Detailed reference viewed: 47 (9 UL)An Iterative Projection method for Synchronization of Invertible Matrices Over Graphs Thunberg, Johan ; Colombo, Nicolo ; Yue, Zuogong et al in 22nd International Symposium on Mathematical Theory of Networks and Systems (2016) This paper addresses synchronization of invertible matrices over graphs. The matrices represent pairwise transformations between n euclidean coordinate systems. Synchronization means that composite ... [more ▼] This paper addresses synchronization of invertible matrices over graphs. The matrices represent pairwise transformations between n euclidean coordinate systems. Synchronization means that composite transformations over loops are equal to the identity. Given a set of measured matrices that are not synchronized, the synchronization problem amounts to fining new synchronized matrices close to the former. Under the assumption that the measurement noise is zero mean Gaussian with known covariance, we introduce an iterative method based on linear subspace projection. The method is free of step size determination and tuning and numerical simulations show significant improvement of the solution compared to a recently proposed direct method as well as the Gauss-Newton method. [less ▲] Detailed reference viewed: 72 (7 UL)Consensus and Formation Control on SE(3) for Switching Topologies Thunberg, Johan ; Goncalves, Jorge ; in Automatica (2016), 66 This paper addresses the consensus problem and the formation problem on SE(3) in multi-agent systems with directed and switching interconnection topologies. Several control laws are introduced for the ... [more ▼] This paper addresses the consensus problem and the formation problem on SE(3) in multi-agent systems with directed and switching interconnection topologies. Several control laws are introduced for the consensus problem. By a simple transformation, it is shown that the proposed control laws can be used for the formation problem. The design is first conducted on the kinematic level, where the velocities are the control laws. Then, for rigid bodies in space, the design is conducted on the dynamic level, where the torques and the forces are the control laws. On the kinematic level, first two control laws are introduced that explicitly use Euclidean transformations, then separate control laws are defined for the rotations and the translations. In the special case of purely rotational motion, the consensus problem is referred to as consensus on SO(3) or attitude synchronization. In this problem, for a broad class of local representations or parameterizations of SO(3), including the Axis–Angle Representation, the Rodrigues Parameters and the Modified Rodrigues Parameters, two types of control laws are presented that look structurally the same for any choice of local representation. For these two control laws we provide conditions on the initial rotations and the connectivity of the graph such that the system reaches consensus on SO(3). Among the contributions of this paper, there are conditions for when exponential rate of convergence occurs. A theorem is provided showing that for any choice of local representation for the rotations, there is a change of coordinates such that the transformed system has a well known structure. [less ▲] Detailed reference viewed: 61 (11 UL)Optimal output consensus for linear systems: a topology free approach Thunberg, Johan ; in Automatica (2016), 68 In this paper, for any homogeneous system of agents with linear continuous time dynamics, we formulate an optimal control problem. In this problem a convex cost functional of the control signals of the ... [more ▼] In this paper, for any homogeneous system of agents with linear continuous time dynamics, we formulate an optimal control problem. In this problem a convex cost functional of the control signals of the agents shall be minimized, while the outputs of the agents shall coincide at some given finite time. This is an instance of the rendezvous or finite time consensus problem. We solve this problem without any constraints on the communication topology and provide a solution as an explicit feedback control law for the case when the dynamics of the agents is output controllable. It turns out that the communication graph topology induced by the solution is complete. Based on this solution for the finite time consensus problem, we provide a solution to the case of infinite time horizon. Furthermore, we investigate under what circumstances it is possible to express the controller as a feedback control law of the output instead of the states. [less ▲] Detailed reference viewed: 34 (4 UL)Linear Shape Deformation Models with Local Support using Graph-based Structured Matrix Factorisation Bernard, Florian ; ; et al in Linear Shape Deformation Models with Local Support using Graph-based Structured Matrix Factorisation (2016) Representing 3D shape deformations by linear models in high-dimensional space has many applications in computer vision and medical imaging, such as shape-based interpolation or segmentation. Commonly ... [more ▼] Representing 3D shape deformations by linear models in high-dimensional space has many applications in computer vision and medical imaging, such as shape-based interpolation or segmentation. Commonly, using Principal Components Analysis a low-dimensional (affine) 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. In this paper, a method to obtain deformation factors with local support is presented. The benefits of such models include better flexibility and interpretability as well as the possibility of interactively deforming shapes locally. For that, based on a well-grounded theoretical motivation, we formulate a matrix factorisation problem employing sparsity and graph-based regularisation terms. We demonstrate that for brain shapes our method outperforms the state of the art in local support models with respect to generalisation ability and sparse shape reconstruction, whereas for human body shapes our method gives more realistic deformations. [less ▲] Detailed reference viewed: 30 (8 UL)Global Optimality Bounds for ICA Algorithms Colombo, Nicolo ; Thunberg, Johan ; Goncalves, Jorge in 22nd International Symposium on Mathematical Theory of Networks and Systems (2016) Independent Component Analysis is a popular statistical method for separating a multivariate signal into additive components. It has been shown that the signal separation problem can be reduced to the ... [more ▼] Independent Component Analysis is a popular statistical method for separating a multivariate signal into additive components. It has been shown that the signal separation problem can be reduced to the joint diagonalization of the matrix slices of some higher-order cumulants of the signal. In this approach, the unknown mixing matrix can be computed directly from the obtained joint diagonalizer. Various iterative algorithms for solving the non-convex joint diagonalization problem exist, but they usually lack global optimality guarantees. In this paper, we introduce a procedure for computing an optimality gap for local optimal solutions. The optimality gap is then used to obtain an empirical error bound for the estimated mixing matrix. Finally, a class of simultaneous matrix decomposition problems that admit such relaxation procedure is identified. [less ▲] Detailed reference viewed: 101 (6 UL)Inverse Problems for Matrix Exponential in System Identification: System Aliasing Yue, Zuogong ; Thunberg, Johan ; Goncalves, Jorge in 22nd International Symposium on Mathematical Theory of Networks and Systems (2016) This note addresses identification of the A-matrix in continuous time linear dynamical systems on state-space form. If this matrix is partially known or known to have a sparse structure, such knowledge ... [more ▼] This note addresses identification of the A-matrix in continuous time linear dynamical systems on state-space form. If this matrix is partially known or known to have a sparse structure, such knowledge can be used to simplify the identification. We begin by introducing some general conditions for solvability of the inverse problems for matrix exponential. Next, we introduce “system aliasing” as an issue in the identification of slow sampled systems. Such aliasing give rise to nonunique matrix logarithms. As we show, by imposing additional conditions on and prior knowledge about the A-matrix, the issue of system aliasing can, at least partially, be overcome. Under conditions on the sparsity and the norm of the A-matrix, it is identifiable up to a finite equivalence class. [less ▲] Detailed reference viewed: 36 (5 UL)Shape-aware 3D Interpolation using Statistical Shape Models Bernard, Florian ; Salamanca Mino, Luis ; Thunberg, Johan et al in Symposium on Statistical Shape Models and Applications, Delemont, Switzerland, October 2015 (2015, October) Detailed reference viewed: 98 (31 UL)Distributed high-gain attitude synchronization using rotation vectors ; Thunberg, Johan ; et al in Journal of Systems Science and Complexity (2015), 28(2), 289-304 This paper addresses attitude synchronization problems for systems of multiple rigid-body agents with directed interconnection topologies. Two scenarios which differ in available information are ... [more ▼] This paper addresses attitude synchronization problems for systems of multiple rigid-body agents with directed interconnection topologies. Two scenarios which differ in available information are considered. In the first scenario the agents can obtain their rotations and angular velocities relative to an inertial reference frame and transmit these information to their neighbors, while in the second scenario the agents can only obtain their own angular velocities and measure the relative rotations and relative angular velocities of their neighbors. By using rotation vectors and the high gain control, the authors provide torque control laws asymptotically synchronizing the rotations of the system almost globally for the first scenario and with initial rotations of the agents contained in a convex ball of SO(3) for the second scenario. An illustrative example is provided to show the synchronization results for both scenarios. [less ▲] Detailed reference viewed: 14 (3 UL)A solution for Multi-Alignment by Transformation Synchronisation Bernard, Florian ; Thunberg, Johan ; et al in The proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2015) 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 ... [more ▼] 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 lie in 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 globally consistent. Simulations demonstrate that for a high amount of noise, a large proportion of missing data and even for wrong correspondence assignments the method delivers encouraging results. [less ▲] Detailed reference viewed: 88 (30 UL)Transitively Consistent and Unbiased Multi-Image Registration Using Numerically Stable Transformation Synchronisation Bernard, Florian ; Thunberg, Johan ; Salamanca Mino, Luis et al in MIDAS Journal (2015) Abstract. Transitive consistency of pairwise transformations is a desir- able property of groupwise image registration procedures. The transfor- mation synchronisation method [4] is able to retrieve ... [more ▼] Abstract. Transitive consistency of pairwise transformations is a desir- able property of groupwise image registration procedures. The transfor- mation synchronisation method [4] is able to retrieve transitively con- sistent pairwise transformations from pairwise transformations that are initially not transitively consistent. In the present paper, we present a numerically stable implementation of the transformation synchronisa- tion method for a ne transformations, which can deal with very large translations, such as those occurring in medical images where the coor- dinate origins may be far away from each other. By using this method in conjunction with any pairwise (a ne) image registration algorithm, a transitively consistent and unbiased groupwise image registration can be achieved. Experiments involving the average template generation from 3D brain images demonstrate that the method is more robust with re- spect to outliers and achieves higher registration accuracy compared to reference-based registration. [less ▲] Detailed reference viewed: 99 (16 UL)Dynamical Structure Function and Granger Causality: Similarities and Differences Yue, Zuogong ; Thunberg, Johan ; et al in 54th IEEE Conference on Decision and Control, Osaka, Japan, December 15-18, 2015 (2015) Detailed reference viewed: 45 (11 UL)Distributed attitude synchronization using backstepping and sliding mode control Thunberg, Johan ; ; et al in Control Theory and Technology (2014), 12(1), 48-55 We consider the problem of attitude synchronization for systems of rigid body agents with directed topologies. Two different scenarios for the rotation matrices of the agents are considered. In the first ... [more ▼] We consider the problem of attitude synchronization for systems of rigid body agents with directed topologies. Two different scenarios for the rotation matrices of the agents are considered. In the first scenario, the rotations are contained in a convex subset of SO(3), which is a ball of radius less than π/2, whereas in the second scenario the agents are contained in a subset of SO(3), which is a ball of radius less than π. Using a control law based on backstepping and sliding mode control, we provide distributed, semi-global, torque control laws for the agents so that the rotations asymptotically synchronize. The control laws for the agents in the first scenario only depend on the relative rotations between neighboring agents, whereas the control laws in the second scenario depend on rotations defined in a global coordinate frame. Illustrative examples are provided where the synchronization is shown for both scenarios. [less ▲] Detailed reference viewed: 28 (0 UL)Consensus and Pursuit-Evasion in Nonlinear Multi-Agent Systems Thunberg, Johan Doctoral thesis (2014) Within the field of multi-agent systems theory, we study the problems of consensus and pursuit-evasion. In our study of the consensus problem, we first provide some theoretical results and then consider ... [more ▼] Within the field of multi-agent systems theory, we study the problems of consensus and pursuit-evasion. In our study of the consensus problem, we first provide some theoretical results and then consider the problem of consensus on SO(3) or attitude synchronization. In Chapter 2, for agents with states in Rm, we present two theorems along the lines of Lyapunov’s second method that, under different conditions, guarantee asymptotic state consensus in multi-agent systems where the interconnection topologies are switching. The first theorem is formulated by using the states of the agents in the multiagent system, whereas the second theorem is formulated by using the pairwise states for pairs of agents in the multi-agent system. In Chapter 3, the problem of consensus on SO(3) for a multi-agent system with directed and switching interconnection topologies is addressed. We provide two different types of kinematic control laws for a broad class of local representations of SO(3). The first control law consists of a weighted sum of pairwise differences between positions of neighboring agents, expressed as coordinates in a local representation. The structure of the control law is well known in the consensus community for being used in systems of agents in the Euclidean space, and here we show that the same type of control law can be used in the context of consensus on SO(3). In a later part of this chapter, based on the kinematic control laws, we introduce torque control laws for a system of rigid bodies in space and show that the system reaches consensus when these control laws are used. Chapter 4 addresses the problem of consensus on SO(3) for networks of uncalibrated cameras. Under the assumption that each agent uses a camera in order to measure its rotation, we prove convergence to the consensus set for two types of kinematic control laws, where only conjugate rotation matrices are available for the agents. In these conjugate rotations, the rotation matrix can be seen as distorted by the (unknown) intrinsic parameters of the camera. For the conjugate rotations we introduce distorted versions of well known local parameterizations of SO(3) and show consensus by using control laws that are similar to the ones in Chapter 3, with the difference that the distorted local representations are used instead. In Chapter 5, we study the output consensus problem for homogeneous systems of agents with linear continuous time-invariant dynamics. We derive control laws that solve the problem, while minimizing a cost functional of the control signal. Instead of considering a fixed communication topology for the system, we derive the optimal control law without any restrictions on the topology. We show that for all linear output controllable homogeneous systems, the optimal control law uses only relative information but requires the connectivity graph to be complete and in general requires measurements of the state errors. We identify cases where the optimal control law is only based on output errors. In Chapter 6, we address the multi-pursuer version of the visibility pursuit-evasion problem in polygonal environments. By discretizing the problem and applying a Mixed Integer Linear Programming (MILP) framework, we are able to address problems requiring so-called recontamination and also impose additional constraints, such as connectivity between the pursuers. The proposed MILP formulation is less conservative than solutions based on graph discretizations of the environment, but still somewhat more conservative than the original underlying problem. It is well known that MILPs, as well as multi-pursuer pursuit-evasion problems, are NP-hard. Therefore we apply an iterative Receding Horizon Control (RHC) scheme, where a number of smaller MILPs are solved over shorter planning horizons. The proposed approach is illustrated by a number of solved examples. [less ▲] Detailed reference viewed: 63 (10 UL)Distributed attitude synchronization control of multi-agent systems with switching topologies Thunberg, Johan ; ; et al in Automatica (2014), 50(3), 832-840 This paper addresses the attitude synchronization problem in multi-agent systems with directed and switching interconnection topologies. Two cases for the synchronization problem are discussed under ... [more ▼] This paper addresses the attitude synchronization problem in multi-agent systems with directed and switching interconnection topologies. Two cases for the synchronization problem are discussed under different assumptions about the measurable information. In the first case the agents can measure their rotations relative to a global reference coordinate frame, whilst in the second case they can only measure the relative rotations between each other. Two intuitive distributed control laws based on the axis–angle representations of the rotations are proposed for the two cases, respectively. The invariance of convex balls in SO(3) is guaranteed. Moreover, attitude synchronization is ensured under the well-known mild switching assumptions, the joint strong connection for the first case and joint quasi-strong connection for the second case. To show the effectiveness of the proposed control schemes, illustrative examples are provided. [less ▲] Detailed reference viewed: 14 (2 UL) |
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