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"Multiply." We throw that term around quite a bit without always understanding it. What exactly is being multiplied?

To understand digital image multiplication, we must first understand that it is only useful when images are represented not as 8-bit 0 to 255 values, but as "floating point" 0.0 to 1.0 fractional values — akin to percentages, if you prefer.

Suppose we have an image that is too pink. That is to say, the image is too red and too blue, but its green channel is just fine. To reduce the red and blue channels, we need to multiply them by a percentage which is less than 100%. For example, to pull 20% from the red and blue channels of an image, one must multiply it by the color (0.8, 1.0, 0.8). Since multiplication by a color is analogous to looking through a filter of that color, we need to look at our overly-magenta image through (as any photographer will have guessed) a "pale green" filter.

Image multiplication makes sense when pixel values range from 0 (black) to 1 (white)

The multiplication has no useful meaning when the color values are expressed in 0 to 255 space, but in 0 to 1 space it all makes sense. Since anything times zero is zero, black (0.0) times anything is black. Since anything times 1 leaves that thing unchanged, white (1.0) times anything has no effect. A point five gray (0.5) times anything will make that thing half as bright. Pale red (1.0, 0.9, 0.9) times anything makes that thing just a bit redder looking. Traditional photographers call this process the subtractive color model which is unfortunate, because no subtraction occurs.

If subtraction did occur, then the physical world would have "negative colors" which, of course, it does not. A better, more clearly understandable term would be "multiplicative color model," or "filtration color model," but the notion of color "subtraction" is deeply ingrained and won't die anytime soon.

Muliplication provides a good way to color line drawings. Here you can really see the "black times anything is black, white times anything is that thing unchanged" rule in action. The sepia regions of the line drawing end up simply tinting the corresponding pixels of the color image with sepia.

Since multiplication is commutative, the order of the layering does not matter.

also provides a good way of "shading" artwork. You can use it to introduce a sense of diffuse lighting into your paintings or 3D CG objects. Notice in the example below how the warm and cool grays not only darken the final 3D rendering, but also influence its color temperature.

Understanding of multiplication can lead to a better understanding of color mixing in pigments. Like photographers, artists also believe there exist something called subtractive color mixing. That's unfortunate. Colors are colors, and light is light, and mixing light can only lead to the production of lighter colors.

So-called subtractive color mixing is really a process akin to filtering. When an artists mixes one color with another color, he is really layering one colored filter over another one. Every filter produces less light on its output side than it receives on its input side. How much less light? It's best to think in terms of percentages.

The end result is still a mixing of light reflected by various pigments, it's just that the pigments restrict the wavelengths of the light which will ultimately be added together.

We can see when we express color mixing in terms of simple image arithmetic that when two pigments are mixed (or when two color filters are sandwiched together) or when two digital colors are multiplied, the resulting color is that color which is present in some measure in both source colors. Pure yellow mixed with pure blue would actually yield black, not green, because pure blue filters only allow pure blue to pass, and pure yellow filters only allow pure yellow light to pass, and so there is no green in the final result because neither filter allows green to pass through.

A greenish blue filter allows some green to pass through. So does a greenish yellow filter. That's why Cerulean blue mixed with Hansa yellow yields a green color. If you'd like to learn more about color mixing and pigments, I recommend you buy and follow the exercises in the book Blue and Yellow Don't Make Green by Michael Wilcox.

** Remember in the example above that 35 * 255 = 35 only when the math is done in zero-to-one space — the number 35 is really 35 out of 255 which is really 0.137, and 255 is really 255 out of 255 which is really 1.0 — so 35 * 255 = 35 only because 0.137 * 1.0 = 0.137 — When Adobe expresses colors in 8-bit 0 to 255 space, they foster a poor understanding of many aspects of image arithmetic.