Abstract
Fractional-order differential operators accentuate weak edges and fine texture more effectively than their integer-order counterparts, but in image enhancement the fractional machinery is confined almost entirely to the spatial stage, while the tonal stage remains integer-order and heuristic. This paper places both stages on fractional foundations and derives what has so far been assumed. The tonal stage is built from the fractional-order Chebyshev and Legendre functions \(\varphi_k^{(\alpha)}(I)=p_k(2I^{\alpha}-1)\) generated by the substitution \(x\mapsto x^{\alpha}\); the spatial stage is a Grünwald–Letnikov fractional differential mask whose weights follow from the definition together with a zero-sum requirement, and which reduces to the classical eight-neighbour Laplacian sharpener exactly at order one. Five analytic results characterise the construction: a uniform bound on the deviation from the identity; a criterion on the slope profile that is necessary and sufficient for strict monotonicity simultaneously at every order \(\alpha>0\), so that intensity folding is excluded; the invariance of the total contrast gain under \(\alpha\), which shows that the fractional order relocates contrast rather than creating it; a closed-form law for where it is relocated; and the exact admissibility threshold \(\nu^{\ast}=3/2\), proved in both directions, beyond which the spatial mask ceases to amplify a band of frequencies and begins to attenuate it. A further proposition shows that monotonicity at every order is incompatible with preservation of the intensity range, so that clipping is intrinsic to this class and must be bounded rather than assumed away. Coefficients are obtained from a linear program instead of by trial. The resulting algorithm is deterministic and cheap: a monotone 256-entry lookup table followed by a fixed \(5\times5\) convolution, \(O(N)\) in the pixel count, with no image-dependent parameter and with monotonicity enforced only at intensities an 8-bit sensor can represent. On 18 images under four acquisition conditions (72 test cases, with a disjoint tuning split), the fractional family reaches operating points that the integer-order operator does not reach anywhere in the swept budget and order ranges, by a margin of +0.14 dB on average and at most +0.85 dB of no-reference contrast at matched fidelity. All code, parameters and metrics are released, and every reported number is regenerated by that code.