Black M J, Sapiro G, Marimont D H, Heeger D
Xerox Palo Alto Res. Center, CA 94304, USA.
IEEE Trans Image Process. 1998;7(3):421-32. doi: 10.1109/83.661192.
Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edge-stopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edge-stopping" function based on Tukey's biweight robust estimator that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the line processes allows us to develop new anisotropic diffusion equations that result in a qualitative improvement in the continuity of edges.
本文描述了各向异性扩散与稳健统计之间的关系。具体而言，我们表明各向异性扩散可被视为一种稳健估计程序，它从有噪声的输入图像中估计出分段平滑的图像。各向异性扩散方程中的“边缘停止”函数与稳健估计框架中的误差范数和影响函数密切相关。这种联系导致了一种基于Tukey双权稳健估计器的新“边缘停止”函数，它比以前的公式能保留更清晰的边界，并改善了扩散的自动停止。稳健的统计解释还提供了一种方法，用于检测经过各向异性扩散平滑处理的图像中分段平滑区域之间的边界（边缘）。此外，我们推导了各向异性扩散与线过程正则化之间的关系。对线过程的空间组织施加约束使我们能够开发新的各向异性扩散方程，从而在边缘连续性方面实现定性改进。
