Aligning a group of linearly correlated images is an important task in computer vision. In this paper, we propose a combination of transformed tensor nuclear norm and tensor \(\ell _1\) norm to deal with this image alignment problem, where the observed images, stacked into a third-order tensor, are deformed by unknown domain transformations and corrupted by sparse noise like impulse noise, partial occlusions, and illumination variation. The key advantage of the proposed method is that both spatial correlation and images variation can be captured by the use of transformed tensor nuclear norm. We show that when the underlying of correlated images is a low multi-rank tensor, an upper error bound of the estimator of the proposed model can be established and this bound can be better than the previous result. Besides the proposed convex transformed tensor model, the method can be further studied by incorporating nonconvex functions in the transformed tensor nuclear norm and the sparsity norm. Both the proposed convex and nonconvex optimization models are solved by generalized Gauss–Newton algorithms. Also the global convergence of the numerical methods for solving the subproblems of convex and nonconvex optimization models can be provided under very mild conditions. Extensive numerical experiments on real images with misalignment and sparse corruptions demonstrate the performance of our proposed methods is better than that of several state-of-the-art methods in terms of accuracy and efficiency.

对齐一组线性相关的图像是计算机视觉中的重要任务。在本文中，我们提出了变换张量核规范和张量\（\ ell _1 \）的组合规范来处理此图像对齐问题，其中观察到的图像堆叠成三阶张量，由于未知域变换而变形，并由于脉冲噪声，部分遮挡和照明变化之类的稀疏噪声而损坏。所提出方法的主要优点是可以通过使用变换张量核范数来捕获空间相关性和图像变化。我们表明，当相关图像的基础是低的多秩张量时，可以建立所提出模型的估计量的误差上限，并且该界限可以比先前的结果更好。除了提出的凸变换张量模型外，还可以通过将非凸函数合并到变换张量核规范和稀疏规范中来进一步研究该方法。提出的凸优化模型和非凸优化模型都可以通过广义Gauss-Newton算法求解。同样，可以在非常温和的条件下提供用于求解凸和非凸优化模型子问题的数值方法的全局收敛性。在未对准和稀疏损坏的真实图像上进行的大量数值实验表明，在准确性和效率方面，我们提出的方法的性能优于几种最新方法。