# Application of Image Fusion in Diagnosis and Treatment of Liver Cancer
## 4 Imaging Fusion Algorithms
### Wavelet Transformation Based Methods  
In the field of image fusion, the wavelet transform-based method, which was initially developed for signal processing [60], is widely used as high-pass filtering. The detailed image is the result of high contrast corresponding to high values in the frequency domain. The detail of image is the result of high contrast, which corresponds to high values in the frequency domain. By DWT, we can detect these details in the image, using functions that are localized in both space and frequency. For image fusion, these detailed pieces of information from input images can then be extracted and fused into a new image using certain fusion rules, such as maximum selection, weighted average and PCA. The low frequency part of the images can then be fused in a similar way.  
DWT has been widely used in CT/MRI and MRI/PET medical image fusion [61; 62; 63; 64; 65; 66; 67; 68]. However, DWT is known to be sensitive to the translation/shift of input signals, and therefore, translation among signals may exert a negative impact on effectiveness. Contourlet transform is a two-dimensional image representation based on wavelet multiscale analysis known as Pyramidal Directional Filter  
Figure 1: Representation of an image pyramid with four levels.  
Bank (PDFB) [69]. Compared with DWT, its basis functions are characterized by multiscale features, directionality, anisotropy and locality.  
Such basis functions effectively represent edge and curve singularity, and allow the efficient extraction of geometric and texture information in the image to obtain a better fusion effect. The multiscale geometric analytical tool used in Contourlet transform demonstrates the excellent spatial and frequency domain localization properties of wavelet analysis, as well as the bonus of multidirectional and multiscale characteristics, good anisotropy and suitability to describe the geometric characteristics of an image [70; 71]. However, for the Contourlet transform, shift invariance is lost as a result of its subsampling scheme for the multiscale partition. To overcome this difficulty, researchers have introduced the improved version of Contourlet transform, the Non-Subsampled Contourlet transform (NNCT) [72]. In the medical field, the Contourlet transform and the NNCT have also been to fuse MRI/PET and CT/MRI [73; 74; 70]. Another deficit of wavelet transformation is the lack of ability to represent edges and geometric structures of the image. Curvelet transform [75; 76; 77], a multiresolution and multi-direction pyramid that can preserve geometric regularity along edges [78], has been proposed to overcome this difficulty. [PERSON] et al. [79] proposed a Curvelet transform (CVT)-based method for the combination of CT and MRI. However, as highlighted in other studies [80], CVT is not built directly in the discrete domain, and thus does not provide a multiresolution representation of geometry. Shearlet transform (ST) and non-subsampled Shearlet transform (NNCT) [81; 82; 83; 84] are other sets of state-of-the-art tools with optimal use in sparse directional image representation. Based on composite wavelets, an optimal approximation of 2D functions is obtained. Compared to the Contourlet method, these methods have the advantage of directional selectivity and computational efficiency. Due to no restrictions on the number of directions for shearing, ST is used for the fusion of 2D and 3D medical images [82; 83] and NSST has application in CT/MRI image fusion [85; 86].