Department of Mathematics, University of Bergen, 5020 Bergen, Norway.
IEEE Trans Image Process. 2011 May;20(5):1199-210. doi: 10.1109/TIP.2010.2090533. Epub 2010 Nov 1.
Minimization of total variation (TV) is a well-known method for image denoising. Recently, the relationship between TV minimization problems and binary MRF models has been much explored. This has resulted in some very efficient combinatorial optimization algorithms for the TV minimization problem in the discrete setting via graph cuts. To overcome limitations, such as staircasing effects, of the relatively simple TV model, variational models based upon higher order derivatives have been proposed. The Euler's elastica model is one such higher order model of central importance, which minimizes the curvature of all level lines in the image. Traditional numerical methods for minimizing the energy in such higher order models are complicated and computationally complex. In this paper, we will present an efficient minimization algorithm based upon graph cuts for minimizing the energy in the Euler's elastica model, by simplifying the problem to that of solving a sequence of easy graph representable problems. This sequence has connections to the gradient flow of the energy function, and converges to a minimum point. The numerical experiments show that our new approach is more effective in maintaining smooth visual results while preserving sharp features better than TV models.
总变差（TV）最小化是一种用于图像去噪的知名方法。最近，TV 最小化问题与二进制马尔可夫随机场（MRF）模型之间的关系得到了广泛探索。这导致了一些非常有效的组合优化算法，可用于离散设置中的 TV 最小化问题，这些算法通过图割实现。为了克服相对简单的 TV 模型的局限性，例如阶梯效应，已经提出了基于更高阶导数的变分模型。Euler 弹性模型就是这样一个具有重要意义的高阶模型，它使图像中所有水平线上的曲率最小化。用于最小化此类高阶模型中的能量的传统数值方法很复杂，计算量也很大。在本文中，我们将提出一种基于图割的有效最小化算法，通过将问题简化为一系列易于图形表示的问题，从而最小化 Euler 弹性模型中的能量。这个序列与能量函数的梯度流有关，并收敛到一个最小值。数值实验表明，我们的新方法在保持平滑视觉效果的同时，比 TV 模型更好地保留了锐利特征，因此更有效。
