Law Yan Nei, Lee Hwee Kuan, Yip Andy M
Department of Mathematics, National University of Singapore, Singapore.
IEEE Trans Image Process. 2008 Dec;17(12):2289-300. doi: 10.1109/TIP.2008.2005823.
The Mumford-Shah model is one of the most successful image segmentation models. However, existing algorithms for the model are often very sensitive to the choice of the initial guess. To make use of the model effectively, it is essential to develop an algorithm which can compute a global or near global optimal solution efficiently. While gradient descent based methods are well-known to find a local minimum only, even many stochastic methods do not provide a practical solution to this problem either. In this paper, we consider the computation of a global minimum of the multiphase piecewise constant Mumford-Shah model. We propose a hybrid approach which combines gradient based and stochastic optimization methods to resolve the problem of sensitivity to the initial guess. At the heart of our algorithm is a well-designed basin hopping scheme which uses global updates to escape from local traps in a way that is much more effective than standard stochastic methods. In our experiments, a very high-quality solution is obtained within a few stochastic hops whereas the solutions obtained with simulated annealing are incomparable even after thousands of steps. We also propose a multiresolution approach to reduce the computational cost and enhance the search for a global minimum. Furthermore, we derived a simple but useful theoretical result relating solutions at different spatial resolutions.
芒福德-沙模型是最成功的图像分割模型之一。然而，该模型的现有算法通常对初始猜测的选择非常敏感。为了有效地利用该模型，开发一种能够高效计算全局或近似全局最优解的算法至关重要。虽然基于梯度下降的方法众所周知只能找到局部最小值，但即使是许多随机方法也不能为这个问题提供实际解决方案。在本文中，我们考虑多相分段常数芒福德-沙模型全局最小值的计算。我们提出了一种混合方法，将基于梯度的方法和随机优化方法相结合，以解决对初始猜测敏感的问题。我们算法的核心是一个精心设计的盆地跳跃方案，该方案使用全局更新以比标准随机方法更有效的方式逃离局部陷阱。在我们的实验中，经过几次随机跳跃就能获得非常高质量的解，而即使经过数千步，模拟退火得到的解也无法与之相比。我们还提出了一种多分辨率方法来降低计算成本并增强对全局最小值的搜索。此外，我们推导了一个简单但有用的理论结果，该结果涉及不同空间分辨率下的解。
