PurposeIn the surgical treatment for lower-leg intra-articular fractures, the fragments have to be positioned and aligned to reconstruct the fractured bone as precisely as possible, to allow the joint to function correctly again. Standard procedures use 2D radiographs to estimate the desired reduction position of bone fragments. However, optimal correction in a 3D space requires 3D imaging. This paper introduces a new navigation system that uses pre-operative planning based on 3D CT data and intra-operative 3D guidance to virtually reduce lower-limb intra-articular fractures. Physical reduction in the fractures is then performed by our robotic system based on the virtual reduction.Methods3D models of bone fragments are segmented from CT scan. Fragments are pre-operatively visualized on the screen and virtually manipulated by the surgeon through a dedicated GUI to achieve the virtual reduction in the fracture. Intra-operatively, the actual position of the bone fragments is provided by an optical tracker enabling real-time 3D guidance. The motion commands for the robot connected to the bone fragment are generated, and the fracture physically reduced based on the surgeon’s virtual reduction. To test the system, four femur models were fractured to obtain four different distal femur fracture types. Each one of them was subsequently reduced 20 times by a surgeon using our system.ResultsThe navigation system allowed an orthopaedic surgeon to virtually reduce the fracture with a maximum residual positioning error of (translational) and (rotational). Correspondent physical reductions resulted in an accuracy of 1.03 ± 0.2 mm and , when the robot reduced the fracture.ConclusionsExperimental outcome demonstrates the accuracy and effectiveness of the proposed navigation system, presenting a fracture reduction accuracy of about 1 mm and , and meeting the clinical requirements for distal femur fracture reduction procedures.Electronic supplementary materialThe online version of this article (doi:10.1007/s11548-016-1418-z) contains supplementary material, which is available to authorized users.

This paper is devoted to the estimation of the Lipschitz constant of neural networks using semidefinite programming. For this purpose, we interpret neural networks as time-varying dynamical systems, where the k-th layer corresponds to the dynamics at time k. A key novelty with respect to prior work is that we use this interpretation to exploit the series interconnection structure of neural networks with a dynamic programming recursion. Nonlinearities, such as activation functions and nonlinear pooling layers, are handled with integral quadratic constraints. If the neural network contains signal processing layers (convolutional or state space model layers), we realize them as 1-D/2-D/N-D systems and exploit this structure as well. We distinguish ourselves from related work on Lipschitz constant estimation by more extensive structure exploitation (scalability) and a generalization to a large class of common neural network architectures. To show the versatility and computational advantages of our method, we apply it to different neural network architectures trained on MNIST and CIFAR-10.

We establish a layer-wise parameterization for 1D convolutional neural networks (CNNs) with built-in end-to-end robustness guarantees. Herein, we use the Lipschitz constant of the input-output mapping characterized by a CNN as a robustness measure. We base our parameterization on the Cayley transform that parameterizes orthogonal matrices and the controllability Gramian for the state space representation of the convolutional layers. The proposed parameterization by design fulfills linear matrix inequalities that are sufficient for Lipschitz continuity of the CNN, which further enables unconstrained training of Lipschitz-bounded 1D CNNs. Finally, we train Lipschitz-bounded 1D CNNs for the classification of heart arrythmia data and show their improved robustness.

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