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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Veröffentlicht in:	IEEE transactions on image processing Jg. 21; H. 4; S. 1488 - 1499
Hauptverfasser:	Brunet, D., Vrscay, E. R., Zhou Wang
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	United States IEEE 01.04.2012
Schlagworte:	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
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Abstract	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.
AbstractList	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. 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.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.
Author	Zhou Wang Vrscay, E. R. Brunet, D.
Author_xml	– sequence: 1 givenname: D. surname: Brunet fullname: Brunet, D. email: dbrunet@uwaterloo.ca organization: Dept. of Appl. Math- ematics, Univ. of WaterlooWaterloo, Waterloo, ON, Canada – sequence: 2 givenname: E. R. surname: Vrscay fullname: Vrscay, E. R. email: ervrscay@uWaterloo.ca organization: Dept. of Appl. Math- ematics, Univ. of WaterlooWaterloo, Waterloo, ON, Canada – sequence: 3 surname: Zhou Wang fullname: Zhou Wang organization: Dept. of Electr. & Comput. Eng., Univ. of Waterloo, Waterloo, ON, Canada
BackLink	https://www.ncbi.nlm.nih.gov/pubmed/22042163$$D View this record in MEDLINE/PubMed
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SubjectTerms	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
Title	On the Mathematical Properties of the Structural Similarity Index
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