Defining the SNR or PSNR for color images (3 channel RGB files)
Your PSNR formula has in the denominator a sum of squared distances in a color intensity 1-D space between original and reconstructed pixel values, the sum is taken over all the image pixels. Therefore, you question can be re-formulated as &quot;how one defines a distance in a color 3-D space&quot;.
You did give an implicit answer, one of many possible and still valid, in your question: your elaboration on the definition of color space (&quot;each pixel is associated with a 3-tuple (R,G,B)&quot;) suggests that you consider the RGB color space. The Euclidian metric can be selected and you have the formula for a color distance between two color vectors F1 and F2
$$
Δ_{RGB}(F_1, F_2) = \sqrt{(R_1-R_2)^2+(G_1-G_2)^2+(B_1-B_2)^2}
$$
and it is true, but not the whole truth.
The practical usefulness of the color distance definition is task specific. From one point of view, it depends on what is the purpose you assign to your image analysis instrumentation. Considering the image impairment by a reconstruction noise, you may want to select a color space adapted to human vision, a perception based color space, as a hue-saturation-intensity (HSI) color space. The metric in HSI used for color distance calculations is
$$
Δ_{RGB}(F_1, F_2) = \sqrt{Δ_I^2+Δ_C^2} \\
Δ_I = |I_1-I_2|; Δ_C=\sqrt{S_1^2+S_2^2-2S_1S_2cosθ} \\
θ = (|H_1-H_2|&lt;π)?(|H_1-H_2|):(2π-|H_1-H_2|)
$$
In the other project, you may want to construct an image from measurement data for presentation in pseudo colors. It will require to work with the proprietary color space that reflects transfer functions of your detectors.
Summing up, your project may require a standard color space, and you simply select the best color space for your task, or it is a sophisticated research work, and to define a PSNR formula for your image processing is a work of its own.
About &quot;color spaces&quot;: a search for &quot;distance in color image space&quot; gives as a first entry an article that I readily recommend to you: https://profs.info.uaic.ro/~ancai/DIP/laborator/lab7/color%20spaces%20&amp;%20distances.pdf
In the article, a section 3.4.3 Opponent Color Spaces (page 62, or 26/34 in the pdf) may be what is of special interest for you.