X06 Delta E used equations

CIE76 equation

$$\LARGE \Delta E_{76} = \sqrt{\left ( \Delta L \right )^{2} + \left ( \Delta a \right )^{2} + \left ( \Delta b \right )^{2}}$$

$$\text{where}$$

$$\Delta L = L_{2} - L_{1}$$

$$\Delta a = a_{2} - a_{1}$$

$$\Delta b = b_{2} - b_{1}$$

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CIE94 equation

$$\LARGE \Delta E_{94} = \sqrt{\left (\frac{\Delta L}{K_{L} \times S_{L}}\right )^{2} + \left (\frac{\Delta C}{K_{C} \times S_{C}}\right )^{2} + \left (\frac{\Delta H}{K_{H} \times S_{H}}\right )^{2}}$$

$$\text{where}$$

$$K_{L} = \begin{cases} 1 & \textup{default} \\ 2 & \textup{textiles} \end{cases}$$

$$K_{C} = 1$$

$$K_{H} = 1$$

$$K_{1} = \begin{cases} .045 & \textup{graphic arts} \\ .048 & \textup{textiles} \end{cases}$$

$$K_{2} = \begin{cases} .015 & \textup{graphic arts} \\ .014 & \textup{textiles} \end{cases}$$

$$C_{1} = \sqrt{{a_{1}}^{2} + {b_{1}}^{2}}$$

$$C_{2} = \sqrt{{a_{2}}^{2} + {b_{2}}^{2}}$$

$$\Delta a = a_{1} - a_{2}$$

$$\Delta b = b_{1} - b_{2}$$

$$\Delta C_{ab} = C_{1} - C_{2}$$

$$\Delta H = \sqrt{ \Delta a^{2} + \Delta b^{2} - \Delta {C_{ab}}^{2} }$$

$$S_{L} = 1$$

$$S_{C} = 1 + \left (K_{1} \times C_{1}\right )$$

$$S_{h} = 1 + \left (K_{2} \times C_{1}\right )$$

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CIEDE2000 equation

$$\LARGE \Delta E_{00} = \sqrt{{\left (\frac{\Delta {L}'}{k_{L} \times S_{L}}\right )}^{2} + {\left (\frac{\Delta {C}'}{k_{C} \times S_{C}}\right )}^{2} + {\left (\frac{\Delta {H}'}{k_{H} \times S_{H}}\right )}^{2} + R_{T} \times \left (\frac{\Delta {C}'}{k_{C} \times S_{C}}\right )\left (\frac{\Delta {H}'}{k_{H} \times S_{H}}\right )}$$

$$\text{where}$$

$$\forall i \in (1,2)$$

$$C^{\ast }_{i,ab} = \sqrt{{a^{\ast}_i}^2 + {b^{\ast}_i}^2}$$

$$\bar{C}^{\ast}_{ab} = \frac{C^{\ast }_{1,ab} + C^{\ast }_{2,ab}}{2}$$ $$G = .5 \times \left ( 1 - \sqrt{\frac{{C^{\ast }_{ab}}^{7}}{{C^{\ast }_{ab}}^{7} + 25^7}} \right )$$ $${a}'_{i} = (1 + G) \times a^{\ast }_{i}$$ $${C}'_{i} = \sqrt{{{a}'_{i}}^2 + {{b}'_{i}}^2}$$ $${h}'_{i} = \begin{cases} 0 & \text{ if } b^{\ast}_{i} = {a}'_{i} = 0 \\ \tan^{-1}(b^{\ast}_{i}, {a}'_{i}) & \text{ otherwise } \end{cases}$$

$$\Delta {L}' = L^{\ast}_2 - L^{\ast}_1$$

$$\Delta {C}' = {C}'_2 - {C}'_1$$

$$\Delta {h}' = \begin{cases} 0 & \text{ if } {C}'_{1}{C}'_{2} = 0 \\ {h}'_{2} - {h}'_{1} & \text{ if } {C}'_{1}{C}'_{2}\neq0; \left | {h}'_{2} - {h}'_{1} \right | \leq 180^{\circ} \\ \left ( {h}'_{2} - {h}'_{1} \right ) - 360 & \text{ if } {C}'_{1}{C}'_{2}\neq0; \left ( {h}'_{2} - {h}'_{1} \right ) > 180^{\circ} \\ \left ( {h}'_{2} - {h}'_{1} \right ) + 360 & \text{ if } {C}'_{1}{C}'_{2}\neq0; \left ( {h}'_{2} - {h}'_{1} \right )< -180^{\circ} \end{cases}$$ $$\Delta {H}' = 2 \times \sqrt{{C}'_{1}{C}'_{2}} \times \sin{\left ( \frac{\Delta {h}'}{2} \right )}$$ $${\bar{L}}' = \frac{\left( L^{\ast }_{1} + L^{\ast }_{2} \right )}{2}$$ $${\bar{C}}' = \frac{\left( {C}'_{1} + {C}'_{2} \right )}{2}$$

$${\bar{h}}' = \begin{cases} \frac{{h}'_1 + {h}'_2}{2} & \text{ if } \left | {h}'_1 - {h}'_2 \right | \leq 180^{\circ}; {C}'_1 {C}'_2 \neq 0 \\ \frac{{h}'_1 + {h}'_2}{2} + 360^{\circ} & \text{ if } \left | {h}'_1 - {h}'_2 \right | > 180^{\circ}; \left ({h}'_1 + {h}'_2 \right ) < 360^{\circ}; {C}'_1 {C}'_2 \neq 0 \\ \frac{{h}'_1 + {h}'_2}{2} - 360^{\circ} & \text{ if } \left | {h}'_1 - {h}'_2 \right | > 180^{\circ}; \left ({h}'_1 + {h}'_2 \right ) \geq 360^{\circ}; {C}'_1 {C}'_2 \neq 0 \\ {h}'_1 + {h}'_2 & \text{ if } {C}'_1 {C}'_2 \neq 0 \end{cases}$$

$$T = 1 - .17 \times \cos{\left ({\bar{h}}' - 30^{\circ} \right )} + .24 \times \cos{\left (2{\bar{h}}' \right )} + .32 \times \cos{\left (3{\bar{h}}' + 6^{\circ} \right )} - .20 \times \cos{\left (4{\bar{h}}' - 63^{\circ} \right )}$$

$$\Delta \theta = 30 \times \exp \left ( - {\left [ \frac{{\bar{h}' - 275^{\circ}}}{25} \right ] }^2 \right )$$

$$R_{C} = 2 \times \sqrt{ \frac{ ({\bar{C}}')^7 }{ ({\bar{C}}')^7 + 25^7 } }$$

$$S_{C} = 1 + .045 \times {\bar{C}}'$$

$$S_{H} = 1 + .015 \times {\bar{C}}' \times T$$

$$R_{T} = -1 \times \sin(2\Delta \theta) \times R_{C}$$

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CMC l:c equation

$$\LARGE \Delta E_{CMC} = \sqrt{ \left ( \frac{\Delta L}{l S_{L}} \right ) ^ 2 + \left ( \frac{\Delta C}{c S_{C}} \right ) ^ 2 + \left ( \frac{\Delta H}{S_{H}} \right ) ^ 2 }$$

$$\text{where}$$

$$l = \begin{cases} 2 & \text{ acceptability } \\ 1 & \text{ perceptibility } \end{cases}$$

$$c = 1$$

$$\Delta C = C_{1} - C_{2}$$

$$C_{1} = \sqrt{{a_{1}}^2 + {b_{1}}^2}$$

$$C_{2} = \sqrt{{a_{2}}^2 + {b_{2}}^2}$$

$$\Delta H = \sqrt{ \Delta a ^ 2 + \Delta b ^ 2 - \Delta C ^ 2}$$

$$\Delta L = L_{1} - L_{2}$$

$$\Delta a = a_{1} - a_{2}$$

$$\Delta b = b_{1} - b_{2}$$

$$S_{L} = \begin{cases} .511 & \text{ if } L_{1} < 16 \\ \frac{.040975 \times L_{1}}{ 1 + .01765 \times L_{1}} & \text{ if } L_{1} \geq 16 \end{cases}$$

$$S_{C} = \frac{.0638 \times C_{1}}{1 + .0131 \times C_{1}} + .638$$

$$S_{H} = S_{C} \times (FT + 1 - F)$$

$$T = \begin{cases} .56 + \left | .2 \times \cos(H_{1} + 168^{\circ})\right | & \text{ if } 164^{\circ} \leq H_{1} \leq 345^{\circ} \\ .36 + \left | .4 \times \cos(H_{1} + 35^{\circ})\right | & \text{ otherwise } \end{cases}$$

$$F = \sqrt{ \frac{ C_{1}^4 }{ C_{1}^4 + 1900 } }$$

$$H = \arctan \left ( \frac{ b_{1} }{ a_{1} } \right )$$

$$H_{1} = \begin{cases} H | & \text{ if } 164^{\circ} \leq H \geq 0 \\ H + 360^{\circ} | & \text{ otherwise } \end{cases}$$