# A Weighted Histogram-Based Tone Mapping Algorithm for CT Images
## 5 Discussion  
The main difference between the proposed algorithm and traditional histogram equalization methods is the power-law-based distance-weighted contribution to the local histograms. LHE and CLAHE are special cases in our proposed generalized framework with constant spatial weights with radial cut-off (CLAHE) or without it (HE). While a similar approach is used for other types of weighting functions, we are not aware of power-law-based weighting. However, power-law was proposed as part of a generalized cumulation function-based histogram equalization [14]. This method is orthogonal to our approach because the Fourier series-based convolution approach is used on the intensity axis of the 3D histogram, while our method uses convolution along the spatial axes.  
Many novel histogram-based methods try to modify or prescribe the shape of the local histograms but leave the spatial part intact. These methods, for instance, the bi-histogram equalization, could be easily combined with our approach, and therefore, we see these algorithms as complementary.  
Traditional TMOs offer dynamic range compression with low noise but good local contrast. These operators perform well for selected sub-problems, usually either for low densities (lung) or for high densities (bones). The challenge for these algorithms is to reproduce good local contrast for medium densities (soft tissues) where the contrast is already low in the source. This situation is common in head CTs where the soft tissue contrast is poor.  
As can be seen in Table 2, the proposed algorithm yields good structural similarity scores. The structural similarity score is from the TMQI algorithm's structural similarity part. TMQI was developed for natural images, and it contains a 'naturalness' component. This does not seem applicable to medical images, and we only present the structural score. However, if naturalness was taken into account, it would not change the conclusions.  
A shortcoming of all of the scores, not only TMQI, is that they do not take into account the image content. Photographic images usually use the whole pixel domain to present information. In CT scans, information from outside of the human body has a limited role, and pixels outside of the scan field of view do not contribute to the image quality, but they are present in the DICOM files as zero pixels. Neither TMQI score nor the tone mapping algorithms take this into account.  
Structural similarity is not equally important in all parts of CT scans. It can be argued that a content-aware structural similarity metric should be developed in the future which takes the field-of-view and tissue properties into account. We are not aware of such a content-dependent performance metric for CT images, and it is beyond the scope of this paper to introduce one, but we recognize this limitation.  
TMQI offers structural similarity maps at various resolution levels. Figure 9 shows the finest resolution structural similarity maps of the different tone mapping operators using the head CT example. The advantage of the proposed method is that it has better structural similarity for medium densities, e.g., white and gray matter in the brain, than the alternative methods which mostly perform well outside of the skull.  
Figure 9: \(Cont\).