# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 3 Applications
### Pattern Recognition and Classification  
In 2016, a survey of advances in image analysis and multi-channel classification for terahertz (THz) pulse imaging and dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) was provided in [45]. The extension of SVM to Clifford SVM [8; 9] is analyzed as an image classification method that uses the high-grade elements of the multivector (elements of grade \(>0\), i.e., elements that are not the scalar element of the multivector) to describe the multi-channel image, as well as the application of extreme learning machine (ELM). In [46], the authors show the use of geometrical algebra in the most convenient way to deal with pattern description for higher dimensional spaces. A GA-SVM was applied to classify substances based on their THz spectra, and the results were compared with those obtained using a real-valued SVM. Using the GA framework, it was proved that substances with different thicknesses have coplanar multivectors, which allowed the authors to classify them. The method showed to be dispersion-independent and improved the accuracy and robustness regarding the real-valued SVM. A modification of the CSVM of [9] involving a multi-output support vector regression (MSVR) model with geometric rotation invariance using GA was proposed in [47]. The main difference between [9] and [47] is that in [9], the primal equation is considered one independent slack variable for each input data multivector and for each multivector component (i.e., one slack variable for each different-grade element of the multivector per each input multivector). This could lead to treating each input sample unfairly, according to the authors of [47], although it is the CSVM key to solving one optimization problem per different-grade element of each input multivector and using the hypercomplex kernel to obtain the direct sum of the linear spaces to obtain multi-classification. Conversely, in the proposal [47], one slack variable is defined for each input multivector data. The experiments show an application of the MSVR to remove clouds in remote-sensing images. A multi-output SVM was used to detect ground objects, thin clouds, and thick clouds. After the detection, the SVM is combined with a support vector value contourlet transform (SVVCT) to achieve multi-scale, multi-direction, and multi-resolution decomposition of remote sensing images. The algorithm presented is effective in removing clouds, but very complex and proportional to the number of decomposed subbands. Tasks of pattern recognition using iris data and breast cancer data were performed in [48] and respective recognition rates of 96% and 97.8% were obtained by an extension of a CSVM [9]. Object recognition problems using synthetic and real-object data were solved in [9]. Each object was described using a sequence of quaternions, and each quaternion contained the information of 3D key points of the object surface, in such a way that the object shape is entirely represented by a sequence of quaternions. A CSVM with a quaternion-valued Gaussian kernel was used to classify the objects. A six-class of synthetic objects and another six real objects sensed by a stereo camera were recognized, obtaining a minimum of 87.87% in recognition accuracy and a maximum of 96.15% for the synthetic objects in test phase. Meanwhile, 60% as minimum recognition accuracy for real objects and 84% as maximum recognition accuracy were obtained for test phase. In [14] two multiclass classification experiments were conducted using the MNIST database of handwritten digits. In both experiments, the images of each digit were preprocessed using a Fourier transform, and the 100 most significant complex coefficients of each image were used as the input data. In the first experiment, the one-versus-all real-valued SVM was compared with a quaternary complex SVM. The hypercomplex-valued SVM approach outperformed the real one by obtaining an error rate of 3.46% vs. 3.79%.