Per-Channel Naka-Rushton Response (Bradford LMS basis)
| Channel | Input I | R (raw) | r = R/Rmax | I/σ | Operating region |
|---|---|---|---|---|---|
| L (red cone) | — | — | — | — | — |
| M (green cone) | — | — | — | — | — |
| S (blue cone) | — | — | — | — | — |
I = signal normalised to white (0–100) · r<0.25 = linear · 0.25–0.75 = transition · r>0.75 = saturation
CIECAM02 Parameterisation Comparison
| Channel | I (0–100) | NR (custom) | CIECAM02 Ra | Hunt Ra |
|---|---|---|---|---|
| L cone | — | — | — | — |
| M cone | — | — | — | — |
| S cone | — | — | — | — |
CIECAM02: Ra=400·(FL·I/100)^0.42/((FL·I/100)^0.42+27.13)+0.1 · Hunt: Ra=40·x^0.73/(x^0.73+2)+1
Active Parameter Summary
| Parameter | Custom | CIECAM02 | Hunt (1994) |
|---|---|---|---|
| n (Hill exponent) | — | 0.42 | 0.73 |
| σ (semi-saturation) | — | 27.13 (in FL-domain) | 2 (in FL-domain) |
| Rmax | — | 400 (+0.1 offset) | 40 (+1 offset) |
| Half-max I (at current σ) | — | — | — |
| Response at I=σ | — | 200.1 (≈½ of 400.1) | 21 (≈½ of 41) |
| Slope at I=σ (dR/dI) | — | — | — |
Naka-Rushton Response Curve (interactive — move I slider)
Gold = custom (σ,n) · Blue = CIECAM02 (n=0.42) · Teal = Hunt (n=0.73) · Dot = current I · L/M/S = cone channel positions
Cone Channel Response Bars
Left 3 = raw LMS (0-1) · Right 3 = NR compressed · White ticks = CIECAM02 response level
Derivative dR/dI (Gain Function)
dR/dI = instantaneous gain (sensitivity). Peak gain occurs at the inflection point. Lower n → broader, flatter gain. σ shifts the peak.
Multi-σ Adaptation Overlay
Six σ values overlaid at current n. Increasing σ = higher adaptation level. Rushton: σ tracks background luminance, keeping the system at half-max.
Weber Fraction ΔI/I
Weber = ΔI/I for a threshold ΔR. Constant Weber ↔ Weber's Law. NR predicts deviation at low I (deVries-Rose) and high I (saturation).
Naka & Rushton (1966) — Retinal Physiology and the Original Paper
K.I. Naka and W.A.H. Rushton published "S-potentials from colour units in the retina of fish (Cyprinidae)" in the Journal of Physiology (1966). They recorded horizontal cell responses to light stimuli in carp and goldfish retinas and needed a mathematical description of how neural response R relates to stimulus intensity I.
They found the response follows Michaelis-Menten kinetics:
Generalised Hill form: R = Rmax · Iⁿ / (Iⁿ + σⁿ)
Normalised: r = Iⁿ / (Iⁿ + σⁿ)
Key properties:
- At I=0: R=0 (no response to no light)
- At I=σ: R = Rmax/2 (half-saturation — definition of σ)
- As I→∞: R→Rmax (finite maximum response)
- Inflection at I=σ·((n−1)/(n+1))^(1/n) for n>1, at I=0 for n≤1
Rushton (1961) established that σ represents the bleaching constant — the intensity at which half the photopigment is bleached at steady state. This is the molecular mechanism behind adaptation.
The Hill Exponent n — Cooperativity and Psychophysical Calibration
The generalised form with exponent n was introduced by A.V. Hill (1910) for oxygen-haemoglobin binding. In vision, n controls the sigmoid shape:
| n value | Shape | Application |
|---|---|---|
| n<1 | Sub-hyperbolic (gradual) | CIECAM02 (n=0.42) |
| n=1 | Michaelis-Menten hyperbola | Original NR, enzyme kinetics |
| n=0.73 | Intermediate | Hunt (1994), CIECAM97s |
| n≈0.74 | Similar to Hunt | Rod photoreceptors |
| n=2–4 | Steep sigmoid | Multi-subunit binding |
Why CIECAM02 uses n=0.42: The overall psychophysical brightness response integrates responses over many neural stages, each compressive. Cascading compressions lower effective n. Fairchild and Luo showed n=0.42 fits the Breneman, Luo-Rigg, and Braun-Fairchild appearance datasets.
Semi-Saturation σ and Luminance Adaptation
Rushton (1965) proposed the sensitivity control hypothesis: the visual system sets σ ≈ LA, keeping the system at its most sensitive operating point:
At adaptation: r(L_A) = L_A^n / (L_A^n + L_A^n) = 0.5
In CIECAM02, this is implemented through the FL factor:
FL = 0.2·k⁴·(5·LA) + 0.1·(1−k⁴)²·(5·LA)^(1/3)
x = FL·(I_LMS/I_white)·100
Ra = 400·x^0.42 / (x^0.42 + 27.13) + 0.1
By embedding FL inside the argument, increasing LA effectively scales σ down, implementing Rushton's sensitivity control without explicitly changing σ.
CIECAM02 and Hunt Parameterisations
| Model | n | σ (FL-domain) | Rmax | Offset | Equation |
|---|---|---|---|---|---|
| NR (1966) | 1 | σ (free) | Rmax | 0 | R=Rmax·I/(I+σ) |
| Hunt (1994) | 0.73 | ≈2.5 | 40 | +1 | Ra=40·x^0.73/(x^0.73+2)+1 |
| CIECAM02 | 0.42 | ≈670 | 400 | +0.1 | Ra=400·x^0.42/(x^0.42+27.13)+0.1 |
Offsets (+1 Hunt, +0.1 CIECAM02) ensure minimum response for darkness — the physiological fact of spontaneous neural activity in the dark.
From Single-Cell Physiology to Colour Appearance Models
| Year | Milestone | Contribution |
|---|---|---|
| 1966 | Naka & Rushton | S-potential sigmoid in fish retina |
| 1965–71 | Rushton | Sensitivity regulation: σ tracks LA |
| 1972 | Boynton & Whitten | NR applied to human cone threshold |
| 1978 | Valeton & van Norren | Modified NR for cones vs rods; n≈0.7 |
| 1982 | Hunt | NR compression (n=0.73) in CAM |
| 1994 | Hunt (1994) | Consolidated model → CIECAM97s |
| 1997 | CIECAM97s | First CIE standard, Hunt n=0.73 |
| 2002 | CIECAM02 | n→0.42, better psychophysical fit |
| 2016 | CAM16 | Unified CAT + CIECAM02 compression |
The effective psychophysical exponent (n≈0.33–0.5) is lower than single-cell (n≈1) because compressive stages multiply: ntotal = n₁ × n₂ × … × nk. This explains why Stevens' power law, CIELAB cube-root, and CIECAM02's 0.42 converge around ⅓.
Generalised Naka-Rushton (Hill) Equation
R = Rmax · Iⁿ / (Iⁿ + σⁿ)
Normalised response:
r = R / Rmax = Iⁿ / (Iⁿ + σⁿ)
Parameters:
I — stimulus intensity (cd/m² or normalised to white)
Rmax — maximum response (asymptotic)
σ — semi-saturation constant (I at R = ½Rmax)
n — Hill exponent (curve steepness / cooperativity)
Inverse (I from R):
I = σ · (R / (Rmax − R))^(1/n)
Gain Function and Operating Point Analysis
dR/dI = Rmax · n · σⁿ · I^(n−1) / (Iⁿ + σⁿ)²
At I = σ (operating point):
slope_σ = Rmax · n / (4 · σ) [exact for all n]
Inflection point (n > 1):
I_infl = σ · ((n−1)/(n+1))^(1/n)
For n ≤ 1: inflection at I = 0 (monotonically decreasing gain)
Gain at inflection:
dR/dI_max = Rmax·n·(n−1)^((n−1)/n) / (4σ·((n+1)/2)^((n+1)/n))
Practical: Lower σ → steeper slope at operating point.
Lower n → less steep overall, more gradual saturation.
CIECAM02 and Hunt Response Functions
k = 1 / (5·LA + 1)
FL = 0.2·k⁴·(5·LA) + 0.1·(1−k⁴)²·(5·LA)^(1/3)
CIECAM02 post-adaptation:
x = (FL · LMS_adapted/100)^0.42
Ra = 400 · x / (x + 27.13) + 0.1
Hunt (1994) post-adaptation:
x = FL · LMS_adapted / 100
Ra = 40 · x^0.73 / (x^0.73 + 2) + 1
Mapping to NR:
CIECAM02: n=0.42, σ_eff=27.13^(1/0.42)≈670, Rmax=400, offset=+0.1
Hunt: n=0.73, σ_eff≈2.5, Rmax=40, offset=+1
Weber's Law and Naka-Rushton
From NR, ΔI for threshold ΔR:
ΔI ≈ ΔR / (dR/dI) = ΔR · (Iⁿ + σⁿ)² / (Rmax·n·σⁿ·I^(n−1))
Weber fraction:
W = ΔI/I = ΔR · (Iⁿ + σⁿ)² / (Rmax·n·σⁿ·Iⁿ)
At I ≫ σ (bright): W → ΔR·Iⁿ / (Rmax·n·σⁿ) → rises (saturation)
At I ≈ σ (adapted): W ≈ 4·ΔR / (Rmax·n) → constant (Weber's Law!)
At I ≪ σ (dark): W → ΔR·σⁿ / (Rmax·n·Iⁿ) → rises (deVries-Rose)
NR naturally predicts Weber's Law in the mid-range and its breakdown at both extremes — matching the psychophysical "bathtub" curve.
Limitations and Extensions
2. Single-stage — real vision has 3+ cascading compressive stages
3. No surround suppression or lateral inhibition
4. σ treated as free parameter rather than background-adapted
5. Munsell/sRGB to LMS conversion is approximate
6. n assumed constant across all luminance levels
7. Does not model rod-cone transition (mesopic)
Despite limitations: The NR equation remains the gold-standard single-function description of photoreceptor compression. Its simplicity, closed-form inverse, and analytical derivatives make it irreplaceable in CAM engineering.
Primary Sources — Naka, Rushton, Hill
- Naka, K.I. & Rushton, W.A.H. (1966). S-potentials from colour units in the retina of fish (Cyprinidae). Journal of Physiology, 185(3), 536–555.
- Rushton, W.A.H. (1965). Visual adaptation. Proc. R. Soc. B, 162, 20–46.
- Rushton, W.A.H. (1972). Pigments and signals in colour vision. Journal of Physiology, 220, 1P–31P.
- Hill, A.V. (1910). The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. Journal of Physiology, 40, iv–vii.
Photoreceptor Physiology
- Boynton, R.M. & Whitten, D.N. (1970). Visual adaptation in monkey cones. Science, 170(3965), 1423–1426.
- Valeton, J.M. & van Norren, D. (1983). Light adaptation of primate cones. Vision Research, 23(12), 1539–1547.
- Hood, D.C. & Finkelstein, M.A. (1986). Sensitivity to light. Handbook of Perception and Human Performance, Vol. 1, Ch. 5.
Colour Appearance Models
- Hunt, R.W.G. (1994). An improved predictor of colourfulness in a model of colour vision. Color Research & Application, 19(1), 23–33.
- CIE 159:2004. A colour appearance model for colour management systems: CIECAM02.
- Fairchild, M.D. (2013). Color Appearance Models, 3rd ed. Chichester: Wiley.
- Li, C. et al. (2017). Comprehensive color solutions: CAM16, CAT16, and s-CIECAM16. Color Research & Application, 42(6), 703–718.
- Luo, M.R. & Hunt, R.W.G. (1998). The structure of the CIE 1997 colour appearance model (CIECAM97s). Color Research & Application, 23(3), 138–146.
Weber's Law and Psychophysics
- Stevens, S.S. (1961). To honor Fechner and repeal his law. Science, 133(3446), 80–86.
- Breneman, E.J. (1987). Corresponding chromaticities for different states of adaptation. JOSA A, 4(6), 1115–1129.
Generate 200 random sRGB colours, compute LMS cone signals, apply Naka-Rushton compression at current (σ, n, Rmax), and show the statistical distribution of normalised response. Includes operating region classification and CIECAM02 comparison per channel.
| Colour | r(L) | r(M) | r(S) | r̄ | Region(L) | Comp. ratio |
|---|