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I know that colour temperature is an approximation of the black body emission, so a body with a higher temperature emits bluish light while at a lower temperature emits yellowish light. However in photo-editing software, the image becomes bluish when I lower the color temperature. Why is this happening and not the opposite? I guess there is a simple convention, but I can't find it online.

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Because the color temperature that you give in photo editing is the assumed color temperature of the light source used for the original picture. With a low temperature light source the colors are reddish, so to compensate the software adds blue. And vice-versa.

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When you slide the color temperature on your camera (or image editing software) up to 10000K you don't actually change the very cool blue/white light in the real world to make it more orange. You change the way your camera makes the very cool blue/white light that is at 10000K look more orange in the picture. If the light is very blue/white then you need to amplify the inverse color of blue light, which happens to be orange, to make it look like normal light centered around 5200K in the picture. But you're not changing anything with regard to the actual light in the real world, it's still very cool blue white at 10000K.

Likewise, when you slide the color temperature on your camera down to 2500K you don't actually change the light to make it more blue. You change the way your camera makes the very warm orange light that is at 2500K look bluer in the picture. If the light is very orange then you need to amplify the inverse color of orange light, which happens to be blue, to make it look like normal light centered around 5200K in the picture. But you're not changing anything with regard to the actual light, it's still very orange in the real world. Your picture is just making it look bluer in the picture.

Another way to look at it is to think of the color temperature setting on your camera or in your editing program as a filter. If the light is tinted towards orange you need to use a blue filter to make the light look more normal. If the light is very blue then you would need to use an orange filter to remove the blue tint. Since 2500K is very orange, one needs to use a blue filter to compensate for it. Since 10000K is very blue, one needs to use an orange filter to compensate for it. If we used an orange filter under orange light it would make the picture even that much more orange!

The change in color that you see when you move the color temperature slider is due to the change in the color of the filter you are applying using the color temperature setting. It is not due to a change in the color of the light that entered the camera when the image was captured.

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