# 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  
The error of 3D medical images DWT occurs as a result of wavelet filters coefficients conversion (quantization noise) by Formula (2). Convolutions, upsampling and the summing of convolution results cause an increase in this error. Rounding down normalized voxel values of the restored image also has an effect. Note the important facts. 1. The analyzing and synthesizing wavelet filters consist of the same coefficients, according to Formula (1), hence, the limited absolute errors of computations will also be equal. Therefore, within the framework of theoretical calculations, wavelet filters are classified only into lowpass \(L\) and highpass \(H\) ones. 2. The sums of the lowpass and highpass wavelet filter coefficients are equal \(\sum\limits_{i=1}^{k}f_{L,i}=\sqrt{2}\) and \(\sum\limits_{i=1}^{k}f_{L,i}=\sqrt{2}\), respectively [27]. We introduce the following notation. 1. \(E_{j,F}\)--limited absolute error (LAE) of calculating the value of the coefficient at the \(j\)-th stage, resulting from convolution with a sequence of wavelet filters \(F\). 2. \(S_{F}\)--the exact value of the sum of the coefficients of the wavelet filter \(F\). 3. \(T_{j,F}\)--the exact value of the calculations in the \(j\)-th stage, after convolution with a sequence of wavelet filters \(F\). The errors \(a\) of all image decomposition coefficients \(LLL,LLH,LHL,LHH,HLL,HLH,HHL\) and \(HHH\) are separated into two groups \(a_{\varepsilon}\)\((\varepsilon=1,2)\) as a result of upsampling \(\uparrow\) 2. Figure 3 shows an example of the errors separation \(a_{\varepsilon}\)\((\varepsilon=1,2,3,4)\) at the upsampling by frames and columns, where \(Y^{*}=(Y+k)/2-1\) and \(Z^{*}=(Z+k)/2-1\). This situation is similar for upsampling by strings. Upsampling is applied three times during image restoration. We got eight groups of errors \(a_{\varepsilon}\)\((\varepsilon=1,2,3,4,5,6,7,8)\) as a result. Thus, we would add an additional index \(\varepsilon\) to the introduced notations, which denotes calculations by the spatial characteristics of wavelet filters coefficients. Next, we carried out analysis calculations for an estimation of the maximum error of the 3D medical images DWT. **Stage 1. Wavelet filters coefficients quantization. ** Let us calculate the exact values of the coefficients sums \(S_{F}\), \(S_{F,F}\) and errors \(E_{1,F}\), \(E_{1,F,F}\) of rounding up filters \(L\) and \(H\) scaled coefficients. \[S_{L}=\sum\limits_{i=1}^{k}2^{n}f_{L,i}=2^{n}{\sum\limits_{i=1}^{k}f_{L,i}}=2^{ n}\cdot\sqrt{2}=2^{n+\frac{1}{2}},S_{H}=\sum\limits_{i=1}^{k}2^{n}f_{H,i}=2^{n} {\sum\limits_{i=1}^{k}f_{H,i}}=2^{n}\cdot 0=0,\]  
\[S_{L,1}=\sum\limits_{i=1}^{k}2^{n}f_{L,2(i-1)},S_{L,2}=\sum\limits_{i=1}^{k}2^ {n}f_{L,2i-1},S_{H,1}=\sum\limits_{i=1}^{k}2^{n}f_{H,2(i-1)},S_{H,2}=\sum\limits _{i=1}^{k}2^{n}f_{H,2i-1},\]  
\[E_{1,L}=\sum\limits_{i=1}^{k}\Bigl{(}\left[2^{n}f_{L,i}\right]-2^{n}f_{L,i} \Bigr{)},E_{1,H}=\sum\limits_{i=1}^{k}\Bigl{(}\left[2^{n}f_{H,i}\right]-2^{n} f_{H,i}\Bigr{)},\]  
\[E_{1,L,1}=\sum\limits_{i=1}^{\frac{k}{2}}\Bigl{(}\left[2^{n}f_{L,2(i-1)} \right]-2^{n}f_{L,2(i-1)}\Bigr{)},E_{1,L,2}=\sum\limits_{i=1}^{\frac{k}{2}} \Bigl{(}\left[2^{n}f_{L,2i-1}\right]-2^{n}f_{L,2i-1}\Bigr{)},\]  
\[E_{1,H,1}=\sum\limits_{i=1}^{\frac{k}{2}}\Bigl{(}\left[2^{n}f_{H,2(i-1)} \right]-2^{n}f_{H,2(i-1)}\Bigr{)},E_{1,H,2}=\sum\limits_{i=1}^{\frac{k}{2}} \Bigl{(}\left[2^{n}f_{H,2i-1}\right]-2^{n}f_{H,2i-1}\Bigr{)}.\]  
**Stage 2. Row decomposition. ** Let us calculate the exact values \(T_{2,F}\) and errors \(E_{2,F}\) of row decomposition with filters \(L\) and \(H\). \[T_{2,L}=S_{L}\cdot M,E_{2,L}=E_{1,L}\cdot M,E_{2,H}=E_{1,H}\cdot M.\]  
All convolution results \(T_{j,F}\) with filter \(H\) are zero since \(T_{i,F}\) for all voxels are equal and \(\sum\limits_{i=0}^{k-1}f_{H,i}=0\)[27]. **Stage 3. Column decomposition. ** Let us calculate the exact values \(T_{3,F}\) and errors \(E_{3,F}\) of column decomposition with filters \(L\) and \(H\).