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In order to understand the capabilities of measuring morphologies from<jats:italic>Euclid<\/jats:italic>-detected galaxies and to help implement measurements in the pipeline of the Organisational Unit MER of the Euclid Science Ground Segment, we have conducted the Euclid Morphology Challenge, which we present in two papers. While the companion paper focusses on the analysis of photometry, this paper assesses the accuracy of the parametric galaxy morphology measurements in imaging predicted from within the Euclid Wide Survey. We evaluate the performance of five state-of-the-art surface-brightness-fitting codes,<jats:monospace>DeepLeGATo<\/jats:monospace>,<jats:monospace>Galapagos-2<\/jats:monospace>,<jats:monospace>Morfometryka<\/jats:monospace>,<jats:monospace>ProFit<\/jats:monospace>and<jats:monospace>SourceXtractor++<\/jats:monospace>, on a sample of about 1.5 million simulated galaxies (350\u2006000 above 5<jats:italic>\u03c3<\/jats:italic>) resembling reduced observations with the<jats:italic>Euclid<\/jats:italic>VIS and NIR instruments. The simulations include analytic S\u00e9rsic profiles with one and two components, as well as more realistic galaxies generated with neural networks. We find that, despite some code-specific differences, all methods tend to achieve reliable structural measurements (&lt; 10% scatter on ideal S\u00e9rsic simulations) down to an apparent magnitude of about<jats:italic>I<\/jats:italic><jats:sub>E<\/jats:sub>\u2004=\u200423 in one component and<jats:italic>I<\/jats:italic><jats:sub>E<\/jats:sub>\u2004=\u200421 in two components, which correspond to a signal-to-noise ratio of approximately 1 and 5, respectively. We also show that when tested on non-analytic profiles, the results are typically degraded by a factor of 3, driven by systematics. We conclude that the official<jats:italic>Euclid<\/jats:italic>Data Releases will deliver robust structural parameters for at least 400 million galaxies in the Euclid Wide Survey by the end of the mission. We find that a key factor for explaining the different behaviour of the codes at the faint end is the set of adopted priors for the various structural parameters.<\/jats:p>","DOI":"10.1051\/0004-6361\/202245042","type":"journal-article","created":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T08:57:39Z","timestamp":1668589059000},"page":"A102","source":"Crossref","is-referenced-by-count":23,"title":["<i>Euclid<\/i>preparation"],"prefix":"10.1051","volume":"671","author":[{"name":"Euclid Collaboration","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9935-9109","authenticated-orcid":false,"given":"H.","family":"Bretonni\u00e8re","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0035-5202","authenticated-orcid":false,"given":"U.","family":"Kuchner","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1416-8483","authenticated-orcid":false,"given":"M.","family":"Huertas-Company","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6870-8900","authenticated-orcid":false,"given":"E.","family":"Merlin","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9875-8263","authenticated-orcid":false,"given":"M.","family":"Castellano","sequence":"additional","affiliation":[]},{"given":"D.","family":"Tuccillo","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2861-9812","authenticated-orcid":false,"given":"F.","family":"Buitrago","sequence":"additional","affiliation":[]},{"given":"C. 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