Preprocessing Quaternion Data in Quaternion Spaces Using the Quaternion Domain Fourier Transform

Recently a new type of hypercomplex Fourier transform has been suggested. It consequently transforms quaternion valued signals (for example electromagnetic scalar-vector potentials, color data, space-time data, etc.) defined over a quaternion domain (space-time or other 4D domains) from a quaternion ”position” space to a quaternion ”frequency” space. Therefore the quaternion domain Fourier transform (QDFT) uses the full potential provided by hypercomplex algebra in higher dimensions, such as 3D and 4D transformation covariance. The QDFT is explained together with its main properties relevant for applications such as quaternionic data preprocessing.