Converting RGB to RGBW

Question

I know this has already been asked, but the anwser given there doesn't work. I've spent over an hour looking for a formula or algorithm, but have found nothing. As a result, I've started writing my own algorithm to convert RGB to RGBW in the most efficient way possible. This is what I've currently got:

        //'Ri', 'Gi', and 'Bi' correspond to the Red, Green, and Blue inputs.
        var M = Math.Max(Ri, Math.Max(Gi, Bi)); //The maximum value between R,G, and B.
        int Wo =0; //White output
        int Ro=0; //Red output
        int Go=0; //Green output
        int Bo=0; //Blue output

        int av = 0; //Average between the two minimum values
        int hR = 0; //Red with 100% hue
        int hG = 0; //Green with 100% hue
        int hB = 0; //Blue with 100% hue

        //These 4 lines serve to figure out what the input color is with 100% hue.
        float multiplier = 255.0f / M;
        hR = Convert.ToInt32(Ri * multiplier);
        hG = Convert.ToInt32(Gi * multiplier);
        hB = Convert.ToInt32(Bi * multiplier);

        //Depending on the maximum value, get an average of the least used colors, weighted for their importance in the overall hue.
        //This is the problematic part
        if (M == Ri)
           av = (Bi*hB + Gi*hG) / (hB+hG);
        else if (M == Gi)
            av = (Ri*hR + Bi*hB) / (hR+hB);
        else if (M == Bi)
            av = (Gi*hG + Ri*hR) / (hG+hR);

        //Set the rgbw colors
        Wo = av;
        Bo = Bi - av;
        Ro = Ri - av;
        Go = Gi - av;
        if (Wo < 1) Wo = 0;
        if (Bo < 1) Bo = 0;
        if (Ro < 1) Ro = 0;
        if (Go < 1) Go = 0;
        if (Wo > 255) Wo = 255;
        if (Bo > 255) Bo = 255;
        if (Ro > 255) Ro = 255;
        if (Go > 255) Go = 255;

It works fine if the color I'm dealing with is a primary color, but not in any other case. What would make it work everywhere? Am I even on the right track?

EDIT: Here's a .gif of the issue I'm running into. the RGBW values are all the way at the bottom

Answer 1

I've finally figured out how to convert RGB to RGBW, turns out my previous method was completely wrong:

//Get the maximum between R, G, and B
float tM = Math.Max(Ri, Math.Max(Gi, Bi));

//If the maximum value is 0, immediately return pure black.
if(tM == 0)
   { return new rgbwcolor() { r = 0, g = 0, b = 0, w = 0 }; }

//This section serves to figure out what the color with 100% hue is
float multiplier = 255.0f / tM;
float hR = Ri * multiplier;
float hG = Gi * multiplier;
float hB = Bi * multiplier;  

//This calculates the Whiteness (not strictly speaking Luminance) of the color
float M = Math.Max(hR, Math.Max(hG, hB));
float m = Math.Min(hR, Math.Min(hG, hB));
float Luminance = ((M + m) / 2.0f - 127.5f) * (255.0f/127.5f) / multiplier;

//Calculate the output values
int Wo = Convert.ToInt32(Luminance);
int Bo = Convert.ToInt32(Bi - Luminance);
int Ro = Convert.ToInt32(Ri - Luminance);
int Go = Convert.ToInt32(Gi - Luminance);

//Trim them so that they are all between 0 and 255
if (Wo < 0) Wo = 0;
if (Bo < 0) Bo = 0;
if (Ro < 0) Ro = 0;
if (Go < 0) Go = 0;
if (Wo > 255) Wo = 255;
if (Bo > 255) Bo = 255;
if (Ro > 255) Ro = 255;
if (Go > 255) Go = 255;
return new rgbwcolor() { r = Ro, g = Go, b = Bo, w = Wo };

Any optimization ideas are more than welcome :)

Answer 2

I made an algorithm that takes into account the color temperature of your "white" LED (as this can vary based on the strip - mine was warm-white, which is 4500k). Gist is here, code is below, and blog post with more context is Ouch! My Eyes! Arduino WS2812B Setup & RGBW Transformation (Cabinet Light Pt. 3).

// Reference, currently set to 4500k white light:
// https://andi-siess.de/rgb-to-color-temperature/
const uint8_t kWhiteRedChannel = 255;
const uint8_t kWhiteGreenChannel = 219;
const uint8_t kWhiteBlueChannel = 186;

// The transformation has to be normalized to 255
static_assert(kWhiteRedChannel >= 255 ||
              kWhiteGreenChannel >= 255 ||
              kWhiteBlueChannel >= 255);

CRGBW GetRgbwFromRgb2(CRGB rgb) {
  uint8_t r = rgb.r;
  uint8_t g = rgb.g;
  uint8_t b = rgb.b;

  // These values are what the 'white' value would need to
  // be to get the corresponding color value.
  double whiteValueForRed = r * 255.0 / kWhiteRedChannel;
  double whiteValueForGreen = g * 255.0 / kWhiteGreenChannel;
  double whiteValueForBlue = b * 255.0 / kWhiteBlueChannel;

  // Set the white value to the highest it can be for the given color
  // (without over saturating any channel - thus the minimum of them).
  double minWhiteValue = min(whiteValueForRed,
                             min(whiteValueForGreen,
                                 whiteValueForBlue));
  uint8_t Wo = (minWhiteValue <= 255 ? (uint8_t) minWhiteValue : 255);

  // The rest of the channels will just be the original value minus the
  // contribution by the white channel.
  uint8_t Ro = (uint8_t)(r - minWhiteValue * kWhiteRedChannel / 255);
  uint8_t Go = (uint8_t)(g - minWhiteValue * kWhiteGreenChannel / 255);
  uint8_t Bo = (uint8_t)(b - minWhiteValue * kWhiteBlueChannel / 255);

  return CRGBW(Ro, Go, Bo, Wo);
}