# Prospects of Structural Similarity Index for Medical Image Analysis
## 3 Current Improvement in SSIM
### Complex-Wavelet SSIM  
After dealing with the SSIM issues in overestimating the similarity of the blurred images, a complex-wavelet SSIM was developed in 2009 to overcome the drawbacks of the original SSIM in the spatial domain related to a high sensitivity when measuring images with a few rotations, translations, and scaling in comparison with the original image [21]. The fundamental aspect behind CW-SSIM is using the wavelet coefficients inspired from [69] over the images to extract the similarity value. Similarly, the main objective of CW-SSIM is to develop an insensitive IQA in evaluating the images under a nonstructured geometric image distortion. CW-SSIM is emphasized more to resolve the drawbacks of the spatial domain by converting the images into a complex wavelet domain and acquiring the wavelet coefficients for determining the similarity score.  
CW-SSIM attempts to model the complex wavelet domain IQA with the ability to separate the magnitude and phase distortion assessment. Moreover, it was developed to be more sensitive to phase distortions than magnitude distortions and insensitive to consistent relative phase distortions. The symmetric complex wavelet can be formulated as \(w(u)=g(u)e^{i\omega_{c}u}\) for a low-pass-filter modulation, where \(\omega_{c}\) denotes the modulated band-pass filter center frequency and \(g(u)\) is a slowly varying and symmetric function. The dilated and translated version of \(w(u)\) can be written as follows:  
\[w_{s,p}(u)=\frac{1}{\sqrt{s}}w\bigg{(}\frac{u-p}{s}\bigg{)}=\frac{1}{\sqrt{s}} g\bigg{(}\frac{u-p}{s}\bigg{)}e^{i\omega_{c}(u-p)/s}, \tag{24}\]  
where \(s\in\mathcal{R}^{+}\) denotes the scale factor, and \(p\in\mathcal{R}\) is the translation factor. In addition, the continuous wavelet transformation of the real signal \(x(u)\) is expressed as  
\[X(s,p)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)\sqrt{s}G(s\omega-\omega_ {c})e^{i\omega p}d\omega, \tag{25}\]  
where \(X(\omega)\) and \(G(\omega)\) are the Fourier transform (FT) of \(x(u)\) and \(g(u)\), respectively. In a complex wavelet domain, we can determine the coefficients within the same spatial domain location represented in the same wavelet subbands of two images under comparison as \(\mathbf{c}_{x}\{c_{x,i}|i=1,2,3\ldots,N\}\) and \(\mathbf{c}_{y}\{c_{y,i}|i=1,2,3\ldots,N\}\), respectively. Then, CW-SSIM can be formulated as  
\[\widetilde{S}\big{(}\mathbf{c}_{x},\mathbf{c}_{y}\big{)}=\frac{2\Big{|}\sum_{i=1}^{N} c_{x,i}c_{y,i}^{*}\Big{|}+K}{\sum_{i=1}^{N}\big{|}c_{x,i}\big{|}^{2}+\sum_{i=1}^{N} \big{|}c_{y,i}\big{|}^{2}+K}\,, \tag{26}\]  
or can be written as a product of two components:  
\[\widetilde{S}\big{(}\mathbf{c}_{x},\mathbf{c}_{y}\big{)}=\frac{2\sum_{i=1}^{N}\big{|} c_{x,i}\big{|}c_{y,i}\big{|}+K}{\sum_{i=1}^{N}\big{|}c_{x,i}\big{|}^{2}+ \sum_{i=1}^{N}\big{|}c_{y,i}\big{|}^{2}+K}\frac{2\Big{|}\sum_{i=1}^{N}c_{x,i}c_ {y,i}^{*}\Big{|}+K}{2\sum_{i=1}^{N}\Big{|}c_{x,i}c_{y,i}^{*}\Big{|}+K}\,, \tag{27}\]  
Figure 9: Demonstration using “Lena” to present (**a**) preserved edge pixel image, (**b**) changed edge pixel image, (**c**) smooth pixel image, and (**d**) texture pixel image.  
where \(c^{*}\) is the complex conjugate of \(c\), and \(K\) denotes the small positive constant to improve the CW-SSIM robustness at an extremely small local SNR. As in the original SSIM, \(\widetilde{S}\big{(}c_{x},c_{y}\big{)}\) is altered from 0 to 1 depending on the degree of similarity. The first component in Equation (27) is a maximum of 1 if \(\left|c_{x,j}\right|=\left|c_{y,j}\right|\). The first component is related to the magnitude. By contrast, the second coefficient is relative to the phase changes.