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The method is derived from a second-order asymptotic expansion in\u00a0the propagation distance\n                    <jats:italic>L<\/jats:italic>\n                    and complemented by the leading Wentzel\u2013Kramers\u2013Brillouin (WKB) wave-optics correction, yielding an efficient iterative correction scheme preconditioned by fast Fourier transform (FFT)-diagonal, energy-dependent inverse operators (Paganin-type filters). To ensure robustness across practical experimental regimes, we combine two complementary solvers: (i) a local\n                    <jats:italic>O<\/jats:italic>\n                    (\n                    <jats:italic>L<\/jats:italic>\n                    <jats:sup>2<\/jats:sup>\n                    ) closure that is accurate when eikonal shifts remain sub-pixel, and (ii) a non-local formulation for multi-pixel shifts, in which intensity is propagated through an explicit eikonal ray mapping using a mass-conserving bilinear redistribution on the detector grid, and detector residuals are transferred back to the object grid by the corresponding discrete adjoint (transpose) operator, implemented as bilinear interpolation, before applying an approximate FFT-diagonal preconditioner to accelerate convergence. The same framework supports polychromatic data through a compact spectral discretization, allowing energy-dependent transport and inversion while keeping the iteration efficient for graphics processing unit (GPU)\/FFT. Overall, this unified approach enables accurate and computationally efficient phase retrieval across propagation conditions relevant to high-throughput propagation-based phase-contrast micro-tomography experiments.\n                  <\/jats:p>","DOI":"10.1107\/s1600577526004273","type":"journal-article","created":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T16:41:16Z","timestamp":1778776876000},"page":"1169-1179","update-policy":"https:\/\/doi.org\/10.1107\/cm_01","source":"Crossref","is-referenced-by-count":0,"title":["Fast eikonal phase retrieval for high-throughput beamlines"],"prefix":"10.1107","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6291-6765","authenticated-orcid":true,"given":"Alessandro","family":"Mirone","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02550n020","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0352-8180","authenticated-orcid":true,"given":"Theresa","family":"Urban","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02550n020","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8424-9510","authenticated-orcid":true,"given":"Joseph","family":"Brunet","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02550n020","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3769-3392","authenticated-orcid":true,"given":"Claire L.","family":"Walsh","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02jx3x895","id-type":"ROR","asserted-by":"publisher"}],"department":["Department of Mechanical Engineering"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3898-8881","authenticated-orcid":true,"given":"Peter D.","family":"Lee","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02jx3x895","id-type":"ROR","asserted-by":"publisher"}],"department":["Department of Mechanical Engineering"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5962-1683","authenticated-orcid":true,"given":"Paul","family":"Tafforeau","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02550n020","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"329","published-online":{"date-parts":[[2026,6,19]]},"reference":[{"key":"gy5091_bb1","unstructured":"Admans, L. 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