Signal Process. Lab., Tampere Univ. of Technol.
IEEE Trans Image Process. 1996;5(6):809-26. doi: 10.1109/83.503901.
A training framework is developed in this paper to design optimal nonlinear filters for various signal and image processing tasks. The targeted families of nonlinear filters are the Boolean filters and stack filters. The main merit of this framework at the implementation level is perhaps the absence of constraining models, making it nearly universal in terms of application areas. We develop fast procedures to design optimal or close to optimal filters, based on some representative training set. Furthermore, the training framework shows explicitly the essential part of the initial specification and how it affects the resulting optimal solution. Symmetry constraints are imposed on the data and, consequently, on the resulting optimal solutions for improved performance and ease of implementation. The case study is dedicated to natural images. The properties of optimal Boolean and stack filters, when the desired signal in the training set is the image of a natural scene, are analyzed. Specifically, the effect of changing the desired signal (using various natural images) and the characteristics of the noise (the probability distribution function, the mean, and the variance) is analyzed. Elaborate experimental conditions were selected to investigate the robustness of the optimal solutions using a sensitivity measure computed on data sets. A remarkably low sensitivity and, consequently, a good generalization power of Boolean and stack filters are revealed. Boolean-based filters are thus shown to be not only suitable for image restoration but also robust, making it possible to build libraries of "optimal" filters, which are suitable for a set of applications.
本文提出了一种用于设计各种信号和图像处理任务的最优非线性滤波器的训练框架。目标非线性滤波器族是布尔滤波器和堆叠滤波器。该框架在实现层面上的主要优点也许是不存在约束模型，使其在应用领域方面几乎具有通用性。我们基于一些代表性的训练集开发了快速设计最优或接近最优滤波器的程序。此外，该训练框架明确显示了初始规范的基本部分以及它如何影响最终的最优解。对数据施加了对称约束，因此也对最终的最优解施加了约束，以提高性能和便于实现。案例研究专门针对自然图像。当训练集中的期望信号是自然场景的图像时，分析了最优布尔和堆叠滤波器的特性。具体来说，分析了改变期望信号（使用各种自然图像）和噪声特征（概率分布函数、均值和方差）的影响。选择了详细的实验条件来使用基于数据集计算的灵敏度度量来研究最优解的鲁棒性。结果表明，布尔和堆叠滤波器具有很低的灵敏度，因此具有很好的泛化能力。这表明布尔滤波器不仅适用于图像恢复，而且还具有鲁棒性，可以构建适合一组应用的“最优”滤波器库。
