Dept. of Electr. Eng. Syst., Univ. of Southern California, Los Angeles, CA.
IEEE Trans Image Process. 1997;6(6):844-61. doi: 10.1109/83.585235.
The parameters of the prior, the hyperparameters, play an important role in Bayesian image estimation. Of particular importance for the case of Gibbs priors is the global hyperparameter, beta, which multiplies the Hamiltonian. Here we consider maximum likelihood (ML) estimation of beta from incomplete data, i.e., problems in which the image, which is drawn from a Gibbs prior, is observed indirectly through some degradation or blurring process. Important applications include image restoration and image reconstruction from projections. Exact ML estimation of beta from incomplete data is intractable for most image processing. Here we present an approximate ML estimator that is computed simultaneously with a maximum a posteriori (MAP) image estimate. The algorithm is based on a mean field approximation technique through which multidimensional Gibbs distributions are approximated by a separable function equal to a product of one-dimensional (1-D) densities. We show how this approach can be used to simplify the ML estimation problem. We also show how the Gibbs-Bogoliubov-Feynman (GBF) bound can be used to optimize the approximation for a restricted class of problems. We present the results of a Monte Carlo study that examines the bias and variance of this estimator when applied to image restoration.
先验参数和超参数在贝叶斯图像估计中起着重要作用。对于吉布斯先验的情况，特别重要的是全局超参数β，它乘以哈密顿量。在这里，我们考虑从不完全数据中进行最大似然（ML）估计β，即从吉布斯先验中得出的图像通过某种退化或模糊过程间接观察到的问题。重要的应用包括图像恢复和从投影重建图像。对于大多数图像处理来说，从不完全数据中进行精确的 ML 估计β是难以处理的。在这里，我们提出了一种近似的 ML 估计器，它与最大后验（MAP）图像估计同时计算。该算法基于均值场逼近技术，通过该技术多维吉布斯分布被逼近为等于一维（1-D）密度乘积的可分离函数。我们展示了如何使用这种方法来简化 ML 估计问题。我们还展示了如何使用吉布斯-玻尔兹曼-费曼（GBF）界来优化对受限类问题的逼近。我们介绍了一项蒙特卡罗研究的结果，该研究考察了该估计器在应用于图像恢复时的偏差和方差。
