# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 3 Applications
### Robotics, Computer Vision  
An approach to predict 3D rotation and 3D transformations using computer vision data is presented in [8]. Using a multivector-valued SVM and the GA framework [16; 17] the authors solve the problem of rigid motion estimation that is necessary for a robot gripper to move along a 3D curve. With the 3D target position computed from the images of a stereo camera system, the SMVM estimates the motion of the gripper as a sequence of multivectors of the motor algebra \(G^{+}_{3,0,1}\). For the problem of 3D pose estimation, a trinocular camera, a triangulation algorithm, and the Hough transform were used to obtain relevant 3D points of an image, from which the 3D lines and their intersection points were computed using a regression to correct the distorted 3D points and to avoid the problem of occluded objects. Once the points, lines, and planes to describe object shapes as multivectors of the motor algebra are represented, they are used to train an SMVM that returns the multivector pose of the described object as two points lying on a 3D line, which were used as reference points on which the robot gripper has to position itself in order to hold the object. Another proposal for predicting 3D transformations is presented in [60]; they used a technique that approximates the error metric for the blending of unit dual quaternions. The authors used the GA framework to represent the 3D rigid transformation as a unit dual quaternion as in [61]. Then, with observations that are pairs of joint angles and rigid transformations of the fingertip relative to its root, regressors were constructed to predict the position and orientation of a fingertip with two rotational joints without any knowledge of its kinematic model. The authors also proved the effectiveness of their method in predicting the elastic deformation markers positions on a balloon as it is squashed, constructing the regressors of 3D rigid transformations of the marker and using those regressors as the 3D shape of the balloon. The regressors used were DQ regression, SVR, and ridge regression. In [57], KRR was used to simulate 3D inertial body sensor data. This is usually used in robotics to teach humanoid robots motion and movement. The paper explores \"the existence and uniqueness conditions of quaternion reproducing kernel Hilbert spaces (QRKHS)\" to aid in the designing of the \begin{table}
\begin{tabular}{c l l} \hline \hline
**Ref./Year** & **Application** & **SVM Extension Used** \\ \hline
[59]/2005 and [58]/2011 & Frequency estimation problem & Complex-valued support vector autoregressive technique \\ \hline
[11]/2004, [23]/2010 and [14]/2015 & Linear and nonlinear channel equalization, channel identification and filtering problems & Complex valued SVMs of [11; 14; 23] \\ \hline
[12]/2007 & Equalization of a 4-symbol quadrature amplitude modulated (4-QAM) signal over a complex linear communication channel that was added with Gaussian noise & Complex valued SVMs \\ \hline
[57]/2014 & Nonlinear channel equalization & Kernel ridge regression with quaternion kernels \\ \hline
[53]/2019 & Handwritten character recognition using the fast Fourier transform (FFT) coefficients of data from MNIST, fashion MNIST, extended MNIST and Latin OCR benchmark databases & Complex-valued neural network with kernel activation functions \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Applications of complex and hypercomplex SVMs for frequency domain processing, linear and nonlinear filtering problems. quaternion-valued kernels that are needed in hypercomplex SVMs and SVRs. Examples of quaternion valued-kernels were designed: the quaternion cubic kernel and the quaternion-valued Gaussian kernel. The first kernel mentioned was used in KRR to perform a one-step-ahead prediction of the limb trajectory in Tai Chi sequences, and a comparison of the results between the KRRs was performed using real, vector, and quaternion cubic kernels. The quaternion-valued kernels were shown to obtain lower mean squared errors than their real-valued counterparts.