CIEDE2000 implementation in VBA

This page presents a reference implementation of the CIEDE2000 color difference formula in VBA. If you want to ensure perfect compatibility (to the tenth decimal place) with some third-party implementations, you may need to modify the comments in the source code. To facilitate this, the following link automates this operation.

The ΔE2000 function in VBA

Let’s consider the more common and academic (Sharma, 2005) of the two formulations.

' This function written in VBA is not affiliated with the CIE (International Commission on Illumination),
' and is released into the public domain. It is provided "as is" without any warranty, express or implied.

' Prevents errors due to typos or undeclared variables.
Option Explicit

' The classic CIE ΔE2000 implementation, which operates on two L*a*b* colors, and returns their difference.
' "l" ranges from 0 to 100, while "a" and "b" are unbounded and commonly clamped to the range of -128 to 127.
Public Function ciede_2000(l_1 As Double, a_1 As Double, b_1 As Double, l_2 As Double, a_2 As Double, b_2 As Double) As Double
	' Working in VBA with the CIEDE2000 color-difference formula.
	' k_l, k_c, k_h are parametric factors to be adjusted according to
	' different viewing parameters such as textures, backgrounds...
	Const M_PI = 3.14159265358979323846264338328, k_l = 1.0, k_c = 1.0, k_h = 1.0
	Dim n As Double, c_1 As Double, c_2 As Double, h_1 As Double, h_2 As Double
	Dim h_m As Double, h_d As Double, p As Double, r_t As Double, l As Double
	Dim t As Double, h As Double, c As Double
	n = (Sqr(a_1 * a_1 + b_1 * b_1) + Sqr(a_2 * a_2 + b_2 * b_2)) * 0.5
	n = n * n * n * n * n * n * n
	' A factor involving chroma raised to the power of 7 designed to make
	' the influence of chroma on the total color difference more accurate.
	n = 1.0 + 0.5 * (1.0 - Sqr(n / (n + 6103515625.0)))
	' Application of the chroma correction factor.
	c_1 = Sqr(a_1 * a_1 * n * n + b_1 * b_1)
	c_2 = Sqr(a_2 * a_2 * n * n + b_2 * b_2)
	' Using 14 lines to simulate atan2, as VBA does not have this built-in.
	If 0.0 < a_1 Then
		h_1 = Atn(b_1 / (a_1 * n)) - (b_1 < 0.0) * 2.0 * M_PI
	ElseIf a_1 < 0.0 Then
		h_1 = Atn(b_1 / (a_1 * n)) + M_PI
	Else
		h_1 = M_PI + ((0.0 < b_1) - (b_1 < 0.0)) * 0.5 * M_PI
	End If
	If 0.0 < a_2 Then
		h_2 = Atn(b_2 / (a_2 * n)) - (b_2 < 0.0) * 2.0 * M_PI
	ElseIf a_2 < 0.0 Then
		h_2 = Atn(b_2 / (a_2 * n)) + M_PI
	Else
		h_2 = M_PI + ((0.0 < b_2) - (b_2 < 0.0)) * 0.5 * M_PI
	End If
	' The atan2 polyfill (customized) is complete.
	n = Abs(h_2 - h_1)
	' Cross-implementation consistent rounding.
	If M_PI - 1E-14 < n And n < M_PI + 1E-14 Then n = M_PI
	' When the hue angles lie in different quadrants, the straightforward
	' average can produce a mean that incorrectly suggests a hue angle in
	' the wrong quadrant, the next lines handle this issue.
	h_m = (h_1 + h_2) * 0.5
	h_d = (h_2 - h_1) * 0.5
	If M_PI < n Then
		h_d = h_d + M_PI
		' 📜 Sharma’s formulation doesn’t use the next line, but the one after it,
		' and these two variants differ by ±0.0003 on the final color differences.
		h_m = h_m + M_PI
		' If h_m < M_PI Then h_m = h_m + M_PI Else h_m = h_m - M_PI
	End If
	p = 36.0 * h_m - 55.0 * M_PI
	n = (c_1 + c_2) * 0.5
	n = n * n * n * n * n * n * n
	' The hue rotation correction term is designed to account for the
	' non-linear behavior of hue differences in the blue region.
	r_t = -2.0 * Sqr(n / (n + 6103515625.0)) _
			* Sin(M_PI / 3.0 * Exp(p * p / (-25.0 * M_PI * M_PI)))
	n = (l_1 + l_2) * 0.5
	n = (n - 50.0) * (n - 50.0)
	' Lightness.
	l = (l_2 - l_1) / (k_l * (1.0 + 0.015 * n / Sqr(20.0 + n)))
	' These coefficients adjust the impact of different harmonic
	' components on the hue difference calculation.
	t = 1.0		+ 0.24 * Sin(2.0 * h_m + M_PI * 0.5) _
			+ 0.32 * Sin(3.0 * h_m + 8.0 * M_PI / 15.0) _
			- 0.17 * Sin(h_m + M_PI / 3.0) _
			- 0.2 * Sin(4.0 * h_m + 3.0 * M_PI / 20.0)
	n = c_1 + c_2
	' Hue.
	h = 2.0 * Sqr(c_1 * c_2) * Sin(h_d) / (k_h * (1.0 + 0.0075 * n * t))
	' Chroma.
	c = (c_2 - c_1) / (k_c * (1.0 + 0.0225 * n))
	' Returning the square root ensures that dE00 accurately reflects the
	' geometric distance in color space, which can range from 0 to around 185.
	ciede_2000 = Sqr(l * l + h * h + c * c + c * h * r_t)
End Function

' GitHub Project : https://github.com/michel-leonard/ciede2000-color-matching
'   Online Tests : https://michel-leonard.github.io/ciede2000-color-matching

' L1 = 85.3   a1 = 21.6   b1 = 3.9
' L2 = 87.7   a2 = 27.3   b2 = -4.2
' CIE ΔE00 = 6.1457265560 (Bruce Lindbloom, Netflix’s VMAF, ...)
' CIE ΔE00 = 6.1457093985 (Gaurav Sharma, OpenJDK, ...)
' Deviation between implementations ≈ 1.7e-5

' See the source code comments for easy switching between these two widely used ΔE*00 implementation variants.

k_l, k_c and k_h parameters

The k_l, k_c, and k_h parameters in CIEDE2000 are weighting factors applied to the terms brightness (ΔL*), chroma (ΔC*), and hue (ΔH*). They are defined as constants in the source code. Their default value is 1, which corresponds to the standard viewing conditions recommended by the International Commission on Illumination (CIE). In practice, you might need to adjust these coefficients to reflect specific conditions: for example, k_l = 2 is sometimes used to give more weight to differences in brightness (a common occurrence in the textile industry), while k_c or k_h can be reduced to increase tolerance for variations in saturation or hue, depending on the requirements. Depending on the context, these coefficients typically range from 0.5 to 2.

Source code accuracy and reliability

The difference between Sharma’s academic formulation and Lindbloom’s simplified formulation does not exceed ±0.0003 on the final ΔE2000. This corresponds to the difference usually measured between two 32-bit implementations and is imperceptible to the human eye. Our 64-bit implementations, all consistent with each other, guarantee at least 10 correct decimal places, so the choice of one formulation over the other is a technical detail. The default formula on this page is the one most often presented in the community, it is slightly easier to vectorize.

How do you convert RGB colors to L*a*b*?

Go to the AWK, C, Dart, Java, JavaScript, Kotlin, Lua, PHP, Python, Ruby or Rust page where such a converter (using D65 illuminant) is already implemented in addition to the color comparison function.

CIELAB value ranges and interpretation of the ΔE2000

In the CIELAB color space, the L* component represents lightness and typically ranges from 0 (black) to 100 (white). The a* and b* components represent color axes: a* goes from green to red, while b* goes from blue to yellow. In practice, a* and b* values usually fall between -128 and +127, although they can slightly exceed these limits depending on the color conversion.

ΔE2000 (CIEDE2000) measures the perceived difference between two colors: 0 means the colors are identical, and higher values (up to around 185 in extreme cases) indicate a larger difference. For example, a ΔE2000 value around 5 means the colors are close, while a value around 15 means they are clearly different.

Example of use in VBA

' Compute the Delta E (CIEDE2000) color difference between two L*a*b* colors in VBA

' Color 1: L1 = 96.4   a1 = 45.4   b1 = -3.0
' Color 2: L2 = 95.9   a2 = 50.4   b2 = 3.7

Dim deltaE As Double
deltaE = ciede_2000(l1, a1, b1, l2, a2, b2)
Debug.Print deltaE

' .................................................. This shows a ΔE2000 of 3.7852068385
' As explained in the comments, compliance with Gaurav Sharma would display 3.7852203702

Test results

The driver, written in the C99 language and featuring 250 precise static tests, has proved that this VBA function is interoperable with the CIEDE2000 function available in other programming languages.

CIEDE2000 Verification Summary :
  First Verified Line : 31.8,56.39,27.1,61,85,111.57,37.83481076432601
             Duration : 43.31 s
            Successes : 10000000
               Errors : 0
      Average Delta E : 62.9544
    Average Deviation : 8.8259009656255217e-15
    Maximum Deviation : 2.9842794901924208e-13

Files to download

Feel free to use these files provided by Michel, even for commercial purposes.

Community

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