On the Mathematical Properties of the Structural Similarity Index
On the Mathematical Properties of the Structural Similarity Index
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DOI：
10.1109/tip.2011.2173206
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发表时间：
2012-04-01
影响因子：
10.6
通讯作者：
Wang, Zhou
中科院分区：
文献类型：
--
作者：
Brunet, Dominique;Vrscay, Edward R.;Wang, Zhou
自2004年引入以来，结构相似性（SSIM）指数作为评估图像质量和评估图像处理算法和系统性能的工具得到了广泛的普及。在各种图像处理应用中，使用SSIM作为优化问题的目标函数也越来越受到关注。可能严重阻碍这些努力取得进展的一个主要问题是对SSIM计量的数学特性缺乏了解。例如，一些高度期望的性质，如凸性和三角不等式，所拥有的均方误差可能不成立。在本文中，我们首先构造了一系列的规范化和广义（向量值）的度量的基础上的重要成分SSIM。然后，我们表明，这样的修改措施是有效的距离度量，并有许多有用的属性，其中最重要的包括准凸性，凸性区域周围的最小值，和距离保持正交或酉变换。这里奠定的基础扩展了SSIM在理论发展和实际应用中的潜力。（一）
Since its introduction in 2004, the structural similarity (SSIM) index has gained widespread popularity as a tool to assess the quality of images and to evaluate the performance of image processing algorithms and systems. There has been also a growing interest of using SSIM as an objective function in optimization problems in a variety of image processing applications. One major issue that could strongly impede the progress of such efforts is the lack of understanding of the mathematical properties of the SSIM measure. For example, some highly desirable properties such as convexity and triangular inequality that are possessed by the mean squared error may not hold. In this paper, we first construct a series of normalized and generalized (vector-valued) metrics based on the important ingredients of SSIM. We then show that such modified measures are valid distance metrics and have many useful properties, among which the most significant ones include quasi-convexity, a region of convexity around the minimizer, and distance preservation under orthogonal or unitary transformations. The groundwork laid here extends the potentials of SSIM in both theoretical development and practical applications. (1)