I noticed that if you perform a supervised classification with a mask, the masked pixels are called 'masked pixels' and assigned a class value that is number_of_classes+1. So these pixels (what I think of as background) should be separate from pixels that remain unclassified during a classification due to the thresholds used or the training data. In this case, you do want to keep the 'background' or masked pixels separate from the 'unclassified' pixels which is a valid class that should be considered.
So it depends on if you have actual 'unclassified' pixels. If unclassified, then yes, include them. If 'background' or masked, then I would not include them and they should not be combined with actual unclassified pixels within the image.
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