Determine font color based on background color

Given a system (a website for instance) that lets a user customize the background color for some section but not the font color (to keep number of options to a minimum), is there a way to programmatically determine if a "light" or "dark" font color is necessary?

I'm sure there is some algorithm, but I don't know enough about colors, luminosity, etc to figure it out on my own.

I encountered similar problem. I had to find a good method of selecting contrastive font color to display text labels on colorscales/heatmaps. It had to be universal method and generated color had to be "good looking", which means that simple generating complementary color was not good solution - sometimes it generated strange, very intensive colors that were hard to watch and read.

After long hours of testing and trying to solve this problem, I found out that the best solution is to select white font for "dark" colors, and black font for "bright" colors.

Here's an example of function I am using in C#:

Color ContrastColor(Color color)
{
    int d = 0;
    
    // Counting the perceptive luminance - human eye favors green color...      
    double luminance = (0.299 * color.R + 0.587 * color.G + 0.114 * color.B)/255;
    
    if (luminance > 0.5)
       d = 0; // bright colors - black font
    else
       d = 255; // dark colors - white font
                
    return  Color.FromArgb(d, d, d);
}

This was tested for many various colorscales (rainbow, grayscale, heat, ice, and many others) and is the only "universal" method I found out.

Edit
Changed the formula of counting a to "perceptive luminance" - it really looks better! Already implemented it in my software, looks great.

Edit 2 @WebSeed provided a great working example of this algorithm: http://codepen.io/WebSeed/full/pvgqEq/

Based on Gacek's answer but directly returning color constants (additional modifications see below):

public Color ContrastColor(Color iColor)
{
    // Calculate the perceptive luminance (aka luma) - human eye favors green color... 
    double luma = ((0.299 * iColor.R) + (0.587 * iColor.G) + (0.114 * iColor.B)) / 255;

    // Return black for bright colors, white for dark colors
    return luma > 0.5 ? Color.Black : Color.White;
}

Note: I removed the inversion of the luma value to make bright colors have a higher value, what seems more natural to me and is also the 'default' calculation method.
(Edit: This has since been adopted in the original answer, too)

I used the same constants as Gacek from here since they worked great for me.

You can also implement this as an Extension Method using the following signature:

public static Color ContrastColor(this Color iColor)

You can then easily call it via
foregroundColor = backgroundColor.ContrastColor().

Thank you @Gacek. Here's a version for Android:

@ColorInt
public static int getContrastColor(@ColorInt int color) {
    // Counting the perceptive luminance - human eye favors green color...
    double a = 1 - (0.299 * Color.red(color) + 0.587 * Color.green(color) + 0.114 * Color.blue(color)) / 255;

    int d;
    if (a < 0.5) {
        d = 0; // bright colors - black font
    } else {
        d = 255; // dark colors - white font
    }

    return Color.rgb(d, d, d);
}

And an improved (shorter) version:

@ColorInt
public static int getContrastColor(@ColorInt int color) {
    // Counting the perceptive luminance - human eye favors green color...
    double a = 1 - (0.299 * Color.red(color) + 0.587 * Color.green(color) + 0.114 * Color.blue(color)) / 255;
    return a < 0.5 ? Color.BLACK : Color.WHITE;
}

My Swift implementation of Gacek's answer:

func contrastColor(color: UIColor) -> UIColor {
    var d = CGFloat(0)

    var r = CGFloat(0)
    var g = CGFloat(0)
    var b = CGFloat(0)
    var a = CGFloat(0)

    color.getRed(&r, green: &g, blue: &b, alpha: &a)

    // Counting the perceptive luminance - human eye favors green color...
    let luminance = 1 - ((0.299 * r) + (0.587 * g) + (0.114 * b))

    if luminance < 0.5 {
        d = CGFloat(0) // bright colors - black font
    } else {
        d = CGFloat(1) // dark colors - white font
    }

    return UIColor( red: d, green: d, blue: d, alpha: a)
}

Javascript [ES2015]

const hexToLuma = (colour) => {
    const hex   = colour.replace(/#/, '');
    const r     = parseInt(hex.substr(0, 2), 16);
    const g     = parseInt(hex.substr(2, 2), 16);
    const b     = parseInt(hex.substr(4, 2), 16);

    return [
        0.299 * r,
        0.587 * g,
        0.114 * b
    ].reduce((a, b) => a + b) / 255;
};