# 結構相似性

## 定義

$\text{SSIM}(\mathbf{x},\mathbf{y})=[l(\mathbf{x},\mathbf{y})]^\alpha[c(\mathbf{x},\mathbf{y})]^\beta[s(\mathbf{x},\mathbf{y})]^\gamma$

$l(\mathbf{x},\mathbf{y})=\frac{2\mu_x\mu_y+C_1}{\mu_x^2+\mu_y^2+C_1}$$c(\mathbf{x},\mathbf{y})=\frac{2\sigma_x\sigma_y+C_2}{\sigma_x^2+\sigma_y^2+C_2}$$s(\mathbf{x},\mathbf{y})=\frac{\sigma_{xy}+C_3}{\sigma_x\sigma_y+C_3}$

$\text{SSIM}(\mathbf{x},\mathbf{x})=\frac{2\mu_x^2+C_1}{\mu_x^2+\mu_x^2+C_1} \times \frac{2\sigma_x^2+C_2}{\sigma_x^2+\sigma_x^2+C_2} \times \frac{\sigma_{xx}+C_3}{\sigma_x\sigma_x+C_3}=1$

## 性質

$\text{SSIM}(\mathbf{x},\mathbf{y})=\text{SSIM}(\mathbf{y},\mathbf{x})$
• 有上下界

## 使用

$\text{SSIM}(\mathbf{x},\mathbf{y})=\frac{(2\mu_x\mu_y+C_1)(2\sigma_{xy}+C_2)}{(\mu_x^2+\mu_y^2+C_1)(\sigma_x^2+\sigma_y^2+C_2)}$

## 參考資料

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2. ^ Zhou Wang and Alan C. Bovik, "Mean squared error: Love it or leave it? - A new look at signal fidelity measures," IEEE Signal Processing Magazine, vol. 26, no. 1, pp 98−117, Jan. 2009.
3. ^ H.R. Sheikh, M.F. Sabir, and A.C. Bovik, "A statistical evaluation of recent full reference image quality assessment algorithms," IEEE Transactions on Image Processing, vol.15, no.11, pp.3440−3451, Nov. 2006.
4. ^ T. Richter, K. J. Kim, "A MS-SSIM optimal JPEG 2000 encoder," in Proc. Data Compression Conf., pp.401−410, Mar. 2009.
5. ^ A. M. Alattar, E. T. Lin, and M. U. Celik, "Digital watermarking of low bit-rate advanced simple profile MPEG-4 compressed video," IEEE Trans. Circuits Syst. Video Technol., vol. 13, no. 8, pp. 787−800, Aug. 2003.
6. ^ V. Vukadinovi and G. Karlsson, "Trade-offs in bit-rate allocation for wireless video streaming," in Proc. ACM Int. Symp. Modeling, Analysis, and Simulation of Wireless and Mobile Systems, Quebec, Canada, 2005, pp. 349−353
7. ^ S. A. Reinsberg, S. J. Doran, E. M. Charles-Edwards, and M. O. Leach, "A complete distortion correction for MR images: II. Rectification of static-field inhomogeneities by similarity-based profile mapping," Phys. Med. Biol., vol. 50, no. 11, pp. 2651−2661, June 2005.
8. ^ Z. Wang and E. P. Simoncelli, "Translation insensitive image similarity in complex wavelet domain," in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, pp. 573−576, Mar. 2005.