Complex Krawtchouk and Hahn Polynomial Operators for Localized and Numerically Stable Discrete Image Enhancement

The Krawtchouk and Hahn families are discrete orthogonal polynomials defined on the integer pixel grid, yet as polynomials, they are entire functions of a complex variable. Adopting this complex variable and complex-valued viewpoint, we develop an image enhancement framework that connects discrete orthogonal polynomial theory with geometric function theory. Two operators are introduced. The Krawtchouk operator exploits the binomial weight, for which the parameter p concentrates the basis around a selectable tonal level x=pN, producing a localized contrast enhancement steerable toward shadows, midtones, or highlights. The Hahn operator uses the two-parameter Hahn polynomials, the discrete analogue of the Jacobi family, in which (α,β) give asymmetric control of dark and light bands. Each operator is the real restriction of a holomorphic near identity map F(z)=z+∑kckϕ˜k(z), with the intensity entering the discrete basis through x=NI, realized as a monotone 256-entry lookup table. We prove a bounded deviation estimate |F−id|≤∑k|ck| on the intensity segment [0, 1] and a positive slope condition that, via the Noshiro–Warschawski criterion, is a univalence condition for the analytic transfer map, ruling out intensity order reversal and oscillatory folding. Because Hahn polynomials lose orthogonality at high order in naive arithmetic, we show that a three-term recurrence with log-gamma weights preserves orthonormality to within about 10−10 on the full 8-bit grid, where single precision computation fails, and that the same scheme remains at the double-precision roundoff level on 10-, 12-, and 16-bit grids (N=1023,4095,65,535). The operators cost O(mL) table construction plus one lookup per pixel (about 3 ms for a 1024×1024 color image), and a histogram-based differential entropy criterion selects the focus parameters automatically. Experiments on imagery of fine art, wildlife, archaeology, and architecture show that the Krawtchouk operator yields stronger localized contrast compared to seven classical methods, while the Hahn operator attains higher PSNR/SSIM, both as deterministic fast slope-controlled transforms. A diffusion MRI example further demonstrates that matching the focus parameter to the tonal mass adapts the same operators to dark-dominated medical scan imagery.