We explore the possibility of remote, from 1 AU, study of the heliopause by an observer outside the geocorona. We argue that the heliopause, a boundary that separates the solar wind and the galactic plasma of the local interstellar medium (LISM), can be imaged by detecting solar extreme ultraviolet (EUV) radiation reflected by interstellar ions. Such EUV imaging would map the heliopause and provide important insight into its three-dimensional structure and the LISM parameters as well. We consider heliopause mapping in the oxygen O+ ion resonance line (83.4 nm); imaging in the helium He+ ion line (30.4 nm) will be considered in a future article. We show that the expected heliopause brightness map at 83.4 nm is essentially different from that of the foreground glow of the solar wind O+ pickup ions. The interstellar plasma glow is brighter in the upwind (with respect to the interstellar wind) direction, while the pickup ion glow dominates in the downwind direction. The spectral characteristics of the radiation scattered by the LISM plasma and by the pickup ions are significantly different. The all-sky images at 83.4 nm are highly sensitive to the ionization state of the LISM and would allow one to probe the asymmetry of the interstellar magnetic field. We briefly discuss the experimental requirements to heliopause EUV mapping, which would require 3 orders of magnitude improvement in instrumentation sensitivity. This is a challenging but not impossible task. ¿ 2000 American Geophysical Union