On the Quadratic Phase Quaternion Domain Fourier Transform and on the Clifford algebra of $R^{3,1}$

We study application of the Clifford algebra and the Grassmann algebra to image recognitions in $(3+1)D$ using quaternions. Following S.L.Adler, we construct a quaternion-valued wave function model with fermions and bosons of equal degrees of freedom, similar to Cartan’s supersymmetric model. The Clifford algebra ${\mathcal{A}}_{3,1}$ is compared with ${\mathcal{A}}_{2,1}$ and the model applied to the $(2+1)D$ non-destructive testing is extended. The fixed point lattice actions are calculated for 7 paths in $(3+1)D$ space with lengths less than or equal to 8 lattice units. Comparison with the quaternion time approach of Ariel, quaternion Fourier transform of Hitzer and the tensor renormalization group approach to classical lattice models are also discussed.