# 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 experiments were conducted using MatLab software version R2018b for the three 3D medical tomographic grayscale images: \"wmri\" is the 8-bit image of size \(128\times 128\times 27\); \"Trufi_COR\" is the 12-bit image of size \(320\times 320\times 30\) and \"Body_1.0\" is the 16-bit image of size \(512\times 512\times 507\). These images have the following histograms (Figure 4). The larger the image bitness, the lower its ratio of the average voxel brightness to the maximum allowed. We show the influence of this factor on the image processing quality further. Images DWT performed as follows: filters coefficients \(f_{\mathrm{F},i}\) of the Daubechies wavelets \(db(k/2)\)\((k=2,4,6,\ldots,20)\), symllets \(sym(k/2)\)\((k=2,4,6,\ldots,20)\) and coiflets \(coif(k/6)\)\((k=6,12,18,24,30)\) were obtained, quantized by multiplying by \(2^{n}\)\((n=1,2,3,\ldots,25)\) and rounding up according to Formula (2) and converted to fixed-point format; DWT of 3D images implemented; the voxels brightness values of the restored images were scaled by dividing by \(2^{6n}\) and rounding down according to Formula (3) and converted to fixed-point format. An example of 3D tomographic images \"wmri\", \"Trufi_COR\" and \"Body_1.0\" DWT with wavelet \(db8\) is shown in Figures 5-7 respectively. Frames in Figures 6 and 7 are selected to illustrate the error effect on the image processing result. Figures show a gradual improvement in the quality of processing with an increase the bit-width \(r\): in Figures 5b, 6b and 7b visible distortion (Figure 5b is darkened in places, and Figures 6b and 7b are lighted); in Figures 5c, 6c and 7c processed images are indistinguishable by eye from the original images; in Figures 5d, 6d and 7d processed images are identical to the corresponding originals. Experimental results are of higher quality compared with the calculation results. The values \(PSNR=47.11\) dB and \(PSNR=\infty\) at \(r=12\) and \(r=15\) respectively (Figure 5) obtained after 8-bit image \"wmri\" DWT with wavelet \(db8\) exceed the corresponding calculated values \(PSNR=37.82\) dB and \(PSNR=\infty\) at \(r=12\) and \(r=15\) respectively (Table 1). The values \(PSNR=64.57\) dB and \(PSNR=\infty\) at \(r=12\) and \(r=17\) respectively (Figure 6) obtained after 12-bit image \"Trufi_COR\" DWT with wavelet \(db8\) exceed the corresponding calculated values \(PSNR=36.67\) dB, \(PSNR=67.30\) dB at \(r=12\) and \(r=17\) respectively (Table 2). Similarly, for \"Body_1.0\"Figure 4: _Cont._  
Figure 4: Histograms of used images: (**a**) “wmri”, average brightness 63.276; (**b**) “Trufi_COR”, average brightness 129.796 and (**c**) “Body_1.0”, average brightness 21.053. Figure 5: Example of 3D tomographic 8-bit image “wmri” DWT by \(d\theta\)8 wavelet: (**a**) original image; processed image: (**b**) \(r=9\), \(PSNR=27.62\) dB; (**c**) \(r=12\), \(PSNR=47.11\) dB and (**d**) \(r=15\), \(PSNR=\infty\). Figure 6: Example of 3D tomographic 12-bit image “Trufi_COR” (15-th frame) DWT by \(d\)8 wavelet: (a) original image; processed image: (b) \(r=7\), \(PSNR=30.27\) dB; (c) \(r=12\), \(PSNR=64.57\) dB and (d) \(r=17\), \(PSNR=\infty\). Figure 7: Example of 3D tomographic 16-bit image “Body_1.0” (1-st frame) DWT by \(d\)8 wavelet: (a) original image; processed image: (b) \(r=7\), \(PSNR=64.05\) dB; (c) \(r=10\), \(PSNR=85.30\) dB and (d) \(r=17\), \(PSNR=\infty\).