Complex cell

A complex cell typically responds selectively to an oriented edge or bar over a finite spatial region with less dependence on spatial phase and contrast polarity than a simple cell. It need not have the simple cell's separated linear ON and OFF subregions. Phase tolerance is local, not invariance to arbitrary displacement outside its receptive field.

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.

Rectify outputs of simple cells with similar orientations and varying phases or positions, then sum or pool them. Opposite-polarity subunits allow response to both light and dark features. Pooling gives approximate phase and position tolerance over the subunit coverage; its extent and contrast dependence depend on the pooling rule.