I was asked to explain what a look up table is.
It's pretty much a series of corrections. As an analogy let's say we have an old set of weighing scales that is constantly out 2 oz so you weigh something that's 4 oz it says 6 oz you weigh something that is 8 oz and it says 10 oz. You could throw out the scales and buy new or just subtract the 2 oz from the answer it gives. Similar to how you might weigh yourself and deduct 5 pounds for clothes.
Let's take a look at the blue filter on a sensor: it mostly lets blue frequency light through and counts the photons (actually the charge on the site). Unfortunately some green and red frequency photons also get through. So maybe 100 photons got counted but 84 were blue and 12 red and 4 green.
So your count says 100 for blue but in reality it was 84. The more photons the brighter the shade so blue gets measured as being a brighter blue than it actually is.
The other problem is the cri of the light. ideally the frequency of light would cover the whole visible spectrum but it doesn't have to be and often isn't. If we were to illuminate our subject with a red laser beam there should be no blue or green recorded but the filters are not perfect and some blue and green does get recorded.
With analogue color film there would be a particular response from a film so your kodak X would look different from your fuji Y but on the whole if your lens wasn't tinted it made little difference what make of camera you used; the nikon guy got the same colors as the canon guy and the pentax guy. It's pretty common these days to emulate a film's look on a digital camera. These days the 'film' is controlled by the camera manufacturer, the sensor they are using and the profiles they apply to the raw data.
We are kind of limited to the best guess that the manufacturers can make for the light we are shooting in.
To do better than that we have calibration charts with tiles of particular colors. Let's say one is an R G B value of 10, 15, 20 and we photograph it and it records as 8, 13, 23. To get the correct value for that tile
in the lut there would be -2 -2 +3.
It's a simplification of course but essentially what a program like 3d lut creator does is create a number of values one for each color tile to create correction curves so each color is represented close to its correct value. We can't provide swatches for every color but the fit will be closer than without it.
The 3D part of a lut is that each color has a red, blue, green component and each needs to be adjusted to get an actual corrected color.
A lut combines contrast, hue, brightness, saturation all in one each pixel value is compared to the lut to see how it should be adjusted.
When you take a photo of your reference in the circumstances you are shooting in and create a lut then those other photos in that session should be able to be corrected with that LUT.
It's kind of a more sophisticated white balance. If you use the white balance dropper to balance a grey in one photo you can apply that white balance adjustment to the next shots you take. With the lut you tune the color to a greater extent. If you have tried to find a good grey in a photo, pretty often different greys will tint your photo green or magenta to different degrees. The best you can do is find a compromise and that may not be completely successful. The LUT will adjust a dark grey differently to a light grey and the more sample points it has in the table the closer the adjustment will be to something close to correct.
OK so that's trying to get to something close to natural but LUTs can be decidedly unnatural too. That's color grading and you might color grade to imitate a particular film stock to give a particular contrast curve. Luts can also Pin some values - usually skin tones so they are stable - but you can do things like change someone's jeans from blue to black while leaving most other colors unaffected. You can also mask areas so that some areas are left as is and others are adjusted by the Lut.
I think there is only so much I can explain with words. Search on you-tube for 3d lut creator and you should find examples of using LUTs to change the look of photos and video, e.g
https://www.youtube.com/watch?v=b68ZqVwIt88 To use this methodology to best effect you need to be consistent, which means keeping the exposure and white balance fixed to gain optimal results. In raw you can choose a white balance after the fact; with jpegs a curve is applied in camera and you are attempting to fit another curve on top which never comes out very good. Film makers will try to get flat or log footage to minimise the curves applied in camera so they can set their own in post processing. Getting a good exposure is important since if that needs adjustment you are going to get distortion and results will be hit and miss. With video you can often use false color to make the majority of your scene green. Different light will change the exposure to achieve that green but the exposure will be consistent, which will make the LUT consistent too.