# A New Adaptive Spatial Filtering Method in the Wavelet Domain for Medical Images
## 2 Materials and Methods
### Theoretical Background  
Based on both inter-band and intra-band correlation, [PERSON] and [PERSON] [64] proposed an algorithm for reducing white Gaussian additive noise in images, whose results are compared by the authors with those obtained by the following methods:  
- Hard truncation with universal threshold;  
- A posteriori maximum probability estimator (MAP), also proposed by the authors in [65];  
- A maximum probability estimator modeled by hidden Markov chains, proposed by [PERSON] in [66];  
- Adaptive Wiener filtering in the wavelet domain, proposed in [67; 68]. The authors compared their proposed method with the maximum probability estimator and adaptive Wiener filtering in the wavelet domain, showing that, depending on the processed image, significant improvements were obtained both in terms of MSE and visually, appreciating them (in terms of MSE) between 8% and 26% for the highest probability estimator, and between 1% and 7% for the Wiener filter. The authors also compared the proposed method with the spatially adaptive method proposed by [PERSON] [69] and presented in the previous paragraph, in which case the results are lower than this (by almost 15% in MSE terms). In this method, the inter-band correlation is used to obtain a map containing the position of those wavelet coefficients that have a high informational content (contours and texture), called the significance map. The wavelet coefficients thus classified are processed by soft truncation using threshold values adapted to each sub-band, obtained by modeling their distribution by a Laplace distribution. The intra-band correlation is used to estimate the local signal scatter in the case of coefficients that do not belong to the significance map, which are processed with a maximum probability estimator. Considering a three-level orthogonal wavelet decomposition, the algorithm proposed by [PERSON] and [PERSON] [70] can be described as follows:  
- As in many other noise reduction methods using dyadic wavelet transformation, the lowest resolution sub-band is not processed (represented in blue in Figure 1);  
- Each of the three low-resolution detail sub-bands (corresponding to wavelet decomposition level 3, red in Figure 1) is processed by soft truncation; the threshold value for each sub-band is determined by modeling the distribution of its coefficients by a Laplace distribution of zero mean and dispersion:  
\[\sigma_{x,j}^{2}~{}=~{}max(0,\sigma_{t}^{2}-\sigma^{2}) \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline
**Original Image** & **PSNR** & **Sigma** \\ \hline CT image (Figure 7) 256 \(\times\) 256 & 20.15 & 0.1 \\ \hline CT image (Figure 8) 256 \(\times\) 256 & 20.13 & 0.1 \\ \hline CT image (Figure 9) 256 \(\times\) 256 & 26.09 & 0.05 \\ \hline CT image (Figure 10) 256 \(\times\) 256 & 20.18 & 0.1 \\ \hline CT image (Figure 11) 256 \(\times\) 256 & 26.15 & 0.05 \\ \hline \end{tabular}
\end{table} 
Table 1: Cont. where:  
\[\sigma_{t}^{2}\ =\ \frac{1}{M}\sum_{i}w^{2}(i) \tag{2}\]  
is the Gaussian noise dispersion, \(M\) being the number of coefficients in the considered sub-band and \(w\) the wavelet coefficients of the noise-degraded image.