Parity Symmetry and Univalence of Complex Lucas, Faber and Gegenbauer Polynomial Operators for Intensity-Domain Image Enhancement

In this study, three global tone mapping operators are built from classical polynomial families to enhance low contrast documentation images, their behaviour fixed in advance by symmetry rather than chosen by taste. Lucas, Faber and Gegenbauer polynomials are generated by a recurrence, a conformal map and an orthogonality weight, yet one parity rule is shared by all three, traced here to a single symmetry of the generating object. The symmetry is carried onto the intensity axis, where mid-grey is recentred to the origin and the photographic negative becomes a reflection; the identity is perturbed by parity symmetric generators pinned at black and white; and the curve is quantized to a 256-entry table applied to the luminance channel by one lookup per pixel. Two properties are thereby imposed rather than observed. Commutation with tonal inversion is obtained exactly when the even-order coefficients vanish, the perturbation splitting into the eigenspaces of a Z2 action, read as contrast and brightness with the latter being deliberately switchable; rounding onto the display grid preserves the commutation away from ties. Order preservation is imposed as a univalence condition through the Noshiro–Warschawski criterion, which yields a closed form ceiling on the gain, whose leading mode value 9/4 is shared by all three families, which are separated thereafter only by the position of their zeros. Validation follows on six images of differing tonal balance against seven classical baselines, the inversion defect staying at the 8-bit quantisation floor.