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The Skin-Restricted Reinhard Transform:Uniqueness under a Lightness-Preserving Constraint

Vijesh KP

arXiv:2609.28424v1cs.CV

TL;DR

Catalog skin recolouring needs to change pigment without altering shading, but classical Reinhard matching rescales lightness and can flatten a limb. The paper constrains a diagonal Lab transform to preserve lightness contrast while matching chroma. On limb pairs, the tested map keeps lightness contrast within 3% of the photograph and chromatic error under one CIE Lab unit.

  • Problem

    Classical Reinhard matching rescales lightness by the reference-to-source standard-deviation ratio, so a low-variance skin reference can flatten the shading in a lit photograph.

    Source: §I · p. 1

  • Method

    The skin-restricted transform fixes the lightness gain at one and shift at the mean difference, while matching chromatic moments with gains clamped to an interval.

    Source: §I · p. 1, 2) Lightness · p. 3

  • Results

    On 27 limb pairs, the map is the only tested method that holds lightness contrast within 3% of the photograph while keeping chromatic error under one CIE Lab unit.

    Source: §VII · p. 8

  • Takeaways & Limitations

    Within the diagonal affine family, the shading rule forces the lightness gain to +1; the result does not claim uniqueness outside that family.

    Source: §I · p. 1, §VII · p. 8

    The uniqueness result is restricted to diagonal affine Lab maps.

    Source: §VII · p. 8

Abstract

from arXiv · show

Catalog skin recolouring has to change pigment and leave shading alone. The classical Reinhard map does not make that split: it rescales lightness by the ratio of standard deviations, and a flat reference swatch therefore flattens the limb. This paper formalises the correction used in our pipeline, the skin-restricted Reinhard transform. It is the diagonal affine map in CIE Lab that translates lightness, matches the chromatic mean, and clamps the chromatic gain to [0.72, 1.18], with moments taken on the central 84% of each channel. A diagonal affine map has six real parameters. The shading constraint forces the lightness gain to +1 and the lightness shift to the difference of means; one-dimensional quadratic optimal transport on each chromatic axis, followed by Euclidean projection onto the gain interval, fixes the other four. Inside that family the four conditions determine every parameter. The content of the result is the forced lightness gain; it is not a uniqueness claim outside the diagonal affine class. For Gaussian marginals the chromatic step is not merely the best affine map: it is the unrestricted Wasserstein-2 map. The same formulae with trimmed moments remain optimal because a positive affine image commutes with quantile trimming. On hands, arms, legs, and feet of nine photographs and three reference tones, the map keeps the lightness contrast ratio at 0.974 +/- 0.029 with chromatic error 0.77 CIE Lab units. Reinhard matching, the linear Monge map, and histogram matching reach a smaller chromatic error only by cutting lightness contrast to about half.

I. Introduction

Catalog skin recolouring must change pigment without changing the lightness patterns that carry shading; classical moment matching instead rescales lightness according to the reference variance. The paper introduces a skin-restricted diagonal map and uses a part parser to limit where it acts.

I. Introduction

  • Classical Reinhard matching rescales lightness by the reference-to-source standard-deviation ratio, flattening lit limbs when the close-up reference has much lower lightness variance.The proposed map translates lightness without rescaling it and clamps chromatic gain to prevent flat or noisy references from washing out the photograph or driving unnatural saturation.Source: §I · p. 1

II. Related Work

  • Prior colour-transfer methods match colour distributions or penalize gradient loss, whereas this paper fixes the lightness Jacobian at 1 within a diagonal map.Skin detectors can include cloth with skin-like chroma; the paper instead uses a part parser for the transfer support, with a guided filter only for matte boundaries.Source: §II · pp. 1–2

III. Skin Support

  • The skin support is built from Sapiens2 body-part labels, with clothing and eyeglasses excluded and ambiguous boundaries refined using a guided filter.The interior likelihood does not reduce the interior matte weight below one, preserving thin fingers from partial transfer.Source: §III · p. 2

IV. Mathematical Formalization

The formalization represents source and reference skin colours in CIE Lab and restricts the transform to a diagonal affine map. Its lightness rule translates the mean without rescaling the spread.

IV. Mathematical Formalization

  • On the skin set, source and reference colours are represented as random vectors in CIE Lab.The vectors have finite means and positive standard deviations.Source: §IV · p. 2
  • The transform family is diagonal affine, so each Lab channel has its own gain and translation without cross-talk.It is written as f_W,k(x) = Wx + k, with W = diag(w_L, w_a, w_b).Source: §IV · p. 2
  • This diagonal affine family has six real parameters, and the skin-restricted Reinhard transform is defined as the member satisfying the catalog constraints.The paper's uniqueness claim concerns this family.Source: §IV · p. 2

A. Lightness

  • The lightness rule shifts the source mean to the reference mean while leaving the lightness spread unchanged.With population means, L′ = L − µ_L + µ_L,Y, so the lightness gain is one.Source: A. Lightness · p. 2
  • The lightness translation is k_L = µ_L,Y − µ_L, with no percentile trim applied to the gain because it is identically one.The implementation estimates the means using the trimmed functional for robustness.Source: A. Lightness · p. 2
  • The implementation uses trimmed means for the two lightness means, and the paper states that this still matches the trimmed mean exactly.The trim is used for robustness, not to change the unit lightness gain.Source: A. Lightness · p. 2

B. Chroma

The chromatic rule matches trimmed means and standard deviations on the central 84% of each channel, then projects each gain onto a fixed interval. The deployed blend applies the endpoint transform at a configured strength.

B. Chroma

  • For each chromatic channel, the rule matches trimmed moments on the central 84% and projects the standard-deviation ratio onto an allowed gain interval.After choosing the gain, a unique translation matches the trimmed reference mean.Source: B. Chroma · p. 3

C. Deployed blend

  • The catalog configuration blends toward the transform at strength 0.7, which is not an extra parameter in the endpoint uniqueness theorem.If the Lab mean gap is at most 0.5, the pipeline returns the image unchanged.Source: C. Deployed blend · p. 3

A. What is being optimised

2) Lightness

The lightness constraint fixes the transform's gain at one and its shift at the difference of means. For Gaussian chromatic marginals, the paper's affine chromatic map is the monotone quadratic optimal-transport map.

2) Lightness

  • The shading constraint specifies no lightness rescaling and matches the lightness mean to the reference within the diagonal affine family.Six conditions on six scalars make the system square, but the lemmas establish existence and identify the solution.Source: 2) Lightness · p. 3

B. Lightness gain and shift

L Var(L). (11)

  • A Pearson correlation of 1 does not establish preserved shading: scaling lightness by ρ also scales its gradients and standard deviation by ρ.Classical Reinhard uses ρ = σ_L,Y / σ_L, while the proposed rule uses ρ = 1.Source: L Var(L). (11) · p. 3

C. Chromatic match as one-dimensional transport

  • For Gaussian chromatic marginals, the monotone quadratic Wasserstein map is affine; for non-Gaussian marginals, the optimal quantile map is nonlinear and reproduces the reference histogram.Matching a different reference lightness variance conflicts with the shading constraint, so histogram matching or other lightness-marginal matching transports are infeasible in that case.Source: C. Chromatic match as one-dimensional… · p. 4

D. The gain clamp is a projection

The chromatic gain is the Euclidean projection of the unclamped transport ratio onto the allowed interval. After clamping, a unique shift still matches the reference mean, while the variance gap may remain.

D. The gain clamp is a projection

  • Small reference chromatic deviation would drive the gain toward zero, while a heavy chroma tail could push it above a reasonable saturation range.The catalog interval prevents either extreme.Source: D. The gain clamp is a projection · p. 4
  • For γ > 0, on any nonempty closed interval [p, q], J(w) = γ(w − w◦)^2 has the unique minimiser w⋆ = Π[p,q](w◦), the Euclidean projection of w◦.Source: D. The gain clamp is a projection · p. 4
  • After the gain is clamped, a unique translation still matches the reference chromatic mean.The shift is μc,Y − wcμc and does not require the unclamped standard-deviation ratio.Source: D. The gain clamp is a projection · p. 4
  • Clamping intentionally leaves a residual variance gap when the unclamped ratio falls outside the allowed interval.The first-moment match remains intact.Source: D. The gain clamp is a projection · p. 4

E. Trimmed moments

Central-window trimmed moments limit the influence of highlights and crushed shadows, and positive affine maps preserve the trimmed-moment relations used by the transform. This lets the implementation substitute trimmed moments without changing the parameter lock.

E. Trimmed moments

  • The central window retains 84% of a channel’s mass, with ordinary moments used instead when fewer than 32 samples survive.An 8% tail replacement cannot move the window, giving the trimmed mean a breakdown point of α = 0.08.Source: E. Trimmed moments · p. 4
  • A positive affine map sends quantiles and the central sample window to their corresponding transformed values, so trimmed means transform affinely.A pure translation also preserves trimmed standard deviation.Source: E. Trimmed moments · p. 5
  • Replacing population moments with trimmed moments leaves the six-parameter lock unchanged and still meets the stated conditions.The gains and shifts are solved from fixed moment values, regardless of which moment functional produced them.Source: E. Trimmed moments · p. 5

F. The six parameters are determined

Within the diagonal affine Lab family, the conditions determine a unique transform, with lightness gain forced to +1. The result does not claim uniqueness outside that family, although the lightness-row constraint also restricts differentiable maps that preserve lightness gradients.

F. The six parameters are determined

  • Theorem 10 gives exactly one diagonal affine map satisfying the definition, including when moments are replaced by trimmed moments.The result’s substantive content is the forced lightness gain wL = +1, not parameter counting alone.Source: F. The six parameters are determined · p. 5, Table II · p. 5

G. Maps outside the diagonal family

  • The lightness gradient constraint forces the lightness row of a differentiable map to be (1, 0, 0) wherever image colours attain an open set.For an affine map, this means unit lightness gain and zero lightness cross-terms.Source: G. Maps outside the diagonal family · p. 5
  • A full 3 × 3 Gaussian Monge map is generally dense, so chromatic changes can alter lightness and violate the definition’s gradient constraint.Its quadratic colour cost does not include a lightness-gradient constraint,.Source: G. Maps outside the diagonal family · p. 5
  • The map is 1.18-Lipschitz, so skin colour differences cannot be amplified by more than 1.18.This bound holds regardless of how small the trimmed chromatic standard deviation becomes.Source: G. Maps outside the diagonal family · p. 5
  • Classical Reinhard lacks the same bound: a source chromatic deviation near zero makes its unclamped gain diverge.The gain ratio is σc,Y /σc.Source: G. Maps outside the diagonal family · p. 5

H. Infeasibility of lightness rescaling

When reference and source lightness deviations differ, classical Reinhard and other lightness-rescaling maps are infeasible under the stated definition. Strength blending and support masking are separate pipeline steps that affect the deployed result without changing the endpoint-map theorem.

H. Infeasibility of lightness rescaling

  • If σL,Y ≠ σL, classical Reinhard, monotone lightness rearrangement, and affine maps with a non-translational lightness row are infeasible under the definition.An infeasible map is not a better solution of that definition, whatever its chromatic error.Source: H. Infeasibility of lightness rescaling · p. 6

I. Strength and support sit outside the six parameters

  • At blend strength s = 0.7, the deployed choice intentionally leaves 30% of the mean gap.For a lightness translation, blending preserves the lightness gradient.Source: §I · p. 6
  • Where the support mask is zero, the output equals the input; clothing labels are assigned zero after each growth step.Identity on cloth is exact up to the three-pixel smoothstep ring and compression.Source: §I · p. 6

VI. Experiments

Across 27 limb pairs, the skin-restricted map preserves lightness contrast while keeping chromatic error low; comparison methods often reduce chromatic error by cutting contrast. The experiments also report deployed-strength results and show that method rankings depend on the scalarization weight.

A. Protocol

  • The limb set comprises nine catalog photographs crossed with three reference tones, yielding 27 photograph–reference pairs.The other two photographs contain only face-band skin and are excluded from the limb tables.Source: A. Protocol · p. 6
  • All methods use the same limb pixels and reference samples; baselines include Reinhard, the linear Monge map, Lab histogram matching, HSV mean shift, and iterative distribution transfer.The proposed method uses trimmed moments at strength s = 1.Source: A. Protocol · p. 6
  • Shading is measured by lightness standard-deviation and Sobel-energy ratios, while chromatic error is the Euclidean gap between trimmed (a, b) means.The scalarization weight β expresses a preference, not an independent measurement.Source: A. Protocol · p. 6

B. Quantitative comparison

  • 0.77 Lab units of chromatic residual accompanies a lightness contrast ratio of 0.974 ± 0.029; Reinhard, the Monge map, and histogram matching sit near 0.5 contrast.Fig. 1 shows the proposed map on the perfect-preservation line; the comparison methods obtain smaller chromatic error by losing lightness contrast.Source: B. Quantitative comparison · p. 6, Figure 1 · p. 7
  • At β = 0 histogram matching ranks first, while the skin-restricted map ranks first at β = 2 and β = 10.The higher weights favor the map because lightness-rescaling methods pay for the contrast they remove.Source: B. Quantitative comparison · p. 6
  • At deployed strength s = 0.7, contrast is 0.986 ± 0.024 and off-mask change is 0.69 ± 0.37 grey levels; global Reinhard changes nearby non-skin pixels by 57.5 ± 16.6 grey levels.The deployed map's chromatic error is 4.82 ± 2.42.Source: B. Quantitative comparison · pp. 6–7

C. Sensitivity of the constants the theorem does not fix

The deployed gain interval and central-84% trim are operating choices: the interval controls chroma contrast, while trimming gives the lowest chromatic residual on this set. The tested map preserves limb lightness contrast while keeping chromatic error under one CIE Lab unit.

D. Face band, without photographs

  • On the face band, the map has ∆Cab = 0.60 ± 0.89 and a lightness-contrast ratio of 0.97 ± 0.04, while classical Reinhard has 0.23 ± 0.08 and 0.41 ± 0.17.These are mask-restricted moments, not a study of eyes, lips, hairlines, or identity.Source: D. Face band, without photographs · p. 7

C. Sensitivity of the constants the theorem does not fix

E. Limb images

  • On the lower leg and foot, the classical methods erase the shin gradient, while the skin-restricted map retains the shading field across tones A, B, and C.The classical methods use tone B, the light reference and hardest contrast case in Table III.Source: E. Limb images · p. 8

F. Scope

  • The evaluation uses eleven catalog photographs and three reference tones, with 27 limb pairs and 33 face-band pairs; it has no held-out public benchmark or preference study.The cloth thresholds and local-fill weights are not ablated, and no labeled set supports a false-positive-rate calculation.Source: F. Scope · p. 8

VII. Conclusion

  • Within the diagonal affine family, the shading rule fixes lightness gain at +1 and shifts it by the difference of means; projected one-dimensional quadratic transport fixes the chromatic parameters.The result does not establish uniqueness outside that family.Source: §VII · p. 8
  • Within 3% of source lightness contrast and under one CIE Lab unit of chromatic error, the map is the only tested method to achieve both on 27 limb pairs.Histogram matching wins on chroma alone when contrast error is not charged.Source: §VII · p. 8

Appendix A Qualitative Comparison across Fourteen References

The appendix plates show the same crop across fourteen references as qualitative comparisons. They depict a portrait and a hand while noting which source details remain unchanged.

Appendix A Qualitative Comparison across Fourteen References

  • The hand and portrait plates are qualitative, use one crop per image, and are separate from the 27 limb pairs in Table III.Each plate transfers one image to fourteen references.Source: Appendix A · p. 10
  • Across the portrait references, hair, eyes, and teeth remain at the source colour.The face plate follows the appendix reference order.Source: Figure 4 · p. 10
  • Across the hand references, knuckle creases and the wrist shadow stay in place while pigment changes.The plate shows a clasped hand and forearm from one catalog photograph.Source: Figure 5 · p. 11
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