Quadrature energy model of a complex cell ($E=(k_e\ast I{)}^2+(k_o\ast I{)}^2$)

Two oriented linear filters in quadrature have responses $A\cos \phi$ and $A\sin \phi$ to a suitably matched narrowband grating. Their squared sum is $A^2$, independent of phase $\phi$, while orientation and spatial-frequency selectivity remain. A nonlinear output function or normalization can transform this energy into firing rate. Ideal quadrature gives an exact phase identity for the matched stimulus; finite approximate filter pairs need not have perfect invariance for arbitrary images.