First-Order Phase Transition and Bayesian Image Processing by Loopy Belief Propagation
First-Order Phase Transition and Bayesian Image Processing by Loopy Belief Propagation
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基于循环置信传播的一阶相变和贝叶斯图像处理
DOI：
10.1143/ptps.157.288
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发表时间：
2005
影响因子：
--
通讯作者：
D. Titterington
中科院分区：
文献类型：
--
作者：
Kazuyuki Tanaka;D. Titterington
提出了基于q态Potts模型的多值图像贝叶斯复原框架。确定概率模型中的超参数，使边际似然最大化。提出了一种实用的基于循环信念传播的多值图像恢复算法。我们的结论是，边际似然的最大化可以提供良好的结果，即使ap -概率模型呈现一阶相变。在贝叶斯图像恢复中，我们必须确定一定的超参数，在实践中，确定超参数以最大化边际似然是很常见的。1)在统计力学处理的一些概率模型中，某些类型的相变发生在临界点。一阶和二阶相变是常见的相变类型。在一阶相变情况下，自由能的一阶导数在临界点处不连续；如果概率模型表现为一阶相变，则自由能在超参数的临界值处不可微。边际似然可以用先验和后验概率分布的自由能来表示。这意味着边际似然在先验概率分布中存在的超参数的临界值处是不可微的。期望最大化算法是一种有效的边际似然最大化方法，其边际似然在超参数的任意值处都是可微的。在本文中，我们研究了采用Q-state Potts模型(2)作为先验概率分布时，以边际似然最大化估计超参数的贝叶斯图像恢复。用先进的平均场近似分析了Q≥3时Q态Potts模型的自由能在超参数的某一值处是不可导的。利用循环信念传播实现了边际似然的最大化
The framework is presented of Bayesian image restoration for multi-valued images based on the Q-state Potts model. Hyperparameters in the probabilistic model are determined so as to maximize the marginal likelihood. A practical algorithm is described for multi-valued image restoration based on loopy belief propagation. We conclude that the maximization of marginal likelihood can provide good results even if the ap rioriprobabilistic model exhibits first-order phase transition. In Bayesian image restoration, we have to determine certain hyperparameters, and in practice it is common for the hyperparameters to be determined so as to maximize a marginal likelihood. 1) In some probabilistic models treated in statistical mechanics, some types of phase transition occur at a critical point. First-order and second-order phase transitions are familiar types of phase transition. In the case of first-order phase transition, the first derivative of the free energy is discontinuous at the critical point; if the probabilistic model exhibits first-order phase transition, the free energy is not differentiable at a critical value of the hyperparameter. The marginal likelihood can be expressed in terms of the free energies of the ap riori and a posteriori probability distributions. This means that the marginal likelihood is not differentiable at a critical value of the hyperparameter present in the ap riori probability distribution. In the context of the expectation-maximization algorithm, which is a powerful method for maximizing the marginal likelihood, the marginal likelihood is differentiable at any value of the hyperparameter. In the present paper, we investigate Bayesian image restoration with hyperparameters estimated by maximizing the marginal likelihood when we adopt a Q-state Potts model 2) as the ap rioriprobability distribution. The free energy of the Q-state Potts model for Q≥3 is not differentiable at a certain value of the hyperparameter when we analyze the model by means of advanced mean-field approximations. The maximization of the marginal likelihood is achieved by using loopy belief propaga