# Analysis of the Quantization Noise in Discrete Wavelet Transform Filters for 3D Medical Imaging
## 3 Results
### Theoretical Analysis of the Maximum Error of the 3D Medical Images DWT
Calculations results (\(PSNR\), dB) obtained by using our method of wavelet filters coefficients quantizing and final Formula (5) for 3D medical grayscale and color images DWT with various BPC, various bit-width \(r\) and numbers \(k=2,4,6,\ldots,20\) of wavelets \(db(k/2)\) filters coefficients are presented in Tables 1-3. 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
10 & 36.79 & 36.67 & 29.87 & 30.08 & 31.59 & 24.60 & 22.29 & 24.46 & 22.19 & 22.08 \\
11 & **44.15** & **43.36** & 39.68 & 34.58 & 36.67 & 32.22 & 28.80 & 31.17 & 28.58 & 27.60 \\
12 & 57.16 & 48.71 & **44.15** & **41.85** & **43.36** & 39.68 & 34.58 & 37.82 & 35.26 & 35.34 \\
13 & \(\infty\) & \(\infty\) & \(51.14\) & 51.14 & 51.14 & **47.16** & **43.18** & **43.36** & **43.36** & 39.68 \\
14 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(57.16\) & 51.14 & 51.14 & **47.16** \\
15 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation results (\(PSNR\), dB) of 3D medical images (with 8 BPC) 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 c} \hline \hline \(r\) & \(db1\) & \(db2\) & \(db3\) & \(db4\) & \(db5\) & \(db6\) & \(db7\) & \(db8\) & \(db9\) & \(db10\) \\ \hline
12 & 49.43 & 45.01 & 42.18 & 40.05 & 41.80 & 38.21 & 33.76 & 36.69 & 34.47 & 34.40 \\
13 & 57.46 & 52.38 & 49.10 & 46.76 & 46.46 & 44.76 & 41.11 & 41.99 & 41.92 & 39.07 \\
14 & **71.28** & **61.23** & 54.39 & 52.38 & 52.27 & 50.35 & 47.91 & 47.73 & 47.65 & 44.90 \\
15 & 70.86 & 71.28 & **61.11** & 59.15 & 57.68 & 57.46 & 55.52 & 52.38 & 53.27 & 52.27 \\
16 & \(\infty\) & 81.28 & 71.28 & **64.37** & **63.79** & **63.79** & **63.15** & 57.97 & 59.85 & 56.33 \\
17 & \(\infty\) & \(\infty\) & \(\infty\) & 70.86 & 75.26 & 71.28 & 68.49 & **67.30** & **64.37** & **63.79** \\
18 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & 81.28 & 81.28 & 75.26 & 75.26 & 75.26 & 71.28 \\
19 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\
20 & \(\infty  
We could make the following conclusions based on calculation results presented in the Tables 10-12. 1. Minimum bit-width \(r\) of wavelet filters coefficients at which the result of 3D medical images with 8 BPC DWT does not contain visible distortions \((PSNR\geq 40\) dB) can be determined by a formula \[r=11+\left\lfloor\sqrt{\frac{k}{2}}\right\rfloor,\] (6) where \(k\) is the number of wavelet filters coefficients. 2. Minimum bit-width \(r\) of wavelet filters coefficients at which the result of 3D medical images with 12 BPC DWT does not contain visible distortions \((PSNR\geq 60\) dB) can be determined by a formula \[r=15+\left\lfloor\sqrt{\frac{k}{4}}\right\rfloor.\] (7)
3. Minimum bit-width \(r\) of wavelet filters coefficients at which the result of 3D medical images with 16 BPC DWT does not contain visible distortions \((PSNR\geq 80\) dB) can be determined by a formula \[r=18+\left\lfloor\sqrt{\frac{k}{3}}\right\rfloor.\] (8)