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Line intensity mapping (LIM) is a promising tool to efficiently probe the three-dimensional large-scale structure by mapping the aggregate emission of a spectral line from all sources that trace the matter density field. Spectral lines from different redshifts can fall in the same observed frequency and be confused, however, which is a major challenge in LIM. In this work, we develop a line de-confusion technique in map space capable of reconstructing the three-dimensional spatial distribution of line-emitting sources. If multiple spectral lines of a source population are observable in multiple frequencies, using the sparse approximation, our technique iteratively extracts sources along a given line of sight by fitting the LIM data to a set of spectral templates. We demonstrate that the technique successfully extracts sources with emission lines present at a few $σ$ above the noise level, taking into account uncertainties in the source modeling and presence of continuum foreground contamination and noise fluctuations. As an example, we consider a TIME/CONCERTO-like survey targeting [C II] at the epoch of reionization, and reliably reconstruct the 3D spatial distribution of the CO interlopers at $0.5\lesssim z\lesssim 1.5$. We also demonstrate a successful de-confusion for the SPHEREx mission in the near-infrared wavelengths. Potentially, the reconstructed maps can be further cross-correlated with a (galaxy) tracer population to estimate the total interloper power in the linear clustering regime. This technique is a general framework to extract the phase-space distribution of low-redshift interlopers, without the need of external information, for any line de-confusion problem.