We compare two approaches for modelling imperfect information in infinite games by using finite-state automata. The first, more standard approach views information as the result of an observation process driven by a sequential Mealy machine. In contrast, the second approach features indistinguishability relations described by synchronous two-tape automata. The indistinguishability-relation model turns out to be strictly more expressive than the one based on observations. We present a characterisation of the indistinguishability relations that admit a representation as a finite-state observation function. We show that the characterisation is decidable, and give a procedure to construct a corresponding Mealy machine whenever one exists. 2012 ACM Subject Classification Theory of computation → Automata over infinite objects; Theory of computation → Representations of games and their complexity; Computing methodologies → Reasoning about belief and knowledge; Computing methodologies → Planning under uncertainty Keywords and phrases Infinite Games on Finite Graphs, Imperfect Information, Automatic Structures Digital Object Identifier 10.4230/LIPIcs.STACS.2020.48