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The trigger system of the ATLAS experiment at the LHC aims at a high selectivity in order to keep the full physics potential while reducing the 40 MHz initial event rate imposed by the LHC bunch crossing down to ∼100 Hz, as required by the data acquisition system. Algorithms working in the final stage of the trigger environment (Event Filter) are implemented to run both in a "wrapped" mode (reconstructing tracks in the entire Muon Spectrometer) and in a "seeded" mode (according to a dedicated strategy that performs pattern recognition only in regions of the detector where trigger hypotheses have been produced at earlier stages). The working principles of the offline muon reconstruction and identification algorithms (MOORE and MuId) implemented and used in the framework of the Event Filter are discussed in this paper. The reconstruction performance of these algorithms is presented for both modes in terms of efficiency, momentum resolution, rejection power and execution times on several samples of simulated single muon events, also taking into account the high background environment expected for ATLAS.