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the pixels in the images are usually viewed as continuous values.

consider an image in RGB color system, each pixel can take on a value from set {(RGB),...} which consists 256*256*256 = 16777216 elements. which is countable.

in this case, is it reasonable to consider the pixels as discrete-value?

in this case, does different perspective lead to different algorithm and different output?

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  • $\begingroup$ yes, this is perfectly valid $\endgroup$
    – shimao
    Commented Nov 10, 2019 at 5:41

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The 8-bits coding for each color level in each pixel is just a convention, you may also use 16-bits scheme, specially if you deal with professional or amateur photography or photo editing. Colours span on a continuous scale, the 256 levels of usual color coding ia just a matter of definition, like when you have some continuous measure rounded to some number of digits.

Anyway, since traditional statistic methods (which put more attention on continuous/discrete variables) are rarely used on this kind of data, almost always there will be no difference in methods deployed. Also notice that even in linear models, predictors are taken as fixed and no importance is given to their domain, so it makes no difference if they are discrete or continuous.

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  • $\begingroup$ Thanks for your answer. Would you please give a concrete example to this statement "in linear models, predictors are taken as fixed". For example, what does "predictors are taken as fixed" mean, in the case of a linear regression model on house price prediction problem? $\endgroup$
    – JJJohn
    Commented Nov 10, 2019 at 12:11
  • $\begingroup$ Predictors X may be designed (as in experiments) or observations of a random variable (as in observational studies - houses, for instance, aren't being built with some designed characteristics just to see how much they will prove worth, they are sampled and then their characteristics are measured). Linear models are used without differences in both cases, but in any case, all the results are conditioned on the matrix X. Hence, even if X is random, it is still considered fixed, in the context of model estimation. $\endgroup$
    – carlo
    Commented Nov 10, 2019 at 14:17
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Following your track of thought, there are no continuous variables, because every value can only be measured up to a certain accuracy (8bit or 16bit color channels in your case) which results in a finite set of possible values.

I would, however, take a different viewpoint: how probable is it that the same value is measured twice? If it is so low that in practice only few values occur more than once, a description by means of a density or, as its poorer approximation, a histogram is more appropriate. That is why image matching based on colors uses histogram cells representing several pixel values in RGB space (or better: HSV space, because it better corresponds to perception).

Concerning color spaces, there is a more important caveat: the H value in HSV space is circular, which requires application of the methods of circular statistics. See e.g.

Jammalamadaka, SenGupta: "Topics in Circular Statstics." World Scientific, 2001

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