Signal Processing Laboratory, École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland.
IEEE Trans Image Process. 2012 Dec;21(12):4722-34. doi: 10.1109/TIP.2012.2202674. Epub 2012 Jun 5.
The level set method is a popular technique for tracking moving interfaces in several disciplines, including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. First, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features, s.a. the curvature or the contour normal. Second, the level set algorithm is slow because the time step is limited by the standard Courant-Friedrichs-Lewy (CFL) condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods and continuous convex relaxation methods provide powerful alternatives to the level set method for image processing problems because they are fast, accurate, and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications, s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient l(1) optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods prove to be 5-6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information.
水平集方法是计算机视觉和流体动力学等多个学科中用于跟踪运动界面的一种流行技术。然而，尽管它具有很高的灵活性，但原始的水平集方法受到两个重要的数值问题的限制。首先，水平集方法没有隐含地将水平集函数作为距离函数来保存，这对于准确估计几何特征（如曲率或轮廓法线）是必要的。其次，水平集算法很慢，因为时间步长受到标准的柯朗-弗里德里希斯-莱维（Courant-Friedrichs-Lewy，CFL）条件的限制，这对于迭代方案的数值稳定性也是至关重要的。最近的图割方法和连续凸松弛方法的进展为图像处理问题提供了替代水平集方法的强大方法，因为它们快速、准确，并且保证可以独立于初始化找到全局最小值。这些最近的技术使用二进制函数来表示轮廓，而不是距离函数，这通常是水平集方法中考虑的。然而，二进制函数不能提供距离信息，这对于一些应用程序可能是必不可少的，例如从散点重建曲面和医学成像中的皮质分割问题。在本文中，我们提出了一种在水平集方法中保持距离函数的快速算法。我们的算法受到最近有效的 l(1) 优化技术的启发，它将提供一种高效且易于实现的算法。有趣的是，我们的算法不受 CFL 条件的限制，并且在演化过程中自然地将水平集函数保持为距离函数，从而避免了水平集方法中的经典重新距离问题。我们将提出的算法应用于图像分割，结果表明，我们的方法比标准的距离保持水平集技术快 5-6 倍。我们还介绍了两个需要保持距离函数的应用程序。然而，我们的方法是通用的，可以应用于任何需要距离信息的水平集方法。
