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Zhifang Liu, Baochen Sun, Xue-Cheng Tai, Qi Wang, Huibin Chang

The Euler Elastica (EE) model with surface curvature can generate artifact-free results compared with the traditional total variation regularization model in image processing. However, strong nonlinearity and singularity due to the curvature term in the EE model pose a great challenge for one to design fast and stable algorithms for the EE model. In this paper, we propose a new, fast, hybrid alternating minimization (HALM) algorithm for the EE model based on a bilinear decomposition of the gradient of the underlying image and prove the global convergence of the minimizing sequence generated by the algorithm under mild conditions. The HALM algorithm comprises three sub-minimization problems and each is either solved in the closed form or approximated by fast solvers making the new algorithm highly accurate and efficient. We also discuss the extension of the HALM strategy to deal with general curvature-based variational models, especially with a Lipschitz smooth functional of the curvature. A host of numerical experiments are conducted to show that the new algorithm produces good results with much-improved efficiency compared to other state-of-the-art algorithms for the EE model. As one of the benchmarks, we show that the average running time of the HALM algorithm is at most one-quarter of that of the fast operator-splitting-based Deng-Glowinski-Tai algorithm.

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Hao Liu, Xue-Cheng Tai, Raymond Chan

Deep neural network is a powerful tool for many tasks. Understanding why it is so successful and providing a mathematical explanation is an important problem and has been one popular research direction in past years. In the literature of mathematical analysis of deep deep neural networks, a lot of works are dedicated to establishing representation theories. How to make connections between deep neural networks and mathematical algorithms is still under development. In this paper, we give an algorithmic explanation for deep neural networks, especially in their connection with operator splitting and multigrid methods. We show that with certain splitting strategies, operator-splitting methods have the same structure as networks. Utilizing this connection and the Potts model for image segmentation, two networks inspired by operator-splitting methods are proposed. The two networks are essentially two operator-splitting algorithms solving the Potts model. Numerical experiments are presented to demonstrate the effectiveness of the proposed networks.

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Xue-Cheng Tai, Hao Liu, Raymond Chan

For problems in image processing and many other fields, a large class of effective neural networks has encoder-decoder-based architectures. Although these networks have made impressive performances, mathematical explanations of their architectures are still underdeveloped. In this paper, we study the encoder-decoder-based network architecture from the algorithmic perspective and provide a mathematical explanation. We use the two-phase Potts model for image segmentation as an example for our explanations. We associate the segmentation problem with a control problem in the continuous setting. Then, multigrid method and operator splitting scheme, the PottsMGNet, are used to discretize the continuous control model. We show that the resulting discrete PottsMGNet is equivalent to an encoder-decoder-based network. With minor modifications, it is shown that a number of the popular encoder-decoder-based neural networks are just instances of the proposed PottsMGNet. By incorporating the Soft-Threshold-Dynamics into the PottsMGNet as a regularizer, the PottsMGNet has shown to be robust with the network parameters such as network width and depth and achieved remarkable performance on datasets with very large noise. In nearly all our experiments, the new network always performs better or as good on accuracy and dice score than existing networks for image segmentation.

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Shousheng Luo, Jinfeng Chen, Yunhai Xiao, Xue-Cheng Tai

This paper focuses on the issue of image segmentation with convex shape prior. Firstly, we use binary function to represent convex object(s). The convex shape prior turns out to be a simple quadratic inequality constraint on the binary indicator function associated with each object. An image segmentation model incorporating convex shape prior into a probability-based method is proposed. Secondly, a new algorithm is designed to solve involved optimization problem, which is a challenging task because of the quadratic inequality constraint. To tackle this difficulty, we relax and linearize the quadratic inequality constraint to reduce it to solve a sequence of convex minimization problems. For each convex problem, an efficient proximal alternating direction method of multipliers is developed to solve it. The convergence of the algorithm follows some existing results in the optimization literature. Moreover, an interactive procedure is introduced to improve the accuracy of segmentation gradually. Numerical experiments on natural and medical images demonstrate that the proposed method is superior to some existing methods in terms of segmentation accuracy and computational time.

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Hao Liu, Xue-Cheng Tai, Ron Kimmel, Roland Glowinski

One classical approach to regularize color is to tream them as two dimensional surfaces embedded in a five dimensional spatial-chromatic space. In this case, a natural regularization term arises as the image surface area. Choosing the chromatic coordinates as dominating over the spatial ones, the image spatial coordinates could be thought of as a paramterization of the image surface manifold in a three dimensional color space. Minimizing the area of the image manifold leads to the Beltrami flow or mean curvature flow of the image surface in the 3D color space, while minimizing the elastica of the image surface yields an additional interesting regularization. Recently, the authors proposed a color elastica model, which minimizes both the surface area and elastica of the image manifold. In this paper, we propose to modify the color elastica and introduce two new models for color image regularization. The revised measures are motivated by the relations between the color elastica model, Euler's elastica model and the total variation model for gray level images. Compared to our previous color elastica model, the new models are direct extensions of Euler's elastica model to color images. The proposed models are nonlinear and challenging to minimize. To overcome this difficulty, two operator-splitting methods are suggested. Specifically, nonlinearities are decoupled by introducing new vector- and matrix-valued variables. Then, the minimization problems are converted to solving initial value problems which are time-discretized by operator splitting. Each subproblem, after splitting either, has a closed-form solution or can be solved efficiently. The effectiveness and advantages of the proposed models are demonstrated by comprehensive experiments. The benefits of incorporating the elastica of the image surface as regularization terms compared to common alternatives are empirically validated.

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Hao Liu, Xue-Cheng Tai, Roland Glowinski

Gaussian curvature is an important geometric property of surfaces, which has been used broadly in mathematical modeling. Due to the full nonlinearity of the Gaussian curvature, efficient numerical methods for models based on it are uncommon in literature. In this article, we propose an operator-splitting method for a general Gaussian curvature model. In our method, we decouple the full nonlinearity of Gaussian curvature from differential operators by introducing two matrix- and vector-valued functions. The optimization problem is then converted into the search for the steady state solution of a time dependent PDE system. The above PDE system is well-suited to time discretization by operator splitting, the sub-problems encountered at each fractional step having either a closed form solution or being solvable by efficient algorithms. The proposed method is not sensitive to the choice of parameters, its efficiency and performances being demonstrated via systematic experiments on surface smoothing and image denoising.

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Lingfeng Li, Xue-Cheng Tai, Jiang Yang

We study gradient-based regularization methods for neural networks. We mainly focus on two regularization methods: the total variation and the Tikhonov regularization. Applying these methods is equivalent to using neural networks to solve some partial differential equations, mostly in high dimensions in practical applications. In this work, we introduce a general framework to analyze the generalization error of regularized networks. The error estimate relies on two assumptions on the approximation error and the quadrature error. Moreover, we conduct some experiments on the image classification tasks to show that gradient-based methods can significantly improve the generalization ability and adversarial robustness of neural networks. A graphical extension of the gradient-based methods are also considered in the experiments.

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Bin Wu, Xue-Cheng Tai, Talal Rahman

The paper presents a fully coupled TV-Stokes model, and propose an algorithm based on alternating minimization of the objective functional whose first iteration is exactly the modified TV-Stokes model proposed earlier. The model is a generalization of the second order Total Generalized Variation model. A convergence analysis is given.

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Bin Wu, Xue-Cheng Tai, Talal Rahman

A complete multidimential TV-Stokes model is proposed based on smoothing a gradient field in the first step and reconstruction of the multidimensional image from the gradient field. It is the correct extension of the original two dimensional TV-Stokes to multidimensions. Numerical algorithm using the Chambolle's semi-implicit dual formula is proposed. Numerical results applied to denoising 3D images and movies are presented. They show excellent performance in avoiding the staircase effect, and preserving fine structures.

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Bin Wu, Xue-Cheng Tai, Talal Rahman

Three dimensional surface reconstruction based on two dimensional sparse information in the form of only a small number of level lines of the surface with moderately complex structures, containing both structured and unstructured geometries, is considered in this paper. A new model has been proposed which is based on the idea of using normal vector matching combined with a first order and a second order total variation regularizers. A fast algorithm based on the augmented Lagrangian is also proposed. Numerical experiments are provided showing the effectiveness of the model and the algorithm in reconstructing surfaces with detailed features and complex structures for both synthetic and real world digital maps.

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