# Analysis of the Quantization Noise in Discrete Wavelet Transform Filters for 3D Medical Imaging
## 3 Results
### Experiments of the 3D Medical Tomographic Images DWT
Thus, the derived Formulas (6)-(9) could be used for determining the minimum bit-width of wavelet filters coefficients, at which the result of 3D medical tomographic images DWT reaches high (\(PSNR\geq 40\) dB for images with 8 BPC, \(PSNR\geq 60\) dB for images with 12 BPC and \(PSNR\geq 80\) dB for images with 16 BPC according to Formula (4)) and maximum (\(PSNR=\infty\)) quality respectively. Tables 13-30 show that _SSIM_ values obtained as a calculation result were set to one when using 4 decimal places in simulating 8-, 12- and 16-bit images when the \(PSNR\) was approximately 45, 65 and 80 dB, respectively. Thus, both \(PSNR\) and _SSIM_ metrics used confirm high-quality image processing. The experiment of 3D 8-bit medical tomographic image DWT required 1-2 bits less for wavelet filters coefficients than the calculations require for high-quality processing since the worst case was predicted in theoretical analysis. An even greater decrease in the bit-width of wavelet filter coefficients led to even greater savings in hardware resources. The difference between the obtained theoretical and experimental values increased significantly in the case of 12-bit and 16-bit images. The 12-bit tomographic image required 4-5 bits and 1-2 bits less for wavelet filters coefficients to achieve high and maximum processing quality respectively. This difference increased to 9-10 and 5-6 bits respectively in the case of a 16-bit image. This is because the range of voxel brightness values significantly increased in 12- and 16-bit images. The average brightness of the image voxels varied insignificantly at this time (was within the 8-bit range) since the high-order bits were rarely used. Thus, the ratio of the average voxel brightness to the maximum allowable value of \(M\) decreased with increasing BPC of images, which were demonstrated by histograms in Figure 4. This led to much faster achievement of high and maximum quality compared with the theoretical analysis results. The darkening and lighting in Figures 5-7 were due to the low accuracy of wavelet filters coefficients quantization used for image processing. The excessive character of quantization error led to an increase in the voxels brightness values of the processed images. Figures 6b and 7b turned out to be lighted since 12- and 16-bit images had a brightness margin, which is shown by the histograms in Figure 4b,c. However, the range of brightness values of the 8-bit image was fully utilized (Figure 4a) and the quantization error led to the computational range overflow. The voxels brightness values that exceeded the range went to zero as a result of this. Table 34 and Figure 8 show the dependence of the 12-bit medical tomographic images processing quality of their average voxels brightness. This dependence had a nonlinear hyperbolic regression form. Equation asymptote exceeded the corresponding theoretical calculation values. The processing quality by \(PSNR\) metric (from 74.57 to 58.39) decreased with an increase in the average voxels brightness (from 16.89 to 187.42). The difference in the image processing quality with the minimum and maximum values of the average brightness according to Table 34 was more than 15 dB. It was commensurate with the difference in the processing quality of the same image by the same wavelet with filter coefficients bit-width that differ by two, according to Tables 15, 21 and 27. That is, we would need 2 bits less for wavelet filter coefficients for high-quality processing of a 12-bit image with an average brightness of 16.89 than for processing a 12-bit image with an average brightness of 187.42. The average voxels brightness of the medical image can vary in different ranges depending on many factors: from the medical image modalities; from the analyzing device type; from specific device settings; from the analyzed organ or group of organs; etc.