We consider sequential and parallel decomposition methods for a dual problem of a general total variation minimization problem with applications in several image processing tasks, like image inpainting, estimation of optical flow and reconstruction of missing wavelet coefficients. The convergence of these methods to a solution of the global problem is analysed in a Hilbert space setting and a convergence rate is provided. Thereby, these convergence result hold not only for exact local minimization but also if the subproblems are just solved approximately. As a concrete example of an approximate local solution process a surrogate technique is presented and analysed. Further, the obtained convergence rate is compared with related results in the literature and shown to be in agreement with or even improve upon them. Numerical experiments are presented to support the theoretical findings and to show the performance of the proposed decomposition algorithms in image inpainting, optical flow estimation and wavelet inpainting tasks.
翻译:我们认为,在两个问题上,相继和平行的分解方法是一个总体的完全变化最小化问题,在几种图像处理任务的应用方面,例如图像油漆、光学流量估计和重建缺失的波子系数等,都存在问题。在希尔伯特空间设置中分析了这些方法与全球问题解决办法的趋同,并提供了一种趋同率。因此,这些趋同结果不仅有利于精确的局部最小化,而且如果次级问题只是近距离解决,这些结果也存在。作为近似当地解决方案进程的一个具体例子,提出并分析一种代用技术。此外,所取得的趋同率与文献中的相关结果相比较,并表明与这些结果一致,甚至表明与它们一致,甚至改进了这些结果。提出了数字实验,以支持理论结论,并展示了在图像油漆、光学流量估计和波子涂料任务中拟议的分解算法的性能。