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

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