International Telecommunication Union，ITU, recommends several standards to construct luminance from colored channel such as ITU-R BT.601, ITU-R BT.601 with Gamma Correction 2.2, ITU/EBU 3213, and ITU-R BT.709 etc.

ITU-R BT.601

In RECOMMENDATION ITU-R BT.601-6 section 2.5.2, the ITU recommends the construction of luminance and colour-difference signals as follows:
L = 0.299 * R + 0.587 * G + 0.114 * B.

Adobe Photoshop uses this standard and actually using the following algorithm:
L = 0.3 * R + 0.59 * G + o.11 * B

   function grayscale_bt601(red,green,blue){
      var l = Math.round(0.299 * red + 0.587 * green + 0.114 * blue);
      return {red:l,green:l,blue:l};
   }
   

ITU-R BT.601 Gamma Correction 2.2

L = (0.299 * R^2.2 + 0.587 * G^2.2 + 0.114 * B^2.2)^1/2.2

   function grayscale_bt601_gamma(red,green,blue){
      var l = Math.round(Math.pow(0.299 * Math.pow(red, 2.2) +
                                  0.587 * Math.pow(green, 2.2) +
                                  0.114 * Math.pow(blue, 2.2), 1 / 2.2));
      return {red:l,green:l,blue:l};
   }
   

ITU-R BT.709

Another recommendation came from ITU-R BT.709 standard. It defines the derivation of luminance signals using algorithm:
L = 0.2126 * R + 0.7152 * G + 0.0722 * B
The original document can be downloaded from Here: BT.709 : Parameter values for the HDTV standards for production and international programme exchange

   function grayscale_bt709(red,green,blue){
      var l = Math.round(0.213 * red + 0.715 * green + 0.072 * blue);
      return {red:l,green:l,blue:l};
   }
   

ITU/EBU 3213

ITU/EBU is defined as standard for European PAL television systems, it recommends the algorithm as:
L = 0.222 * R + 0.707 * G + 0.071 * B

   function grayscale_itu3213(red,green,blue){
      var l = Math.round(0.222 * red + 0.707 * green + 0.071 * blue);
      return {red:l,green:l,blue:l};
   }
   

You can download the source code here:
Source Code (.zip)

Except using single channel from colored image as the grayscale luminance value,applying arithmetical algorithm to red, green and blue channel are also seen in image processing program. The most common methods used are max, min, average of the red, green and blue channel values as grayed value.

Using maximum of red, green, blue

   function grayscale_max(red,green,blue){
      var l = Math.max(red,green,blue);
      return {red:l,green:l,blue:l};
   }
   

Using minimum of red, green, blue

   function grayscale_min(red,green,blue){
      var l = Math.min(red,green,blue);
      return {red:l,green:l,blue:l};
   }
   

Using average of red, green, blue

   function grayscale_avg(red,green,blue){
      var l = Math.avg(red + green + blue);
      return {red:l,green:l,blue:l};
   }
   

Using average of maximum of red, green, blue and minimum of red, green, blue

   function grayscale_avg_min_max(red,green,blue){
      var l = Math.round((Math.max(red, green, blue) +
                          Math.min(red, green, blue)) / 2);
      return {red:l,green:l,blue:l};
   }
   

You can download the source code here:
Source Code (.zip)

A grayscale digital image is an image that carries only intensity information. A common stratege to convert a colored image to grayscale image is to match the luminance of the grayscale image to the luminance of the color image. Many algorithms have been developed to accomplish this. In this post, you will see different methods that has different effects when converting an colored the image to grayscale image. However, using which methods is really up to your selection.

Using single channel. Colored image contains 4 channels: red, green, blue and alpha.
This method simply use the value from one of the r, g, b channel of the color image to be the value of the grayscale image. This method is used in Adobe Photoshop color channel and iPaintPro properties window.

iPaintPro Channel Property using Single Channel Grayscale method

Using red channel

   function grayscale_red(red,green,blue){
      return {red:red,green:red,blue:red};
   }
   

Using green channel

   function grayscale_blue(red,green,blue){
      return {red:green,green:green,blue:green};
   }
   

Using blue channel

   function grayscale_blue(red,green,blue){
      return {red:blue,green:blue,blue:blue};
   }
   

You can download the source code here:
Source Code (.zip)