Colorcheck Appendix

Algorithms and reference formulas

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 Pro 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*/S_C )² + (ΔH*/S_H )² )^1/2     (DCIH (5.37);  omitting constants set to 1 ), where

S_C = 1 + 0.045 C* ;   S_H = 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, C_s* = ( a_s*² + b_s*² )^1/2, is preferred for calculating S_C and S_H. 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 Standards Committee 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).

ΔE*_CMC(l,c) = ( (ΔL*/lS_L)² + (ΔC*/cS_C )² + (ΔH*/S_H )² )^1/2     (DCIH (5.37) ), where

(That's the lowercase letter l in (l,c) and the denominator of (ΔL*/lS_L)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).

S_L = 0.040975 L_s* / (1+0.01765 L_s*) ;   L_s* ≥ 16       (DCIH (1.48) )

     = 0.511 ;   L_s* < 16

S_C = 0.0638 C_s* / (1+0.0131 ) + 0.638 ;   S_H = S_C (T_CMC F_CMC + 1 - F_CMC )    (DCIH (1.49, 1.50) )

F_CMC = ( ( C_s*)⁴ / ( ( C_s*)⁴ + 1900 ) )^1/2     (DCIH (1.51);

T_CMC = 0.56 + | 0.2 cos(h_s* + 168°) |     164° ≤ h_s* ≤ 345°    (DCIH (1.52) )

        = 0.36 + | 0.4 cos(h_s* + 35 °) |       otherwise

ΔH*  and ΔC* have the same formulas as CIE-94.

CIEDE2000

The CIEDE2000 formulas (ΔE_oo and ΔC_oo ) 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 k_L, k_C, and k_H.

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*/S_C )² + (ΔH*/S_H )² )^1/2

ΔC*_CMC = ((ΔC*/S_C )² + (ΔH*/S_H )² )^1/2 

ΔC₀₀ omits the ( ΔL'/k_LS_L )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.

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( (a_i*²+b_i*²)^1/2 ) ) / (Colorchecker mean( (a_i*²+b_i*²)^1/2 ) )
      = 100% * mean (C_measured) / mean (C_ideal) ;   C_i = (a_i*²+b_i*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 a_{i_corr} and b_{i_corr} are substituted for measured (camera) values a_i and b_i in the above equations.

a_{i_corr} = 100 a_i / Chrp ;   b_{i_corr} = 100 b_i / 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) = Σx_i / n   for n values of x.
RMS(x) = σ(x) =  (Σx_i² / 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 exposure^gamc

Gamc is the gamma of the capture device. Monitors also have gamma = gamm defined by

monitor 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 exposure^gamc ) = 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 exposure^gamma, 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 speed

ISO speed is defined in Kodak Image Sensors - ISO Measurement (App note MTD/PS-0234), which is extracted from ISO standard 12232:1998. (Only 73 CHF— about $60 US— for 131 kB; such a deal!)

There are two basic types of ISO measurement: saturation-based (also called "base") and noise-based. The saturation-based ISO, which corresponds to the lowest camera ISO setting, is

ISO = (15.4 * f#²) / (L * t)    [Exposed so an 18% gray card has a level 18/106 of full scale. See below.]

where  f# is the effective f-number of the camera lens, L is the luminance in cd/m² of an 18% reflector (the familiar 18% gray card), and t is the exposure time in seconds. A camera's effective ISO setting may be increased by increasing the electronic gain.

You can use this equation to find the approximate luminance of an object, L =  (15.4 * f#²) / (ISO * t), where f# and t are read from an 18% gray card. For a typical white surface (90% reflectance), L =  (3.08 * f#²) / (ISO * t).

Digital image sensors are linear devices: their output voltage (and the pixel level in a RAW image file) is proportional to the light energy reaching the pixel up to the point where it abruptly saturates. Now here's the heart of the matter.

When an exposure is made according to the above setting, an 18% gray card has a voltage or pixel level 18/106 of full scale. In an 8-bit color space encoded for display at gamma = 2.2 (sRGB, Adobe RGB, WGRGB, etc.), the corresponding pixel level is 114 (from the formula, pixel level = 255*(18/106)^(1/2.2) ). Of course this is an ideal value; pixel levels in real cameras may vary, especially when a tonal response "S" curve is applied.

Any portion of the image with luminance equivalent to more than 106% reflectiance (5.9x brighter than the gray card) will be saturated-- pure white; burnt out; gonzo. Since a white card is about 90% reflectance (density = 0.05), that doesn't leave much margin. The Kodak document says, "Very demanding applications might need more headroom for highlights, and so might use a higher number when calculating base ISO." That's why exposure compensation-- equivalent to increased ISO speed-- is often required to avoid burnt out highlights for "very demanding applications" like ordinary sunlit outdoor scenes.