On the Mathematical Properties of the Structural Similarity Index

By Dominique Brunet, Edward R. Vrscay, Zhou Wang. IEEE Transactions on Image Processing, 2012

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Image Quality Assessment: Unifying Structure and Texture Similarity (2020, unknown): “[58] have proved that dij(x, y) is a metric for αij ≥ 0 and βij ≥ 0.”
Quality Aware Generative Adversarial Networks (2019, methodology): “Further, as in the SSIM index, the image level distance metric for an M × N image is deﬁned as We refer the reader to [15] for a detailed exposition of the properties of this distance metric.”
Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields (2021, unknown): “To enable easier comparison, we also present an “average” error metric that summarizes all three metrics: the geometric mean of MSE = 10−PSNR/10, √ 1− SSIM (as per [5]), and LPIPS.”
Perceptual Video Coding Based on SSIM-Inspired Divisive Normalization (2013, methodology): “In this work, we aim to transform the optimization process in (1) into a perceptually uniform domain by incorporating the divisive normalization framework.”
Image Quality Assessment for Fake Biometric Detection: Application to Iris, Fingerprint, and Face Recognition (2014, unknown): “3.”