Once you convert a picture, the pieces do not disappear, and subsequent gameplays will not award you with another piece for the set you converted.
However, if there are other sets within that world, you may collect picture pieces for them if you have a slot open and have not converted the pictures.
Once you complete all 3 sets, you may clear them and start over, but not before collecting and converting all 3.
Why can't I sell pieces after I convert them into a full picture?
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