Color/Tone and Colorcheck Appendix

Algorithms and reference formulas

This page contains algorithms and reference formulas for Colorcheck, Color/Tone Setup and Color/Tone Auto. It’s mostly in green text because it’s all math.

 

Color difference formulas – CIE 1976 – CIE 1994  – CIEDE2000 – Delta-C –  Color ellipses(ΔC ab, ΔC 94, ΔC 2000)
Color difference visualizer – Chroma correction (corr) – Algorithm
Grayscale and Exposure error (fix in 5.1+)

 

Color difference (error) formulas-

You have two rectangular patches with different colors. How different are they?  

The Wikipedia Color Difference page has a concise introduction to color difference concepts and equations. This page (written before the Wikipedia page was available) covers similar material.

The notation on this page 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, typically by increasing saturation. (That is why Fujichrome Velvia was so successful when it was introduced in 1990.) In calculating color error, you may choose not to use the exact Colorchecker (or other chart) L*a*b* reference values; you may want to substitute your own enhanced values. Imatest Master allows you to enter reference values from a file written in CSV format (you can save the values from a measured chart using Color/Tone Setup).

Imatest users frequently ask about the meaning of “corr” and “uncorr” in the Colorcheck a*b* error plot. Corr means that the mean saturation (mean chroma =  mean(sqrt(a*²+b*²)) of the camera is adjusted to be the same as that of the reference before making the comparison. This is a very easy correction to make; it tells how accurate the color hue would be if the mean chroma were the same as the reference. See below for more detail.

Absolute differences (including luminance)

CIE 1976

The CIELAB (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*,

\(\Delta E^*_{ab} = \sqrt{\Delta L^{*2} +  \Delta a^{*2} + \Delta b^{*2} } \)      (DCIH (1.42, 5.35).

Although ΔE*_ab is relatively simple to calculate and understand, it’s not very accurate, especially 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 ( \(\Delta H^* = \sqrt{\Delta E^{*2}_{ab} – \Delta L^{*2} – \Delta C^{*2}_{ab}}\) ) 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 than ΔE*_ab .

\(\displaystyle \Delta E^*_{94} = \sqrt{(\Delta L^*)^2 + (\Delta C^*/S_C)^2 + (\Delta H^*/S_H)^2}\)   (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) )

[ \(C^* = \sqrt{\sqrt{a_1^{*2} + b_1^{*2}} \ \sqrt{a_2^{*2} + b_2^{*2}}}\)  (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.]

\(\displaystyle \Delta H^* = \sqrt{\Delta E^{*2}_{ab} – \Delta L^{*2} – \Delta C^{*2}_{ab}}\)       (hue difference ; DCIH (5.36) )

\(\displaystyle \Delta C^*_{ab} = \sqrt{a^{*2}_1 + b^{*2}_1} \; – \sqrt{a^{*2}_2 + b^{*2}_2}\)    (chroma difference))

CMC — The CMC color difference formula is used by the textile industry to match bolts of cloth, but it has not gained traction in the photographic industry. 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). We mention it only reference. An early meeting of the CMC Standards Committee ΔE*CMC(l,c) = ( (ΔL* ⁄ lSL)2 + (ΔC* ⁄ cSC )2 + (ΔH* ⁄ SH )2 )1/2 (DCIH (5.37) ), where ΔE*CMC(1,1) (l = c = 1), used for graphic arts perceptibility measurements,.was also used by Imatest. 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.

CIEDE2000

The CIEDE2000 formulas (ΔE_oo and ΔC_oo ) are the best current color difference metric, and may be regarded as more accurate than the previous formulas. We omit the equations here because they are given in the Wikipedia Color Difference page and in Gaurav Sharma’s CIEDE2000 Color-Difference Formula web page. Default values of 1 are used for parameters k_L, k_C, and k_H.

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. The complexity of the CIEDE2000 equations (DCIH, section 1.7.4, pp. 34-40) has slowed its widespread adoption, but it’s now the accepted standard. CIEDE2000 color difference metrics are the best choice.

Chroma differences ΔC (color differences that omit luminance difference)

Exposure errors strongly affect color differences ΔE*_ab, ΔE*₉₄, and ΔE*₀₀. Since it is useful to look at color errors independently of exposure error, we define color differences that omit ΔL*.

\(\Delta C_{ab} = \sqrt{(\Delta a^*)^2 + (\Delta b^*)^2} = \sqrt{(\Delta E^*_{ab})^2 – (\Delta L^*)^2 } \)     (This is a more general form: Delta-E with Delta-L removed.)

\(\Delta C_{94} = \sqrt{(\Delta C^* / S_C)^2 + (\Delta H^* / S_H)^2} \) 

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 a* and b* are affected somewhat by exposure, but they reduce it to a manageable level. Note that the ΔC definition can cause some confusion because it is different from the pure chroma difference, Δ|C*|, where chroma = |C*| = sqrt(a*² + b*²). Δ|C*| is not suitable as a perceptual measurement because it does not include hue.

Other color differences used for Colorcheck grayscale

The bottom of the fourth Colorcheck figure illustrates the White Balance error (deviation from neutral gray), which is quantified in four ways.
 

1. ΔC₂₀₀₀. This is probably the most perceptually meaningful color difference metric.  See the Wikipedia color differences page. It is color-space independent.
2. Saturation S from the HSV color model (burgundy). S can take on values between 0 for a perfect neutral gray and 1 for a totally saturated color. The perceptual color difference corresponding to S depends on the color space. 
3. Correlated Color Temperature (CCT) error in Degrees Kelvin (blue), calculated from a table in Color Science: Concepts and Methods, Quantitative Data and Formulae, by Günther Wyszecki and W. S. Stiles, pp. 224-228. Color temperature in degrees K is universally used to specify lighting. Higher color temperatures are perceptually “cooler” , i.e., more blue. The CCT error is the change in illumination color temperature that would cause the measured deviation from neutral gray.
4. Inverse color temperature in Mireds (micro-reciprocal degrees) (blue), where Mireds = 10⁶/(Degrees Kelvin). A more perceptually uniform metric than CCT error. It was used to specify the strength of optical color correction filters, but isn’t used much in the digital era because such filters are rarely required. 

Color difference representation (ellipses)

Color difference can be represented in Imatest by ΔC circles or ellipses on the a*b* plane of L*a*b* space. It can also be displayed in the xy and uv representations— useful for visualizing the perceptual uniformity of these spaces (uv isn’t as good as claimed), but a*b* is usually the best representation. Three ΔC representations are shown below (click on images to see them full size). ΔC₉₄ and ΔC₂₀₀₀ are close to each other, but very different from (and superior to) ΔC_ab.

Color difference circles or ellipses (shown for ΔC_xx = 4)

ΔCab circles

ΔC94 ellipses

ΔC2000 ellipses

Until these ellipses were added to Imatest they were not generally available. We thought the difference between ΔC₉₄ and ΔC₂₀₀₀ was greater than it actually is. In Color/Tone Setup these ellipses can be set in the Color ellipses dropdown menu. Color/Tone Auto uses settings saved from Color/Tone Setup. In Colorcheck they can be selected in the Settings window.

Color difference visualizer

An app for visualizing color differences has been added to Imatest 5.2. It works best viewed on a high quality calibrated monitor with at least sRGB gamut, though it can provide information on colors on lower quality monitors. To open it from the Imatest main window, press Utility, Color Diff Visualizer or press Color Diff Visualizer in the Utility tab in the box on the right. This opens the window below with a* = b* = 0 in the large display on the left and Δa* = Δb* = 0 in the small display at the bottom, which illustrates the rotation function. 

Color difference visualizer, shown for a* = 52, b* = 44, rotating on ΔC₂₀₀₀ ≅ 4 ellipse

To view a color difference, select an a*,b* reference value by clicking on the large a*b* display on the left. (a* and b* will be rounded to the nearest integer.) Then select Δa* and Δb* by clicking on the small display on the bottom. Color difference results are displayed on the top-right. In both the a*b* and Δa*Δb* displays + indicates the a*, b* reference value and o indicates Δa*, Δb* difference values. The selected values are displayed near the x and y-axes, respectively.

In the results display, the outer region is the a*, b* reference value (shown as magenta) and the inner part is the a*+Δa*, b*+Δb* shifted value (shown above rotating on the ΔC₂₀₀₀ ≅ 4 ellipse).

The small buttons with ^, v, <, and > below the right of the a*b* display shift the a* or b* setting by 1. The small ctr button to the right of the Δa*Δb* display sets Δa* = Δb* = 0.

sRGB color space is assumed. The middle gray curves in the large a*b* image represent the gamut boundaries for sRGB with L (HSL) = 0.5. Note that this is not the same as CIELAB L* = 50 (it’s more difficult to find a curve for this value).

The L* slider lets you set luminance L*. 

The Rotate on ellipse button on the lower-right causes the Δa* and Δb* values to move around the ellipse. The default ellipse is for ΔC2000 = 4.  You can pause or resume rotation by pressing the button. You can check (or uncheck) Fast rotation in the Settings dropdown menu. Color difference visibility may be larger during rotation than for static images because of the way the eye adapts.

The Settings dropdown lets you reset default values ( a* = Δa* = b* = Δb* =0 and L* = 70), set the axes limits in the Δa*Δb* window to ±20 (the default) or ±10, or select Fast rotation.

The Color ellipses dropdown let you set the ellipse type (none, Macadam, ΔC_ab, ΔC₉₄, ΔC₀₀ (ΔC₂₀₀₀) and magnification (typically 4x or 2x for the ΔC ellipses; 10x or 5x for the Macadam ellipses; 1x is just too small to be useful). ΔC₂₀₀₀ is the best available color difference formula, though it’s far from perfect.

Color differences corrected for chroma (saturation) boost/cut

This result has been de-emphasized in recent versions of Imatest because adjusting chroma is not of interest to camera designers, and it is not relevant to the performance of Color Correction Matrices (CCMs).

 

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, though it may improve perceived image quality if it’s not overdone.

The mean chroma percentage is

Chroma(%) = 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

1 ≤ i ≤ 18 (the first three rows of the Colorchecker)

Chroma, which is closely related to the perception of saturation, is boosted when Chroma(%) > 100. Chroma boost increases ΔE* and ΔC* color error measurements. 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 ⁄ Chroma(%) ; b_{i_corr} = 100 b_i ⁄ Chroma(%)

The reference values for the ColorChecker are unchanged. Color differences corrected for chroma are denoted ΔC*_ab(corr), ΔC*₉₄(corr), and ΔC*₂₀₀₀(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.