# Analysis of the Quantization Noise in Discrete Wavelet Transform Filters for 3D Medical Imaging
## 3 Results
### Experiments of the 3D Medical Tomographic Images DWT
The image processing results were analyzed using \(PSNR\) and structure similarity (\(SSIM\)) [52], calculating by formula  
\[SSIM\Big{(}I,\overline{I}\Big{)}=\frac{\Big{(}2\mu_{I}\mu_{T}^{-}c_{1}\Big{)} \Big{(}2\sigma_{\overline{I}\overline{I}}+c_{2}\Big{)}}{\Big{(}\mu_{I}^{2}+\mu_ {\overline{I}}^{2}+c_{1}\Big{)}\Big{(}\sigma_{I}^{2}+\sigma_{\overline{I}}^{2 }+c_{2}\Big{)}},\]  
where: \(\mu_{I}\) is the average of \(I\); \(\mu_{T}^{-}\) is the average of \(\overline{I}\); \(\sigma_{I}^{2}\) is the variance of \(I\); \(\sigma_{I}^{2}\) is the variance of \(\overline{I}\); \(c_{1}=\left(0.01\cdot M\right)^{2}\); \(c_{2}=\left(0.03\cdot M\right)^{2}\) and \(M\) is the maximum brightness of the image voxels. Experimental results (\(PSNR\), dB; \(SSIM\)) of DWT of 3D tomographic grayscale images \"wmri\" (8-bit), \"Trufi_COR\" (12-bit) and \"Body_1.0\" (16-bit) for various bit-width \(r\) and numbers \(k=2,4,6,\ldots,20\) of wavelets \(db(k/2)\) filters coefficients are presented in Tables 13-18. The cells in bold correspond to the minimum bit-widths of the filter coefficients, at which the processing quality achieves a high level according to the formula (4). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(r\) & _db1_ & _db2_ & _db3_ & _db4_ & _db5_ & _db6_ & _db7_ & _db8_ & _db9_ & _db10_ \\ \hline
9 & 0.9996 & 0.9995 & 0.9981 & 0.9979 & 0.9986 & 0.9948 & 0.9902 & 0.9939 & 0.9864 & 0.9884 \\
10 & 0.9999 & 0.9999 & 0.9996 & 0.9996 & 0.9997 & 0.9985 & 0.9972 & 0.9982 & 0.9968 & 0.9967 \\
11 & 1.0000 & 1.0000 & 1.0000 & 0.9999 & 0.9999 & 0.9998 & 0.9994 & 0.9996 & 0.9993 & 0.9991 \\
12 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 0.9999 & 0.9999 & 0.9999 & 0.9999 \\
13 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Experimental results (_SSIM_) of 3D tomographic 12-bit image “Trufi_COR” DWT by using bit-width \(r\) of Daubechies wavelets filters coefficients. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(r\) & _db1_ & _db2_ & _db3_ & _db4_ & _db5_ & _db6_ & _db7_ & _db8_ & _db9_ & _db10_ \\ \hline
7 & 77.99 & 77.69 & 70.86 & 68.90 & 68.96 & 65.38 & 65.39 & 64.05 & 62.04 & 59.98 \\
8 & **87.98** & **81.65** & 78.77 & 76.46 & 76.54 & 71.53 & 70.89 & 72.11 & 68.08 & 68.88 \\
9 & 88.22 & 88.00 & **82.85** & **82.90** & **84.95** & 79.80 & 77.67 & 79.97 & 76.69 & 77.63 \\
10 & 94.84 & 94.62 & 89.15 & 88.98 & 91.67 & **84.80** & **83.03** & **85.30** & **83.27** & **83.27** \\
11 & 101.92 & 101.93 & 98.64 & 93.87 & 96.45 & 92.39 & 89.47 & 91.69 & 89.76 & 89.16 \\
12 & 109.95 & 105.14 & 103.17 & 100.98 & 103.38 & 99.76 & 94.87 & 98.86 & 96.63 & 96.80 \\
13 & 120.74 & 114.03 & 111.17 & 108.38 & 108.91 & 107.49 & 103.36 & 104.87 & 104.83 & 102.16 \\
14 & 167.56 & 127.71 & 118.07 & 115.86 & 116.02 & 115.25 & 111.44 & 111.96 & 112.20 & 109.24 \\
15 & 166.77 & 169.11 & 131.41 & 126.83 & 122.99 & 125.99 & 123.36 & 119.26 & 119.89 & 119.63 \\
16 & \(\infty\) & \(\infty\) & \(\infty\) & 144.12 & 138.29 & 145.66 & 140.29 & 130.67 & 135.37 & 128.20 \\
17 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\
18 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Experimental results (_PSNR_, dB) of 3D tomographic 16-bit image “Body_1.0” DWT by using bit-width \(r\) of Daubechies wavelets filters coefficients.