This page contains algorithms and reference formulas for Colorcheck. It's all in green text because it's all math.
Color difference (error) formulas
The notation in this section is adapted from the Digital Color Imaging Handbook, edited by Gaurav Sharma, published by the CRC Press, referred to below as DCIH. The DCIH online Errata was consulted.
In measuring color error, keep it in mind that accurate color is not necessarily the same as pleasing color. Many manufacturers deliberately alter colors to make them more pleasing, most often by increasing saturation. In calculating color error, you may choose not to use the exact formulas for ColorChecker L*a*b* values; you may want to substitute your own enhanced values. Imatest Master allows you to enter values from a file written in CSV format.
Absolute differences (including luminance)
CIE 1976
The L*a*b* color space was designed to be relatively perceptually uniform. That means that perceptible color difference is approximately equal to the Euclidean distance between L*a*b* values. For colors {L1*, a1*, b1*} and {L2*, a2*, b2*}, where ΔL* = L2* - L1*, Δa* = a2* - a1*, and Δb* = b2* - b1*,
ΔE*ab = ( ( ΔL*)2 + (Δa*)2 + (Δb*)2 )1/2 (DCIH (1.42, 5.35); (...)1/2 denotes square root of (...) ).
Although ΔE*ab is relatively simple to calculate and understand, it's not very accurate expecially for strongly saturated colors. L*a*b* is not as perceptually uniform as its designers intended. For example, for saturated colors, which have large chroma values (C* = ( a*2 + b*2 )1/2 ), the eye is less sensitive to changes in chroma than to corresponding changes for Hue (ΔH* = ( (ΔE*ab)2 - (ΔL*)2 - (ΔC*)2 )1/2 ) or Luminance (ΔL*). To address this issue, several additional color difference formulas have been established. In these formulas, just-noticeable differences (JNDs) are represented by ellipsoids rather than circles.
CIE 1994
The CIE-94 color difference formula, ΔE*94, provides a better measure of perceived color difference.
ΔE*94 = ( (ΔL*)2 + (ΔC* ⁄ SC )2 + (ΔH* ⁄ SH )2 )1/2 (DCIH (5.37); omitting constants set to 1 ), where
SC = 1 + 0.045 C* ; SH = 1 + 0.015 C* (DCIH (1.53, 1.54) )
[ C* = ( ( a1*2 + b1*2 )1/2 ( a2*2 + b2*2 )1/2 )1/2 (the geometrical mean chroma) gives symmetrical results for colors 1 and 2. However, when one of the colors (denoted by subscript s) is the standard, the chroma of the standard, Cs* = ( as*2 + bs*2 )1/2, is preferred for calculating SC and SH. The asymmetrical equation is used by Bruce Lindbloom.]
ΔH* = ( (ΔE*ab)2 - (ΔL*)2 - (ΔC*)2 )1/2 (hue difference ; DCIH (5.36) )
ΔC* = ( a1*2 + b1*2 )1/2 – ( a2*2 + b2*2 )1/2 (chroma difference)
An early meeting of the CMC The CMC color difference formula is widely used by the textile industry to match bolts of cloth. Although it's less familiar to photographers than the CIE 1976 geometric distance ΔE*ab , it's probably the best of the measurement metrics. It is slightly asymmetrical: subscript s denotes the standard (reference) measurement. CMC is the Color Measurement Committee of the Society of Dyers and Colourists (UK). |
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ΔE*CMC(l,c) = ( (ΔL* ⁄ lSL)2 + (ΔC* ⁄ cSC )2 + (ΔH* ⁄ SH )2 )1/2 (DCIH (5.37) ), where
(That's the lowercase letter l in (l,c) and the denominator of (ΔL* ⁄ lSL)2.) ΔE*CMC(1,1) (l = c = 1) is used for graphic arts perceptibility mesurements. l = 2 is used in the textile industry for acceptibility of fabric matching. For now Imatest displays ΔE*CMC(1,1).
SL = 0.040975 Ls* ⁄ (1+0.01765 Ls*) ; Ls* ≥ 16 (DCIH (1.48) )
= 0.511 ; Ls* < 16
SC = 0.0638 Cs* ⁄ (1+0.0131 ) + 0.638 ; SH = SC (TCMC FCMC + 1 - FCMC ) (DCIH (1.49, 1.50) )
FCMC = ( ( Cs*)4 ⁄ ( ( Cs*)4 + 1900 ) )1/2 (DCIH (1.51);
TCMC = 0.56 + | 0.2 cos(hs* + 168°) | 164° ≤ hs* ≤ 345° (DCIH (1.52) )
= 0.36 + | 0.4 cos(hs* + 35 °) | otherwise
ΔH* and ΔC* have the same formulas as CIE-94.
The CIEDE2000 formulas (ΔEoo and ΔCoo ) are the upcoming standard, and may be regarded as more accurate than the previous formulas. We omit the equations here because they are described very well on Gaurav Sharma's CIEDE2000 Color-Difference Formula web page. Default values of 1 are used for parameters kL, kC, and kH.
At the time of this writing (February 2008) the CIE 1976 color difference metrics (ΔE*ab...) are still the most familiar. CIE 1994 is more accurate and robust, and retains a relatively simple equation. ΔE*CMC is more complex but widely used in the textile industry. The complexity of the CIEDE2000 equations (DCIH, section 1.7.4, pp. 34-40) has slowed their widespread adoption, but they are on their way to becomming the accepted standard. For the long run, CIEDE2000 color difference metrics are the best choice.
Color differences that omit luminance difference
Since Colorcheck measures captured images, exposure errors will strongly affect color differences ΔE*ab, ΔE*94, and ΔE*94. Since it is useful to look at color errors independently of exposure error, we define color differences that omit ΔL*.
ΔC*ab = ((Δa*)2 + (Δb*)2 )1/2
ΔC*94 = ( (ΔC* ⁄ SC )2 + (ΔH* ⁄ SH )2 )1/2
ΔC*CMC = ((ΔC* ⁄ SC )2 + (ΔH* ⁄ SH )2 )1/2
ΔC00 omits the ( ΔL' ⁄ kLSL )2 term from the ΔE00 (square root) equation (see Sharma).
These formulas don't entirely remove the effects of exposure error since L* is affected by exposure, but they reduce it to a manageable level.
| Notation changes (made for consistency with textbooks such as DCIH) |
| ΔC*ab = ((Δa*)2 + (Δb*)2 )1/2, formerly called ΔE(a*b*), is the color difference (only), with L* omitted. ΔE*ab = ( ( ΔL*)2 + (Δa*)2 + (Δb*)2 )1/2, formerly called ΔE(L*a*b*), is the total difference, including L*. This notation is somewhat confusing because ΔE*ab includes L* (as well as a and b) in its formula. |
Color differences corrected for chroma (saturation) boost/cut
Many digital cameras deliberately boost chroma, i.e., saturation, to enhance image appearance in digital cameras. This boost increases color error in the ΔE and ΔC formulas, above.
The mean chroma percentage is
Chrp = 100% * (measured mean( (ai*2+bi*2)1/2 ) ) ⁄ (Colorchecker mean( (ai*2+bi*2)1/2 ) )
= 100% * mean (Cmeasured) ⁄ mean (Cideal) ; Ci = (ai*2+bi*2 )1/2
i ≤ 1 ≤ 18 (the first three rows of the Colorchecker)
Chroma, which is closely related to the perception of saturation, is boosted when Chrp > 100. Chroma boost increases color error measurements ΔE*ab, ΔC*ab, ΔE*94, and ΔC*94. Since it is easy to remove chroma boost in image editors (with saturation settings), it is useful to measure the color error after the mean chroma has been corrected (normalized) to 100%. To do so, normalized ai_corr and bi_corr are substituted for measured (camera) values ai and bi in the above equations.
ai_corr = 100 ai ⁄ Chrp ; bi_corr = 100 bi ⁄ Chrp
The reference values for the ColorChecker are unchanged. Color differences corrected for chroma are denoted ΔC*ab(corr), ΔC*94(corr), and ΔC*CMC(corr).
Mean and RMS values
Colorcheck Figure 3 reports the mean and RMS values of ΔC*ab corrected (for saturation) and uncorrected, where
mean(x) = Σxi ⁄ n for n values of x.
RMS(x) = σ(x) = (Σxi2 ⁄ n )1/2 for n values of x.
The RMS value is of interest because it gives more weight to the larger errors.
Algorithm
- Locate the regions of interest (ROIs) for the 24 ColorChecker zones.
- Calculate statistics for the six grayscale patches in the bottom row, including the average pixel levels and a second order polynomial fit to the pixel levels in the ROIs— this fit is subtracted from the pixel levels for calculating noise. It removes the effects of nonuniform illumination. Calculate the noise for each patch.
- Using the average pixel values of grayscale zones 2-5 in the bottom (omitting the extremes: white and black), the average pixel response is fit to a mathematical function (actually, two functions). This requires some explanation.
A simplified equation for a capture device (camera or scanner) response is,normalized pixel level = (pixel level ⁄ 255) = k1 exposuregamc
Gamc is the gamma of the capture device. Monitors also have gamma = gamm defined bymonitor luminance = (pixel level ⁄ 255)gamm
Both gammas affect the final image contrast,System gamma = gamc * gamm
Gamc is typically around 0.5 = 1/2 for digital cameras. Gamm is 1.8 for Macintosh systems; gamm is 2.2 for Windows systems and several well known color spaces (sRGB, Adobe RGB 1998, etc.). Images tend to look best when system gamma is somewhat larger than 1.0, though this doesn't always hold— certainly not for contrasty scenes. For more on gamma, see Glossary, Using Imatest SFR, and Monitor calibration.
Using the equation, density = - log10(exposure) + k,log10(normalized pixel level) = log10( k1 exposuregamc ) = k2 - gamc * density
This is a nice first order equation with slope gamc, represented by the blue dashed curves in the figure. But it's not very accurate. A second order equation works much better:log10(normalized pixel level) = k3 + k4 * density + k5 * density2
k3, k4, and k5 are found using second order regression and plotted in the green dashed curves. The second order fit works extremely well.
- Saturation S in HSV color representation is defined as S(HSV) = (max(R,G,B)-min(R,G,B)) ⁄ max(R,G,B). S correlates more closely with perceptual White Balance error in HSV representation than it does in HSL.
The the equation for saturation boost in the lower image of the third figure is S' = (1-e-4S ) ⁄ (1-e-4), where e = 2.71828...
Grayscale levels and exposure error
The Colorchecker grayscale patch densities (in the bottom row) are specified as 0.05, 0.23, 0.44, 0.70, 1.05, and 1.50. Using the equation, pixel level = 255 * (10–density ⁄ 1.06)(1/2.2) (see ISO speed, below), the ideal pixel levels would be 236, 195, 157, 119, 83, and 52, about 3% lower than the values measured by Bruce Lindbloom (242, 201, 161, 122, 83, and 49 for the Green channel) and provided with a Colorchecker purchased in October 2005 (243, 200, 160, 122, 85, 52). On the average, these measured values fit the equation,
pixel level = 255 * (10–density ⁄ 1.01)(1/2.2)
Exposure error is measured by comparing the measured levels of patches 2-5 in the bottom row (20-23 in the chart as a whole) with the selected reference levels. Patches 1 and 6 (19, 24) are omitted because they frequently clip. Since pixel level is proportional to exposuregamma, and hence log10(exposure) is proportional to log10(pixel level) ⁄ gamma (where gamma is measured from patches 2-5), the log exposure error for an individual patch is
Δ(log exposure) = (log10(measured pixel level) - log10(reference level)) ⁄ gamma
Using the mean value of Δ(log exposure) for patches 2-5 and the equation, f-stops = 3.32 * log exposure,
Exposure error in f-stops = 3 32 * (mean(log10(measured pixel level) - log10(reference level))) ⁄ gamma
ISO sensitivity
Starting with Imatest Master 3.5, two types of ISO sensitivity are calculated: Satuaraton-based Sensitivity and Standard Output Sensitivity. These measurements are described on the ISO Sensitivity page.
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