Per-Channel Stevens Response (Bradford LMS, normalised to white = 1)
| Channel | I (0–1) | S = k·Iⁿ | CIELAB S1/3 | CIECAM02 S0.42 | Fechner ln(I/I₀) |
|---|---|---|---|---|---|
| L (red cone) | — | — | — | — | — |
| M (green cone) | — | — | — | — | — |
| S (blue cone) | — | — | — | — | — |
Fechner: S = k·ln(I/I₀) with I₀ from threshold slider · CIELAB: S = I^(1/3) · CIECAM02: S = I^0.42
Sensation at Itest — All Models
| Model / Modality | n | S at I | S at I=0.01 | S at I=0.5 | S at I=1 |
|---|
All with k=1 for comparability · I normalised 0–1 · S(I=1) = k (by definition)
CIELAB L* vs CIECAM02 J vs Stevens
| Channel | I (norm) | L* (CIELAB) | J proxy (n=0.42) | Custom S | Fechner S |
|---|---|---|---|---|---|
| L | — | — | — | — | — |
| M | — | — | — | — | — |
| S | — | — | — | — | — |
L*: 116·I^(1/3)−16 (CIELAB) · J proxy: 100·I^0.42 (CIECAM02 shape) · Fechner: k·ln(I/I₀)
Active Parameter Summary
| Parameter | Custom | CIELAB (n=1/3) | CIECAM02 (n=0.42) |
|---|---|---|---|
| n (exponent) | — | 0.333 | 0.420 |
| k (scale) | — | 1 (normalised) | 1 (normalised) |
| S at I=0.5 | — | — | — |
| dS/dI at I=0.5 | — | — | — |
| Perceived range (1000:1 input) | — | 10.0 (cube-root) | 15.1 (1000^0.42) |
| Classification | — | compressive | compressive |
S = k·In — Power Law Family (interactive — move I slider)
Gold = current n · Blue = CIELAB (1/3) · Teal = CAM02 (0.42) · Purple = Hunt (0.73) · Amber = Fechner (log) · Coloured dots = L/M/S
Sensation at Current I — Modality Comparison
Each bar = one modality at Itest · Gold outline = matches current n
Derivative dS/dI (Sensitivity Function)
dS/dI = k·n·I^(n−1) — sensitivity at each intensity. For n<1 sensitivity is highest at low I and decreases. Fechner predicts dS/dI = k/I.
Multi-n Exponent Overlay
Eight n values overlaid (0.20–3.50). n<1 compressive (brightness), n=1 linear (length), n>1 expansive (weight, pain).
Weber Fraction ΔI/I — Stevens vs Fechner
Weber = ΔI/I for threshold ΔS. Fechner predicts constant Weber fraction (Weber's Law). Stevens predicts it varies with I for n≠1.
S.S. Stevens (1957) — Magnitude Estimation and the Power Law
Stanley Smith Stevens at Harvard published "On the psychophysical law" (Psychological Review, 1957). He argued that sensation S is proportional to a power of the physical stimulus:
S = perceived sensation magnitude
I = physical stimulus intensity
n = modality-specific exponent
k = scaling constant (depends on measurement units)
Stevens developed the method of magnitude estimation: a standard stimulus is assigned a number (e.g. "10"), then participants assign proportional numbers to other stimuli. This ratio scale method differs from Fechner's partition scaling and consistently gives power functions, not logarithms.
Key property: Scale invariance — multiplying I by any constant multiplies S by a fixed ratio. log(2I) ≠ 2·log(I), but (2I)ⁿ = 2ⁿ·Iⁿ. The nervous system encodes ratios, not differences.
The Exponent n — Sensory Modalities and Published Values
| Modality | Stimulus | n | Notes |
|---|---|---|---|
| Brightness | Luminance | 0.33 | Binocular; basis of CIELAB L* |
| Brightness (monocular) | Luminance | 0.50 | Hunter Lab |
| Brightness (CIECAM02) | FL-adapted | 0.42 | Multi-observer fit |
| Loudness | Sound pressure | 0.60 | Free field; ISO 226 |
| Smell (coffee) | Concentration | 0.55 | Olfactory |
| Vibration (60 Hz) | Amplitude | 0.95 | Nearly linear |
| Line length | Spatial extent | 1.00 | Exactly linear |
| Apparent weight | Mass | 1.45 | Expansive |
| Electric shock | Current | 3.50 | Highly expansive |
For colour science: 1000:1 luminance range → 1000^(1/3) = 10:1 perceived brightness. This is exactly the compression encoded in CIELAB's cube-root.
Stevens vs Weber-Fechner — The Great Psychophysics Debate
| Property | Fechner (log) | Stevens (power) |
|---|---|---|
| Equation | S = k·log(I/I₀) | S = k·Iⁿ |
| Scale type | Interval | Ratio |
| Method | JND counting | Magnitude estimation |
| At low I | −∞ (breaks at threshold) | 0 (continuous) |
| At high I | Underestimates | Matches data |
| CIE adopted? | No | Yes (L* ≈ I^1/3) |
Fechner's mistake (per Stevens): assuming all JNDs are perceptually equal. Stevens showed JND size varies with modality. The debate remains philosophically alive, but for colour appearance modelling the power law won.
Stevens and CIELAB — The Cube-Root as a Psychophysical Power Law
CIE (1976): L* = 116·f(Y/Yn) − 16
f(t) = t^(1/3) for t > 0.008856
f(t) = 7.787·t + 16/116 for t ≤ 0.008856
CIECAM02 refinement: 0.42 instead of 0.33
At I=0.1: I^(1/3) = 0.464 vs I^0.42 = 0.380
At I=0.5: I^(1/3) = 0.794 vs I^0.42 = 0.727
n=0.42 is slightly more compressive at mid-tones
The CIELAB cube-root was chosen because it perfectly fits magnitude estimation data for brightness over a 100:1 luminance range — exactly what Stevens predicted.
Neural Mechanism — From Naka-Rushton to Stevens to CIECAM02
| Neural stage | Transform | Effective n |
|---|---|---|
| Photoreceptor (NR) | n≈0.73–1.0 | 0.73–1.0 |
| Bipolar → ganglion | n≈0.6–0.8 | 0.44–0.73 |
| LGN | n≈0.8 | 0.35–0.58 |
| V1 cortex | Divisive norm | 0.33–0.47 |
| Psychophysical | Magnitude est. | 0.33 |
The cascade model: ntotal = n₁·n₂·…·nk. Compressive stages multiply, pushing the effective psychophysical exponent toward 0.33–0.42. Stevens' power law thus serves as both the psychophysical target and the design constraint for every CAM.
Stevens' Power Law (Generalised)
S = k · I^n
Parameters:
S — perceived sensation magnitude
I — physical stimulus intensity
k — scaling constant (unit-dependent)
n — modality-specific exponent
Key properties:
S(0) = 0 (no stimulus → no sensation)
S(1) = k (unit stimulus → sensation = k)
Scale invariance: S(aI) = a^n · S(I)
For n<1: compressive (brightness, loudness)
For n=1: linear (line length)
For n>1: expansive (weight, pain)
Sensitivity and Gain Analysis
dS/dI = k · n · I^(n−1)
For n<1 (compressive):
dS/dI → ∞ as I→0 (very high sensitivity at low I)
dS/dI → 0 as I→∞ (saturating)
For n=1 (linear):
dS/dI = k (constant sensitivity)
For n>1 (expansive):
dS/dI → 0 as I→0 (low sensitivity at low I)
dS/dI → ∞ as I→∞ (accelerating)
Weber fraction:
ΔI/I = ΔS / (k·n·I^n) [for threshold ΔS]
Fechner predicts ΔI/I = const (Weber's Law)
Stevens predicts ΔI/I varies with I unless n=1
Weber-Fechner Logarithmic Law
S = k · ln(I / I₀)
I₀ = absolute detection threshold
Derivation from Weber's Law:
Weber: ΔI/I = c (constant)
Fechner assumed: ΔS = w (all JNDs equal)
Integrating: S = (w/c)·ln(I) + C
With S=0 at I=I₀: S = k·ln(I/I₀)
Stevens' critique (1961):
All JNDs are NOT perceptually equal
A JND in loudness ≠ a JND in brightness
Magnitude estimation gives power, not log
CIELAB and CIECAM02 as Stevens Power Laws
f(t) = t^(1/3) for t > 0.008856
f(t) = 7.787·t + 16/116 for t ≤ 0.008856
L* = 116·f(Y/Yn) − 16
This is Stevens with n=1/3, linearly extended
near black to avoid infinite cube-root slope.
CIECAM02:
k = 1/(5·LA+1)
FL = 0.2·k⁴·(5·LA) + 0.1·(1−k⁴)²·(5·LA)^(1/3)
x = (FL·LMS/100)^0.42
Ra = 400·x / (x + 27.13) + 0.1
The 0.42 exponent is a Stevens power law,
embedded inside a Naka-Rushton saturation.
Total: S ~ I^0.42 × Naka-Rushton compression.
Limitations and Context
2. n depends on experimental method and range
3. Regression bias in magnitude estimation
4. Cross-modality matching gives different n than direct
5. Does not account for adaptation (CIECAM02 adds FL)
6. Fechner better near threshold; Stevens better supra
7. Single-parameter (n) may not capture all observers
Despite these: the power law is the accepted psychophysical
basis for CIELAB, CIECAM02, and all modern CAMs. n ≈ 1/3
for brightness is one of the most replicated results in
psychophysics.
Primary Sources — Stevens
- Stevens, S.S. (1957). On the psychophysical law. Psychological Review, 64(3), 153–181.
- Stevens, S.S. (1961). To honor Fechner and repeal his law. Science, 133(3446), 80–86.
- Stevens, S.S. (1970). Neural events and the psychophysical law. Science, 170(3962), 1043–1050.
- Stevens, S.S. (1975). Psychophysics: Introduction to Its Perceptual, Neural, and Social Prospects. New York: Wiley.
Weber-Fechner and Historical Context
- Fechner, G.T. (1860). Elemente der Psychophysik. Leipzig: Breitkopf & Härtel.
- Weber, E.H. (1834). De Pulsu, Resorptione, Auditu et Tactu. Leipzig: Koehler.
- Gescheider, G.A. (1997). Psychophysics: The Fundamentals, 3rd ed. Mahwah: Erlbaum.
CIELAB, CIECAM02, and Colour Appearance
- CIE (1976). Colorimetry — Part 4: CIE 1976 L*a*b* Colour Space. CIE 15:2004.
- CIE 159:2004. A colour appearance model for colour management systems: CIECAM02.
- Fairchild, M.D. (2013). Color Appearance Models, 3rd ed. Chichester: Wiley.
- Glasser, L.G., McKinney, A.H., Reilly, C.D. & Schnelle, P.D. (1958). Cube-root color coordinate system. JOSA, 48(10), 736–740.
- Li, C. et al. (2017). Comprehensive color solutions: CAM16, CAT16, and s-CIECAM16. Color Research & Application, 42(6), 703–718.
Neural Basis and Naka-Rushton
- Naka, K.I. & Rushton, W.A.H. (1966). S-potentials from colour units in the retina of fish. Journal of Physiology, 185(3), 536–555.
- Hood, D.C. & Finkelstein, M.A. (1986). Sensitivity to light. Handbook of Perception and Human Performance, Vol. 1, Ch. 5.
- Breneman, E.J. (1987). Corresponding chromaticities for different states of adaptation. JOSA A, 4(6), 1115–1129.
Generate 200 random sRGB colours, compute Bradford LMS, apply Stevens' power law at current (n, k), and show the statistical distribution of perceived sensation. Includes CIELAB comparison and compression ratio.
| Colour | S(L) | S(M) | S(S) | S̄ | Class | Compress |
|---|