Skip to main content

RGB and HSI

Red, Green, Blue ⇄ Hue, Saturation, Intensity

RGB to HSI​

I=R+G+B3S={1−min(R,G,B)I if I>00 otherwise H={atan2(32(G−B),12(2R−G−B)) if I>00 otherwise \begin{align*} I &= \frac{R + G + B}{3} \\ \\ S &= \begin{cases} 1 - \frac{min(R, G, B)}{I} & \text{ if } I > 0 \\ 0 & \text{ otherwise } \\ \end{cases} \\ \\ H &= \begin{cases} \mathrm{atan2}(\frac{\sqrt{3}}{2}(G - B), \frac{1}{2}(2R - G - B)) & \text{ if } I > 0 \\ 0 & \text{ otherwise } \end{cases} \end{align*}

atan2\mathrm{atan2} determines the counterclockwise angle in radians between the positive X axis and the point (x,y). The output range is [−π,π][-\pi,\pi].

atan2(y,x)={atan(yx) if x>0atan(yx)+π if x<0 and y≥0atan(yx)−π if x<0 and y<0π2 if x=0 and y>0−π2 if x=0 and y<0undefined if x=0 and y=0\mathrm{atan2}(y,x) = \begin{cases} \mathrm{atan}(\frac{y}{x}) & \text{ if } x > 0 \\ \mathrm{atan}(\frac{y}{x}) + \pi & \text{ if } x < 0 \text{ and } y \geq 0 \\ \mathrm{atan}(\frac{y}{x}) - \pi & \text{ if } x < 0 \text{ and } y < 0 \\ \frac{\pi}{2} & \text{ if } x = 0 \text{ and } y > 0 \\ -\frac{\pi}{2} & \text{ if } x = 0 \text{ and } y < 0 \\ \text{undefined} & \text{ if } x = 0 \text{ and } y = 0 \end{cases}

HSI to RGB​

f(a)=cos⁡((H−a)π180)cos⁡(60−(H−a))π180R={I⋅(1+S⋅f(0)) if 0≤H<120I⋅(1−S) if 120≤H<2403I−G−B if 240≤H<360G={3I−R−B if 0≤H<120I⋅(1+S⋅f(120)) if 120≤H<240I⋅(1−S) if 240≤H<360B={I⋅(1−S) if 0≤H<1203I−R−G if 120≤H<240I⋅(1+S⋅f(240)) if 240≤H<360\begin{align*} f(a) &= \frac{\cos((H-a)\frac{\pi}{180})}{\cos(60 - (H - a))\frac{\pi}{180}} \\ \\ R &= \begin{cases} I \cdot (1 + S \cdot f(0)) & \text{ if } 0 \leq H < 120 \\ I \cdot (1 - S) & \text{ if } 120 \leq H < 240 \\ 3I - G - B & \text{ if } 240 \leq H < 360 \end{cases} \\ \\ G &= \begin{cases} 3I - R - B & \text{ if } 0 \leq H < 120 \\ I \cdot (1 + S \cdot f(120)) & \text{ if } 120 \leq H < 240 \\ I \cdot (1 - S) & \text{ if } 240 \leq H < 360 \end{cases} \\ \\ B &= \begin{cases} I \cdot (1 - S) & \text{ if } 0 \leq H < 120 \\ 3I - R - G & \text{ if } 120 \leq H < 240 \\ I \cdot (1 + S \cdot f(240)) & \text{ if } 240 \leq H < 360 \end{cases} \\ \\ \end{align*}