一种二阶非光滑变分模型用于恢复流形值图像
A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images
AI总结：
本文提出了一种新的非光滑变分模型，用于恢复流形值数据，该模型在正则化项中包含二阶差分。虽然此类模型已成功应用于实值图像，但本文引入了二阶差分及对应的变分模型，适用于流形数据，此前仅在循环数据中存在。该方法结合了数值分析、凸优化和微分几何的技巧，通过定义适合的绝对二阶差分，引入基于流形设置中一阶和二阶差分的变分去噪模型，并开发了使用近似循环近点算法的算法来最小化相应功能。对于n-球面和对称正定矩阵流形，展示了算法的实际性能，并证明了所提出的精确和近似循环近点算法变体在Hadamard空间中的收敛性。
摘要：
我们介绍了一种新的非光滑变分模型，用于恢复流形值数据，该模型在正则化项中包含二阶差分。虽然此类模型已成功应用于实值图像，但本文引入了二阶差分及对应的变分模型，适用于流形数据，此前仅在循环数据中存在。该方法结合了数值分析、凸优化和微分几何的技巧。首先，我们建立了适合信号和流形值图像的绝对二阶差分的合适定义。利用这一定义，我们引入了基于流形设置中一阶和二阶差分的变分去噪模型。为了最小化相应函数，我们开发了使用近似循环近点算法的算法。我们提出了一种高效的策略，利用Jacobi场的 machinery 来计算相应的近点映射，以对称空间为用。对于n-球面和对称正定矩阵流形，我们展示了算法的实际性能。我们证明了所提出的精确和近似循环近点算法变体在Hadamard空间中的收敛性。这些结果本身具有兴趣，例如对称正定矩阵流形。
英文摘要：
We introduce a new non-smooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the n-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.
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