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I am looking into the colour science regarding lookup tables, and their application to images.

I am trying to find out what the values within the LUT actually represent.

LUT_3D_SIZE 33
0.000000 0.000000 0.000000
0.000214 0.000031 0.000046
0.001465 0.000031 0.000092
0.004562 0.000000 0.000031
0.010178 0.000000 0.000000
0.018982 0.000000 0.000000
0.031540 0.000000 0.000000
0.048173 0.000000 0.000000
0.068177 0.000000 0.000000
0.087922 0.000000 0.000000
...
...
...

I know these values are numerical representations of RGB values, post mathematical transform.

For example, to correctly view an an image in Log-C(ArriWideGamut) we need to use a transform LUT ArriLogC->Rec709. A mathematical function is applied to transform the image into a viewable colour space.

What I want to know is how are the numbers in the LUT encoded in relation to the actual pixel values that make up the image? Is there a mathematical standard that pixel values must be converted to in order to apply the transform function and vice versa once the LUT is applied?

  • You might take a look at the 3D LUTs at www.color.org which is the website of the International Color Consortium. While not specific to the cinematography industry, it has a considerable set of published standards some of which detail the way LUTs (both 1D and 3D) are created and used in photography/printing/displays/etc. – doug Aug 2 at 22:14
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The LUT evaluates a given colour transformation function or series of colour transformation function over a given domain to produce the resulting range of colour values.

At the simplest, to generate your ARRI Log-C to BT.709 3D LUT, one would start by producing a cube/table of linearly spaced RGB samples according to the precision required, commonly 33 (or even 65 samples) per cube dimension. The resulting linear cube would maybe have a 4D shape as follows: (33, 33, 33, 3), i.e. 35937 RGB samples. Then the ARRI Log-C to BT.709 function needs to be applied onto the cube RGB samples and the LUT is ready to use. Importantly, the order of the samples is what determines how the cube should be indexed. Now, with a given ARRI Log-C input value, it simply a matter of calculating its index in the cube to find the corresponding BT.709 output value.

Things get a bit more complicated if the ARRI Log-C input value is between multiple indices, which happens systematically, then interpolation is required, 3D LUT are typically interpolated with Trilinear Interpolation or Tetrahedral Interpolation.

Paul Bourke has an easy to grasp Trilinear Interpolation article. If you are Python and Numpy savvy, we have vectorised Trilinear and Tetrahedral Interpolation algorithms in Colour.

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  • Thanks. Could you explain how one might create the linearly spaced RGB samples? Would the Log-C function be applied to each row or index of the linear sample LUT? Colour library looks good, there is much to learn in this realm of digital imaging, apologies for my ignorance – hdcdigi Aug 3 at 17:36
  • Generating the samples can be done with np.meshgrid, here is the code in Colour: github.com/colour-science/colour/blob/develop/colour/io/luts/…, if you look at the docstring example, the Blue channel changes the fastest and Red the slowest. As for applying the function, you would apply it on each RGB vector in the last axis of the cube, you could also imagine it being a large 2D table with a shape as follows: (35937, 3), and yes, indeed you would apply the function on each row! – Kel Solaar Aug 4 at 4:34
  • Thanks, I have sent you an email from your website. – hdcdigi Aug 6 at 10:00
  • Just found it in my spam directory, I will get back to you! – Kel Solaar Aug 6 at 20:07

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