CIEDE2000 implementation in Lua

This page presents a reference implementation of the CIEDE2000 color difference formula written in the Lua programming language. If you want to ensure perfect compatibility (to the tenth decimal place) with certain third-party implementations, it may be necessary to modify the comments in the source code; the following link automates this operation for you.

The ΔE2000 function in Lua

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

-- This function written in Lua 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.

-- 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.
function ciede_2000(l_1, a_1, b_1, l_2, a_2, b_2)
	-- Working in Lua/LuaJIT 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...
	local k_l, k_c, k_h = 1.0, 1.0, 1.0;
	local n = (math.sqrt(a_1 * a_1 + b_1 * b_1) + math.sqrt(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 - math.sqrt(n / (n + 6103515625.0)));
	-- Simulate atan2, as Lua does not originally have this built-in.
	local c = a_1 * n;
	local c_1 = math.sqrt(c * c + b_1 * b_1);
	if 0.0 < a_1 then
		h_1 = math.atan(b_1 / c) + (b_1 < 0.0 and 2.0 or 0.0) * math.pi;
	elseif a_1 < 0.0 then
		h_1 = math.atan(b_1 / c) + math.pi;
	else
		h_1 = (b_1 < 0.0 and 1.5 or 0.0 < b_1 and 0.5 or 1.0) * math.pi;
	end
	c = a_2 * n;
	local c_2 = math.sqrt(c * c + b_2 * b_2);
	if 0.0 < a_2 then
		h_2 = math.atan(b_2 / c) + (b_2 < 0.0 and 2.0 or 0.0) * math.pi;
	elseif a_2 < 0.0 then
		h_2 = math.atan(b_2 / c) + math.pi;
	else
		h_2 = (b_2 < 0.0 and 1.5 or 0.0 < b_2 and 0.5 or 1.0) * math.pi;
	end
	-- The atan2 polyfill (customized) is complete.
	n = math.abs(h_2 - h_1);
	-- Cross-implementation consistent rounding.
	if math.pi - 1E-14 < n and n < math.pi + 1E-14 then n = math.pi end;
	-- 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.
	local h_m = (h_1 + h_2) * 0.5;
	local h_d = (h_2 - h_1) * 0.5;
	if math.pi < n then
		h_d = h_d + math.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 + math.pi;
		-- h_m = h_m + (h_m < math.pi and math.pi or -math.pi)
	end
	local p = 36.0 * h_m - 55.0 * math.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.
	local r_t = -2.0 * math.sqrt(n / (n + 6103515625.0)) *
			math.sin(math.pi / 3.0 * math.exp(p * p / (-25.0 * math.pi * math.pi)));
	n = (l_1 + l_2) * 0.5;
	n = (n - 50.0) * (n - 50.0);
	-- Lightness.
	local l = (l_2 - l_1) / (k_l * (1.0 + 0.015 * n / math.sqrt(20.0 + n)));
	-- These coefficients adjust the impact of different harmonic
	-- components on the hue difference calculation.
	local t = 1.0	+ 0.24 * math.sin(2.0 * h_m + math.pi * 0.5)
			+ 0.32 * math.sin(3.0 * h_m + 8.0 * math.pi / 15.0)
			- 0.17 * math.sin(h_m + math.pi / 3.0)
			- 0.20 * math.sin(4.0 * h_m + 3.0 * math.pi / 20.0);
	n = c_1 + c_2;
	-- Hue.
	local h = 2.0 * math.sqrt(c_1 * c_2) * math.sin(h_d) / (k_h * (1.0 + 0.0075 * n * t));
	-- Chroma.
	local 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.
	return math.sqrt(l * l + h * h + c * c + c * h * r_t);
end

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

-- L1 = 42.5   a1 = 23.6   b1 = -4.4
-- L2 = 45.2   a2 = 18.5   b2 = 3.5
-- CIE ΔE00 = 6.4300403439 (Bruce Lindbloom, Netflix’s VMAF, ...)
-- CIE ΔE00 = 6.4300224381 (Gaurav Sharma, OpenJDK, ...)
-- Deviation between implementations ≈ 1.8e-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 parameters k_l, k_c and k_h in the CIEDE2000 formula are weighting factors applied to the brightness (ΔL*), chroma (ΔC*) and hue (ΔH*) components respectively. In the source code, they are defined as constants with a default value of 1, corresponding to the standard observation conditions laid down 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. In summary, these coefficients typically range from 0.5 to 2, with 1 being the most common value.

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. The implementation presented here is 64-bit and guarantees more than 10 decimal places of precision; the choice of one formulation over another is, therefore, a technical detail. At the top of this page, you can choose between the two formulations; the one currently displayed is the simplified formulation.

How to determine if a given implementation of CIEDE2000 is academic or simplified?

• Evaluate ciede_2000(56.6, 43.6, 41.1, 68.4, 9.4, -8.6)
• If the result is 30.0001, then it is the academic type (such as Sharma, OpenJDK, etc.)
• If the result is 29.9999, then it is the simplified type (such as Lindbloom, Netflix VMAF, etc.)

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

You will need to use the XYZ intermediate color space for the conversion, and if you need help, the source code is provided at the bottom of this page (using the D65 white point formalized in 1964).

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, the values of a* and b* are almost always limited to a range between -128 and +127, although the standard does not specify an official limit for these two components.

ΔE2000 (CIEDE2000) measures the perceived difference between two colors: 0 means the colors are identical, and higher values (up to 185 and more) 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. When the ΔE2000 value exceeds 40, the colors being compared have virtually nothing in common, and we can no longer derive any precise information from them.

Example of use in Lua

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

-- Color 1
local L1, a1, b1 = 75.1, 61.9, -3.2

-- Color 2
local L2, a2, b2 = 75.6, 55.9, 3.1

local deltaE = ciede_2000(L1, a1, b1, L2, a2, b2)
print(deltaE)

-- .................................................. This shows a ΔE2000 of 3.3591979531
-- As explained in the comments, compliance with Gaurav Sharma would display 3.3591841825

Test results

Our test program, written in C99, includes 250 precise static tests. These tests ensure that your calculations will be performed without errors, even in critical edge cases, for example, when the arctangent function returns a mathematically undefined value. The results show that this CIEDE2000 function in Lua is interoperable with the 41 other programming languages that we offer.

CIEDE2000 Verification Summary :
  First Verified Line : 25.3,0.28,53.78,65.8,-101.21,-86,67.92949263023695
             Duration : 19.28 s
            Successes : 10000000
               Errors : 0
      Average Delta E : 62.9474
    Average Deviation : 4.2586916049192067e-15
    Maximum Deviation : 1.1368683772161603e-13

Files to download

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

Community

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