On the Mathematical Properties of the Structural Similarity Index

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...

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Bibliographic Details
Published in:IEEE transactions on image processing Vol. 21; no. 4; pp. 1488 - 1499
Main Authors: Brunet, D., Vrscay, E. R., Zhou Wang
Format: Journal Article
Language:English
Published: United States IEEE 01.04.2012
Subjects:
Algorithms
Artificial Intelligence
Cone metrics
Correlation
Distortion measurement
Extraterrestrial measurements
Image Enhancement - methods
Image Interpretation, Computer-Assisted - methods
Indexes
normalized metrics
Optimization
Pattern Recognition, Automated - methods
perceptually optimized algorithms and methods
quality metrics and assessment tools
quasi-convexity and convexity
Reproducibility of Results
Sensitivity and Specificity
Signal Processing, Computer-Assisted
structural similarity (SSIM) index
Subtraction Technique
ISSN:1057-7149, 1941-0042, 1941-0042
Online Access:Get full text
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Summary: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.
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ISSN:1057-7149
1941-0042
1941-0042
DOI:10.1109/TIP.2011.2173206