Algorithm for Additive Color Mixing for RGB Values

To blend using alpha channels, you can use these formulas:

r = new Color();
r.A = 1 - (1 - fg.A) * (1 - bg.A);
if (r.A < 1.0e-6) return r; // Fully transparent -- R,G,B not important
r.R = fg.R * fg.A / r.A + bg.R * bg.A * (1 - fg.A) / r.A;
r.G = fg.G * fg.A / r.A + bg.G * bg.A * (1 - fg.A) / r.A;
r.B = fg.B * fg.A / r.A + bg.B * bg.A * (1 - fg.A) / r.A;

fg is the paint color. bg is the background. r is the resulting color. 1.0e-6 is just a really small number, to compensate for rounding errors.

NOTE: All variables used here are in the range [0.0, 1.0]. You have to divide or multiply by 255 if you want to use values in the range [0, 255].

For example, 50% red on top of 50% green:

// background, 50% green
var bg = new Color { R = 0.00, G = 1.00, B = 0.00, A = 0.50 };
// paint, 50% red
var fg = new Color { R = 1.00, G = 0.00, B = 0.00, A = 0.50 };
// The result
var r = new Color();
r.A = 1 - (1 - fg.A) * (1 - bg.A); // 0.75
r.R = fg.R * fg.A / r.A + bg.R * bg.A * (1 - fg.A) / r.A; // 0.67
r.G = fg.G * fg.A / r.A + bg.G * bg.A * (1 - fg.A) / r.A; // 0.33
r.B = fg.B * fg.A / r.A + bg.B * bg.A * (1 - fg.A) / r.A; // 0.00

Resulting color is: (0.67, 0.33, 0.00, 0.75), or 75% brown (or dark orange).

You could also reverse these formulas:

var bg = new Color();
if (1 - fg.A <= 1.0e-6) return null; // No result -- 'fg' is fully opaque
if (r.A - fg.A < -1.0e-6) return null; // No result -- 'fg' can't make the result more transparent
if (r.A - fg.A < 1.0e-6) return bg; // Fully transparent -- R,G,B not important
bg.A = 1 - (1 - r.A) / (1 - fg.A);
bg.R = (r.R * r.A - fg.R * fg.A) / (bg.A * (1 - fg.A));
bg.G = (r.G * r.A - fg.G * fg.A) / (bg.A * (1 - fg.A));
bg.B = (r.B * r.A - fg.B * fg.A) / (bg.A * (1 - fg.A));

or

var fg = new Color();
if (1 - bg.A <= 1.0e-6) return null; // No result -- 'bg' is fully opaque
if (r.A - bg.A < -1.0e-6) return null; // No result -- 'bg' can't make the result more transparent
if (r.A - bg.A < 1.0e-6) return bg; // Fully transparent -- R,G,B not important
fg.A = 1 - (1 - r.A) / (1 - bg.A);
fg.R = (r.R * r.A - bg.R * bg.A * (1 - fg.A)) / fg.A;
fg.G = (r.G * r.A - bg.G * bg.A * (1 - fg.A)) / fg.A;
fg.B = (r.B * r.A - bg.B * bg.A * (1 - fg.A)) / fg.A;

The formulas will calculate that background or paint color would have to be to produce the given resulting color.

If your background is opaque, the result would also be opaque. The foreground color could then take a range of values with different alpha values. For each channel (red, green and blue), you have to check which range of alphas results in valid values (0 - 1).

It depends on what you want, and it can help to see what the results are of different methods.

If you want

Red + Black        = Red
Red + Green        = Yellow
Red + Green + Blue = White
Red + White        = White 
Black + White      = White

then adding with a clamp works (e.g. min(r1 + r2, 255)) This is more like the light model you've referred to.

If you want

Red + Black        = Dark Red
Red + Green        = Dark Yellow
Red + Green + Blue = Dark Gray
Red + White        = Pink
Black + White      = Gray

then you'll need to average the values (e.g. (r1 + r2) / 2) This works better for lightening/darkening colors and creating gradients.

Fun fact: Computer RGB values are derived from the square root of photon flux. So as a general function, your math should take that into account. The general function for this for a given channel is:

blendColorValue(a, b, t)
    return sqrt((1 - t) * a^2 + t * b^2)

Where a and b are the colors to blend, and t is a number from 0-1 representing the point in the blend you want between a and b.

The alpha channel is different; it doesn't represent photon intensity, just the percent of background that should show through; so when blending alpha values, the linear average is enough:

blendAlphaValue(a, b, t)
    return (1-t)*a + t*b;

So, to handle blending two colors, using those two functions, the following pseudocode should do you good:

blendColors(c1, c2, t)
    ret
    [r, g, b].each n ->
        ret[n] = blendColorValue(c1[n], c2[n], t)
    ret.alpha = blendAlphaValue(c1.alpha, c2.alpha, t)
    return ret

Incidentally, I long for a programming language and keyboard that both permits representing math that (or more) cleanly (the combining overline unicode character doesn't work for superscripts, symbols, and a vast array of other characters) and interpreting it correctly. sqrt((1-t)*pow(a, 2) + t * pow(b, 2)) just doesn't read as clean.

Few points:

I think you want to use min instead of max

I think you want to use 255 instead of 256

This will give:

(r1, g1, b1) + (r2, g2, b2) = (min(r1+r2, 255), min(g1+g2, 255), min(b1+b2, 255))

However, The "natural" way of mixing colors is to use the average, and then you don't need the min:

(r1, g1, b1) + (r2, g2, b2) = ((r1+r2)/2, (g1+g2)/2, (b1+b2)/2)

Javascript function to blend rgba colors

c1,c2 and result - JSON's like c1={r:0.5,g:1,b:0,a:0.33}

    var rgbaSum = function(c1, c2){
       var a = c1.a + c2.a*(1-c1.a);
       return {
         r: (c1.r * c1.a  + c2.r * c2.a * (1 - c1.a)) / a,
         g: (c1.g * c1.a  + c2.g * c2.a * (1 - c1.a)) / a,
         b: (c1.b * c1.a  + c2.b * c2.a * (1 - c1.a)) / a,
         a: a
       }
     } 

PYTHON COLOUR MIXING THROUGH ADDITION IN CMYK SPACE

One possible way to do this is to first convert the colours to CMYK format, add them there and then reconvert to RGB.

Here is an example code in Python:

rgb_scale = 255
cmyk_scale = 100


def rgb_to_cmyk(self,r,g,b):
    if (r == 0) and (g == 0) and (b == 0):
        # black
        return 0, 0, 0, cmyk_scale

    # rgb [0,255] -> cmy [0,1]
    c = 1 - r / float(rgb_scale)
    m = 1 - g / float(rgb_scale)
    y = 1 - b / float(rgb_scale)

    # extract out k [0,1]
    min_cmy = min(c, m, y)
    c = (c - min_cmy) 
    m = (m - min_cmy) 
    y = (y - min_cmy) 
    k = min_cmy

    # rescale to the range [0,cmyk_scale]
    return c*cmyk_scale, m*cmyk_scale, y*cmyk_scale, k*cmyk_scale

def cmyk_to_rgb(self,c,m,y,k):
    """
    """
    r = rgb_scale*(1.0-(c+k)/float(cmyk_scale))
    g = rgb_scale*(1.0-(m+k)/float(cmyk_scale))
    b = rgb_scale*(1.0-(y+k)/float(cmyk_scale))
    return r,g,b

def ink_add_for_rgb(self,list_of_colours):
    """input: list of rgb, opacity (r,g,b,o) colours to be added, o acts as weights.
    output (r,g,b)
    """
    C = 0
    M = 0
    Y = 0
    K = 0

    for (r,g,b,o) in list_of_colours:
        c,m,y,k = rgb_to_cmyk(r, g, b)
        C+= o*c
        M+=o*m
        Y+=o*y 
        K+=o*k 

    return cmyk_to_rgb(C, M, Y, K)

The result to your question would then be (assuming a half-half mixture of your two colours:

r_mix, g_mix, b_mix = ink_add_for_rgb([(r1,g1,b1,0.5),(r2,g2,b2,0.5)])

where the 0.5's are there to say that we mix 50% of the first colour with 50% of the second colour.

Yes, it is as simple as that. Another option is to find the average (for creating gradients).

It really just depends on the effect you want to achieve.

However, when Alpha gets added, it gets complicated. There are a number of different methods to blend using an alpha.

An example of simple alpha blending: http://en.wikipedia.org/wiki/Alpha_compositing#Alpha_blending

Find here the mixing methods suggested by Fordi and Markus Jarderot in one python function that gradually mixes or blends between two colors A and B.

The "mix" mode is useful to interpolate between two colors. The "blend" mode (with t=0) is useful to compute the resulting color if one translucent color is painted on top of another (possibly translucent) color. gamma correction leads to nicer results because it takes into consideration the fact that physical light intensity and perceived brightness (by humans) are related non-linearly.

import numpy as np

def mix_colors_rgba(color_a, color_b, mode="mix", t=None, gamma=2.2):
    """
    Mix two colors color_a and color_b.

    Arguments:
        color_a:    Real-valued 4-tuple. Foreground color in "blend" mode.
        color_b:    Real-valued 4-tuple. Background color in "blend" mode.
        mode:       "mix":   Interpolate between two colors.
                    "blend": Blend two translucent colors.
        t:          Mixing threshold.
        gamma:      Parameter to control the gamma correction.

    Returns: 
        rgba:       A 4-tuple with the result color.

    To reproduce Markus Jarderot's solution:
            mix_colors_rgba(a, b, mode="blend", t=0, gamma=1.)
    To reproduce Fordi's solution:
            mix_colors_rgba(a, b, mode="mix", t=t, gamma=2.)
    To compute the RGB color of a translucent color on white background:
            mix_colors_rgba(a, [1,1,1,1], mode="blend", t=0, gamma=None)
    """
    assert(mode in ("mix", "blend"))
    assert(gamma is None or gamma>0)
    t = t if t is not None else (0.5 if mode=="mix" else 0.)
    t = max(0,min(t,1))
    color_a = np.asarray(color_a)
    color_b = np.asarray(color_b)
    if mode=="mix" and gamma in (1., None):
        r, g, b, a = (1-t)*color_a + t*color_b
    elif mode=="mix" and gamma > 0:
        r,g,b,_ = np.power((1-t)*color_a**gamma + t*color_b**gamma, 1/gamma)
        a = (1-t)*color_a[-1] + t*color_b[-1]
    elif mode=="blend":
        alpha_a = color_a[-1]*(1-t)
        a = 1 - (1-alpha_a) * (1-color_b[-1])
        s = color_b[-1]*(1-alpha_a)/a
        if gamma in (1., None):
            r, g, b, _ = (1-s)*color_a + s*color_b
        elif gamma > 0:
            r, g, b, _ = np.power((1-s)*color_a**gamma + s*color_b**gamma,
                                  1/gamma)

    return tuple(np.clip([r,g,b,a], 0, 1))

See below how this can be used. In "mix" mode the left and right colors match exactly color_a and color_b. In "blend" mode, the left color at t=0 is the color that results if color_a is blended over color_b (and a white background). In the example, color_a then is made increasingly translucent until one arrives at color_b.

Note that blending and mixing are equivalent if the alpha values are 1.0.

https://i.stack.imgur.com/4Y2fQ.png

For completeness, here the code to reproduce the above plot.

import matplotlib.pyplot as plt
import matplotlib as mpl

def plot(pal, ax, title):
    n = len(pal)
    ax.imshow(np.tile(np.arange(n), [int(n*0.20),1]),
              cmap=mpl.colors.ListedColormap(list(pal)),
              interpolation="nearest", aspect="auto")
    ax.set_xticks([])
    ax.set_yticks([])
    ax.set_xticklabels([])
    ax.set_yticklabels([])
    ax.set_title(title)

_, (ax1, ax2, ax3, ax4) = plt.subplots(nrows=4,ncols=1)

n = 101
ts = np.linspace(0,1,n)
color_a = [1.0,0.0,0.0,0.7] # transparent red
color_b = [0.0,0.0,1.0,0.8] # transparent blue

plot([mix_colors_rgba(color_a, color_b, t=t, mode="mix", gamma=None)
      for t in ts], ax=ax1, title="Linear mixing")
plot([mix_colors_rgba(color_a, color_b, t=t, mode="mix", gamma=2.2)
      for t in ts], ax=ax2, title="Non-linear mixing (gamma=2.2)")
plot([mix_colors_rgba(color_a, color_b, t=t, mode="blend", gamma=None)
      for t in ts], ax=ax3, title="Linear blending")
plot([mix_colors_rgba(color_a, color_b, t=t, mode="blend", gamma=2.2)
      for t in ts], ax=ax4, title="Non-linear blending (gamma=2.2)")
plt.tight_layout()
plt.show()

Formulas:
    Linear mixing (gamma=1):
                r,g,b,a:    (1-t)*x + t*y
    Non-linear mixing (gama≠1):
                r,g,b:      pow((1-t)*x**gamma + t*y**gamma, 1/gamma)
                a:          (1-t)*x + t*y
    Blending (gamma=1):
                a:          1-(1-(1-t)*x)*(1-y)
                s:          alpha_b*(1-alpha_a)*a
                r,g,b:      (1-s)*x + s*y
    Blending (gamma≠1):
                a:          1-(1-(1-t)*x)*(1-y)
                s:          alpha_b*(1-alpha_a)/a
                r,g,b:      pow((1-s)*x**gamma + s*y**gamma, 1/gamma)

And finally, here a useful read about gamma correction.

When I came here I didn't find the "additive color mixing" algorithm I was actually looking for, which is also available in Photoshop and is described as "Screen" on Wikipedia. (Aka "brighten" or "invert multiply".) It produces a result similar to two light sources being combined.

With Screen blend mode the values of the pixels in the two layers are inverted, multiplied, and then inverted again. This yields the opposite effect to multiply. The result is a brighter picture.

Here it is:

// (rgb values are 0-255)
function screen(color1, color2) {
    var r = Math.round((1 - (1 - color1.R / 255) * (1 - color2.R / 255)) * 255);
    var g = Math.round((1 - (1 - color1.G / 255) * (1 - color2.G / 255)) * 255);
    var b = Math.round((1 - (1 - color1.B / 255) * (1 - color2.B / 255)) * 255);
    return new Color(r, g, b);
}

Have written/used something like @Markus Jarderot's sRGB blending answer (which is not gamma corrected since that is the default legacy) using C++

//same as Markus Jarderot's answer
float red, green, blue;
alpha = (1.0 - (1.0 - back.alpha)*(1.0 - front.alpha));
red   = (front.red   * front.alpha / alpha + back.red   * back.alpha * (1.0 - front.alpha));
green = (front.green * front.alpha / alpha + back.green * back.alpha * (1.0 - front.alpha));
blue  = (front.blue  * front.alpha / alpha + back.blue  * back.alpha * (1.0 - front.alpha));

//faster but equal output
alpha = (1.0 - (1.0 - back.alpha)*(1.0 - front.alpha));
red   = (back.red   * (1.0 - front.alpha) + front.red   * front.alpha);
green = (back.green * (1.0 - front.alpha) + front.green * front.alpha);
blue  = (back.blue  * (1.0 - front.alpha) + front.blue  * front.alpha);

//even faster but only works when all values are in range 0 to 255
int red, green, blue;
alpha = (255 - (255 - back.alpha)*(255 - front.alpha));
red   = (back.red   * (255 - front.alpha) + front.red   * front.alpha) / 255;
green = (back.green * (255 - front.alpha) + front.green * front.alpha) / 255;
blue  = (back.blue  * (255 - front.alpha) + front.blue  * front.alpha) / 255;

more info: what-every-coder-should-know-about-gamma

I was working on a similar problem and ended up here, but had to write my own implementation in the end. I wanted to basically "overlay" the new foreground color over the existing background color. (And without using an arbitrary midpoint like t. I believe my implementation is still "additive.") This also seems to blend very cleanly in all of my test-cases.

Here, new_argb just converts the int into a struct with 4 unsigned char so I can reduce the amount of bit-shifts.

int blend_argb(int foreground, int background)
{
    t_argb fg;
    t_argb bg;
    t_argb blend;
    double ratio;

    fg = new_argb(foreground);
    bg = new_argb(background);

    // If background is transparent,
    // use foreground color as-is and vice versa.
    if (bg.a == 255)
        return (foreground);
    if (fg.a == 255)
        return (background);

    // If the background is fully opaque,
    // ignore the foreground alpha. (Or the color will be darker.)
    // Otherwise alpha is additive.
    blend.a = ((bg.a == 0) ? 0 : (bg.a + fg.a));

    // When foreground alpha == 0, totally covers background color.
    ratio = fg.a / 255.0;
    blend.r = (fg.r * (1 - ratio)) + (bg.r * ratio);
    blend.g = (fg.g * (1 - ratio)) + (bg.g * ratio);
    blend.b = (fg.b * (1 - ratio)) + (bg.b * ratio);

    return (blend.a << 24 | blend.r << 16 | blend.g << 8 | blend.b);
}

For context, in my environment I'm writing color ints into a 1D pixel array, which is initialized with 0-bytes and increasing the alpha will make the pixel tend towards black. (0 0 0 0 would be opaque black and 255 255 255 255 would be transparent white... aka black.)

Here's a highly optimized, standalone c++ class, public domain, with floating point and two differently optimized 8-bit blending mechanisms in both function and macro formats, as well as a technical discussion of both the problem at hand and how to, and the importance of, optimization of this issue:

https://github.com/fyngyrz/colorblending

Thank you Markus Jarderot, Andras Zoltan and hkurabko; here is the Python code for blending a list of RGB images.

Using Markus Jarderot's code we can generate RGBA color, then i use Andras Zoltan and hkurabko's method to trans RGBA to RGB.

Thank you!

import numpy as np
def Blend2Color(C1,C2):
    c1,c1a=C1
    c2,c2a=C2
    A = 1 - (1 - c1a) * (1 - c2a);
    if (A < 1.0e-6): 
        return (0,0,0) #Fully transparent -- R,G,B not important
    Result=(np.array(c1)*c1a+np.array(c2)*c2a*(1-c1a))/A
    return Result,A
def RGBA2RGB(RGBA,BackGround=(1,1,1)):# whilt background
    A=RGBA[-1]
    RGB=np.add(np.multiply(np.array(RGBA[:-1]),A),
               np.multiply(np.array(BackGround),1-A))
    return RGB

def BlendRGBList(Clist,AlphaList=None,NFloat=2,ReturnRGB=True,
                 RGB_BackGround=(1,1,1)):
    N=len(Clist)
    if AlphaList==None:
        ClistUse=Clist.copy()
    else:
        if len(AlphaList)==N:
            AlphaListUse=np.multiply(AlphaList,10**NFloat).astype(int)
            ClistUse=np.repeat(np.array(Clist), AlphaListUse, axis=0)
        else:
            raise('len of AlphaList must equal to len of Clist!')
    while N!=1:
        temp=ClistUse.copy()
        ClistUse=[]
        for C in temp[:-1]:
            c1,a1=C
            c2,a2=temp[-1]
            ClistUse.append(Blend2Color(C1=(c1,a1*(1-1/N)),C2=(c2,a2*1/N)))
        N=len(ClistUse)
    Result=np.append(ClistUse[0][0],ClistUse[0][1])
    if ReturnRGB:
        Result=RGBA2RGB(Result,BackGround=RGB_BackGround)

    return Result

Test

BlendRGBList([[(1,0,0),1],[(0,1,0),1]],ReturnRGB=True)
#array([0.75, 0.5 , 0.25])
BlendRGBList([[(1,0,0),1],[(0,1,0),1]],ReturnRGB=False)
#array([0.66666667, 0.33333333, 0.        , 0.75      ])