Color Difference Formulas

Dec 12, 2024 · 5 min read · color delta-e color-difference colorimetry color-science ·

Share on:

Quantifying perceptual differences between colors using Delta E metrics.

Overview

Color difference formulas quantify how different two colors appear to the human eye. They're essential for:

Quality control in manufacturing
Color matching and reproduction
Perceptual image processing
Color tolerance specifications

Delta E Formulas

Delta E 1976 (CIE76)

The simplest formula, based on Euclidean distance in LAB space:

$$ \Delta E_{76} = \sqrt{(L_2^* - L_1^)^2 + (a_2^ - a_1^)^2 + (b_2^ - b_1^*)^2} $$

Pros: Simple, fast Cons: Not perceptually uniform, especially for saturated colors

Delta E 1994 (CIE94)

Improved formula with weighting factors:

$$ \Delta E_{94} = \sqrt{ \left(\frac{\Delta L^}{k_L S_L}\right)^2 + \left(\frac{\Delta C^}{k_C S_C}\right)^2 + \left(\frac{\Delta H^*}{k_H S_H}\right)^2 } $$

Where:

$\Delta L^* = L_2^* - L_1^*$
$\Delta C^* = C_2^* - C_1^*$ (chroma difference)
$\Delta H^* = \sqrt{(\Delta E_{76})^2 - (\Delta L^)^2 - (\Delta C^)^2}$ (hue difference)
$S_L = 1$
$S_C = 1 + 0.045 C_1^*$
$S_H = 1 + 0.015 C_1^*$
$k_L, k_C, k_H$ are weighting factors (typically 1 for reference conditions)

Delta E 2000 (CIEDE2000)

Most accurate perceptual formula (current standard):

$$ \Delta E_{00} = \sqrt{ \left(\frac{\Delta L'}{k_L S_L}\right)^2 + \left(\frac{\Delta C'}{k_C S_C}\right)^2 + \left(\frac{\Delta H'}{k_H S_H}\right)^2 + R_T \frac{\Delta C'}{k_C S_C} \frac{\Delta H'}{k_H S_H} } $$

Note: Full formula is complex with many correction terms. See implementation below.

Practical Implementation

Python (Delta E 76)

 1import numpy as np
 2
 3def delta_e_76(lab1, lab2):
 4    """
 5    Calculate Delta E 1976 (CIE76)
 6    Input: lab1, lab2 as (L*, a*, b*) tuples or arrays
 7    Output: Delta E value
 8    """
 9    lab1 = np.asarray(lab1)
10    lab2 = np.asarray(lab2)
11    return np.sqrt(np.sum((lab2 - lab1)**2))
12
13# Example
14color1 = (50, 20, 30)  # L*, a*, b*
15color2 = (55, 25, 35)
16de = delta_e_76(color1, color2)
17print(f"Delta E 76: {de:.2f}")

Python (Delta E 2000)

 1import numpy as np
 2
 3def delta_e_2000(lab1, lab2, kL=1, kC=1, kH=1):
 4    """
 5    Calculate Delta E 2000 (CIEDE2000)
 6    Simplified implementation
 7    """
 8    L1, a1, b1 = lab1
 9    L2, a2, b2 = lab2
10    
11    # Calculate C and h
12    C1 = np.sqrt(a1**2 + b1**2)
13    C2 = np.sqrt(a2**2 + b2**2)
14    C_bar = (C1 + C2) / 2
15    
16    # G factor
17    G = 0.5 * (1 - np.sqrt(C_bar**7 / (C_bar**7 + 25**7)))
18    
19    # Adjusted a*
20    a1_prime = a1 * (1 + G)
21    a2_prime = a2 * (1 + G)
22    
23    # Adjusted C and h
24    C1_prime = np.sqrt(a1_prime**2 + b1**2)
25    C2_prime = np.sqrt(a2_prime**2 + b2**2)
26    
27    h1_prime = np.degrees(np.arctan2(b1, a1_prime)) % 360
28    h2_prime = np.degrees(np.arctan2(b2, a2_prime)) % 360
29    
30    # Differences
31    dL_prime = L2 - L1
32    dC_prime = C2_prime - C1_prime
33    
34    # Hue difference
35    if C1_prime * C2_prime == 0:
36        dh_prime = 0
37    elif abs(h2_prime - h1_prime) <= 180:
38        dh_prime = h2_prime - h1_prime
39    elif h2_prime - h1_prime > 180:
40        dh_prime = h2_prime - h1_prime - 360
41    else:
42        dh_prime = h2_prime - h1_prime + 360
43    
44    dH_prime = 2 * np.sqrt(C1_prime * C2_prime) * np.sin(np.radians(dh_prime / 2))
45    
46    # Averages
47    L_bar_prime = (L1 + L2) / 2
48    C_bar_prime = (C1_prime + C2_prime) / 2
49    
50    if C1_prime * C2_prime == 0:
51        h_bar_prime = h1_prime + h2_prime
52    elif abs(h1_prime - h2_prime) <= 180:
53        h_bar_prime = (h1_prime + h2_prime) / 2
54    elif h1_prime + h2_prime < 360:
55        h_bar_prime = (h1_prime + h2_prime + 360) / 2
56    else:
57        h_bar_prime = (h1_prime + h2_prime - 360) / 2
58    
59    # Weighting functions
60    T = 1 - 0.17 * np.cos(np.radians(h_bar_prime - 30)) + \
61        0.24 * np.cos(np.radians(2 * h_bar_prime)) + \
62        0.32 * np.cos(np.radians(3 * h_bar_prime + 6)) - \
63        0.20 * np.cos(np.radians(4 * h_bar_prime - 63))
64    
65    dTheta = 30 * np.exp(-((h_bar_prime - 275) / 25)**2)
66    
67    RC = 2 * np.sqrt(C_bar_prime**7 / (C_bar_prime**7 + 25**7))
68    
69    SL = 1 + (0.015 * (L_bar_prime - 50)**2) / np.sqrt(20 + (L_bar_prime - 50)**2)
70    SC = 1 + 0.045 * C_bar_prime
71    SH = 1 + 0.015 * C_bar_prime * T
72    
73    RT = -np.sin(np.radians(2 * dTheta)) * RC
74    
75    # Final Delta E 2000
76    dE00 = np.sqrt(
77        (dL_prime / (kL * SL))**2 +
78        (dC_prime / (kC * SC))**2 +
79        (dH_prime / (kH * SH))**2 +
80        RT * (dC_prime / (kC * SC)) * (dH_prime / (kH * SH))
81    )
82    
83    return dE00

Go (Delta E 76)

 1package color
 2
 3import "math"
 4
 5func DeltaE76(lab1, lab2 LAB) float64 {
 6    dL := lab2.L - lab1.L
 7    da := lab2.A - lab1.A
 8    db := lab2.B - lab1.B
 9    return math.Sqrt(dL*dL + da*da + db*db)
10}

Interpretation Guidelines

Delta E 76

Range	Perception
< 1.0	Not perceptible by human eyes
1.0 - 2.0	Perceptible through close observation
2.0 - 10.0	Perceptible at a glance
10.0 - 49.0	Colors are more different than similar
> 49.0	Colors are opposite

Delta E 2000

Range	Perception
< 1.0	Not perceptible
1.0 - 2.0	Perceptible by trained observers
2.0 - 3.5	Perceptible by untrained observers
3.5 - 5.0	Clear difference
> 5.0	Very different colors

Note: Delta E 2000 values are typically smaller than Delta E 76 for the same color pair.

Industry Standards

Tolerances

Printing: $\Delta E_{00} < 2.0$ (good match)
Textiles: $\Delta E_{94} < 1.0$ (critical match)
Automotive: $\Delta E_{00} < 1.0$ (body panels)
Displays: $\Delta E_{76} < 3.0$ (acceptable)
Professional monitors: $\Delta E_{00} < 2.0$ (calibrated)

Choosing the Right Formula

Use Case	Recommended Formula
Quick comparison	Delta E 76
Quality control	Delta E 2000
Textiles	Delta E CMC
General purpose	Delta E 2000
Real-time processing	Delta E 76 or 94
Critical color matching	Delta E 2000

Common Pitfalls

Comparing different Delta E formulas: $\Delta E_{76} \neq \Delta E_{00}$
Ignoring viewing conditions: Lighting affects perceived difference
Using RGB distance: Always convert to LAB first
Not accounting for texture: Surface finish affects perception

Advanced Topics

Delta E CMC (l:c)

British textile industry standard with adjustable lightness and chroma weights:

$$ \Delta E_{CMC} = \sqrt{ \left(\frac{\Delta L^}{l \cdot S_L}\right)^2 + \left(\frac{\Delta C^}{c \cdot S_C}\right)^2 + \left(\frac{\Delta H^*}{S_H}\right)^2 } $$

Common ratios: CMC(2:1) for acceptability, CMC(1:1) for perceptibility.

Related Snippets

CIE LAB Color Space
Perceptually uniform color space for color difference calculations
CIE XYZ Color Space
Device-independent color representation based on human vision
Color Space Conversions - Quick Reference
Quick reference for all color space conversion formulas
HSV/HSL Color Spaces
Intuitive color representations for user interfaces and color pickers
RGB Color Space & Gamma Correction
sRGB color model and gamma encoding/decoding formulas
Standard Illuminants & Chromatic Adaptation
Reference white points, color temperature, and chromatic adaptation transforms