Many convolutional neural networks (CNN’s) have a feed-forward structure. In this paper, we model a general framework for analyzing the Lipschitz bounds of CNN’s and propose a linear program that estimates these bounds. Several CNN’s, including the scattering networks, the AlexNet and the GoogleNet, are studied numerically. In these practical numerical examples, estimations of local Lipschitz bounds are compared to these theoretical bounds. Based on the Lipschitz bounds, we next establish concentration inequalities for the output distribution with respect to a stationary random input signal. The Lipschitz bound is further used to perform nonlinear discriminant analysis that measures the separation between features of different classes.

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Purpose: Motion-induced magnetic resonance imaging (MRI) artifacts can deteriorate image quality and reduce diagnostic accuracy, but motion by human subjects is inevitable and can even be caused by involuntary physiological movements. Deep-learning-based motion correction methods might provide a solution. However, most studies have been based on directly applying existing models, and the trained models are rarely accessible. Therefore, we aim to develop and evaluate a deep-learning-based method (Motion Correction-Net, or MC-Net) for suppressing motion artifacts in brain MRI scans. Methods: A total of 57 subjects, providing 20,889 slices in four datasets, were used. Furthermore, 3T 3D sagittal magnetization-prepared rapid gradient-echo (MP-RAGE) and 2D axial fluid-attenuated inversion-recovery (FLAIR) sequences were acquired. The MC-Net was derived from a UNet combined with a two-stage multi-loss function. T1-weighted axial brain images contaminated with synthetic motions were used to train the network to remove motion artifacts. Evaluation used simulated T1- and T2-weighted axial, coronal, and sagittal images unseen during training, as well as T1-weighted images with motion artifacts from real scans. The performance indices included the peak-signal-to-noise ratio (PSNR), the structural similarity index measure (SSIM), and visual reading scores from three blinded clinical readers. A one-sided Wilcoxon signed-rank test was used to compare reader scores, with p < 0.05 considered significant. Intraclass correlation coefficients (ICCs) were calculated for inter-rater evaluations. Results: The MC-Net outperformed other methods in terms of PSNR and SSIM for the T1 axial test set. The MC-Net significantly improved the quality of all T1-weighted images for all directions (i.e., the mean SSIM of axial, sagittal, and coronal slices improved from 0.77, 0.64, and 0.71 to 0.92, 0.75, and 0.84; the mean PSNR improved from 26.35, 24.03, and 24.55 to 29.72, 24.40, and 25.37, respectively) and for simulated as well as real motion artifacts, both using quantitative measures and visual scores. However, MC-Net performed poorly for images with untrained T2-weighted contrast because the T2 contrast was unseen during training and is different from T1 contrast. Conclusion: The proposed two-stage multi-loss MC-Net can effectively suppress motion artifacts in brain MRI without compromising image quality. Given the efficiency of MC-Net (with a single-image processing time of ~40 ms), it can potentially be used in clinical settings.