How do you prevent the red LEDs to look pink-ish?
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I know this isn't the best photo equipment and more expensive gear probably provides better results, but is there a way to workaround it with cheaper cameras?
It is a problem for even the most expensive cameras and comes down to one critical factor: dynamic range.
The human eye is amazingly adaptable in dealing with bright and dark areas within a scene. Photographic equipment far less so. A typical example I can cite is watching a tennis match as the shadows move across the court as the sun goes down. When play goes between areas in sun and areas in shade, the TV image doesn't cope with the difference very well and following the ball is a challenge - but if you are sitting in the stands, you don't really notice. The difference is the primary image sensor - a piece of silicon in a camera or the human eye.
This difference shows up in photos of LEDs where the centre spot of the LED is very high in brightness compared to the surrounding areas and this can push the numeric values extracted from the silicon quite high, heading towards maximums (which, in the extreme, would represent the brightest white) in a scene where the rest of the image would be properly exposed.
There are cameras that have a dual scanning of the sensor - once with high sensitivity and once with lower sensitivity, with electronics to mix the signals in a way to obtain the best total image. I looked at a Panasonic camera for a project some 20 years ago which had this feature - but I don't know if even it would have been able to render a scene with an LED in it.
Look for cameras that boast High Dynamic Range. My Samsung Galaxy S5 phone has a HDR setting on the camera. With it off, a red LED shows up as a pinkish white spot with a red halo. With it on, the LED shows up as orange with a red halo.
It's been a long time since I was seriously assessing cameras - so I don't know if there is anything on the market that would do this properly ... but I have my doubts. Seems a bit of post production airbrushing is still the best way to achieve a satisfactory result.