A Closed-Form Theory of Shadow-Induced Log-Chromaticity Error and the Ambient-Subtracted Chromaticity Invariant

Illumination-invariant chromaticity representations are widely used for shadow detection and removal, typically under the assumption that a shadow scales the incident illumination multiplicatively, I = αR. Real shadows, however, also receive ambient (sky, interreflection) illumination that does not vanish as the direct component is attenuated, giving the additive model I = αR + a. We derive, for this additive model, a closed-form expression (Theorem 1) for the error that shadows induce in log-chromaticity — a representation widely believed to be illumination-robust — and show it converges, as the direct term αR→0, to a fixed, non-zero limit equal to the log-chromaticity of the ambient illuminant itself (Theorem 2): in deep shadow, reflectance information is provably destroyed and the observed color is determined entirely by the ambient illuminant's chromaticity. We further show (Theorem 3) that, in 2D log-chromaticity space, the trajectories traced by a fixed surface under varying shadow strength α form a pencil of curves through a common vanishing point equal to the ambient illuminant's log-chromaticity — generalizing the parallel-line structure underlying prior invariant-direction methods. Exploiting this structure, we propose Ambient-Subtracted Chromaticity (ASC) and prove it is exactly invariant to α for all α when the ambient vector a is known (Theorem 4), with a closed-form characterization of its error under ambient misestimation (Theorem 5). All five theorems are verified to floating-point precision on synthetic Mondrian scenes. We further show that a least-squares multi-surface line-intersection estimator recovers a to error O(N⁻¹ᐟ²) in the number of surfaces N, and demonstrate on a real photograph with a synthetic additive-ambient shadow gradient that ASC with an estimated ambient (N=8, ∥ϵ∥≈0.018) reduces mean deviation from ground-truth chromaticity by 67% relative to raw chromaticity and 79% relative to log-chromaticity. We additionally derive a closed-form threshold α*(ε,εR)=[maxᵢ|cᵢ(R)·E-εᵢ|+|E|·εR] ]/(εR·S) partitioning every pixel into Recoverable, Erasure, and Unstable recovery regimes (Theorem 6), verified to ratio 0.989–1.000. The Erasure regime (Var(chroma(I))→0 as α→0) is connected to Landauer’s erasure principle and the Recoverable regime is reminiscent of a Feynman-gate information transfer.