{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T18:59:33Z","timestamp":1790362773147,"version":"4.1.0"},"reference-count":29,"publisher":"Optica Publishing Group","issue":"20","license":[{"start":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T00:00:00Z","timestamp":1790294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/doi.org\/10.1364\/OA_License_v2#VOR-OA"},{"start":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T00:00:00Z","timestamp":1790294400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/opg.optica.org\/policies\/opg-tdm-policy.json"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62305272"],"award-info":[{"award-number":["62305272"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"name":"National Natural Science Foundation of China","award":["12504148"],"award-info":[{"award-number":["12504148"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Natural Science Basic Research Program of Shaanxi","award":["2023-JC-QN-0722"],"award-info":[{"award-number":["2023-JC-QN-0722"]}]},{"name":"Chinese Universities Scientific Fund","award":["2452026138"],"award-info":[{"award-number":["2452026138"]}]}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"short-container-title":["Opt. Express"],"published-print":{"date-parts":[[2026,10,5]]},"abstract":"<jats:p>Recent advances in deep learning have introduced new phase-retrieval paradigms for transport-of-intensity-equation (TIE)-based quantitative phase imaging (QPI). However, most existing approaches focus on image-to-image mapping or physics-constrained reconstruction, making it difficult to directly characterize the TIE's continuous inverse operator in function spaces. In contrast, operator-learning methods tailored to this problem remain underexplored, and existing neural operators may still be limited by discretization-grid dependence and spectral aliasing during cross-resolution inference. To address these challenges, we propose a TIE neural operator, termed TIENO, which learns the inverse TIE operator from the in-focus intensity and axial intensity derivative to the phase distribution by preserving continuous\u2013discrete equivalence in band-limited function spaces. TIENO incorporates band-limited spectral regulation and frequency-domain channel attention into an encoder\u2013decoder architecture, thereby suppressing unresolved high-frequency components and enhancing the representation of phase-relevant features. Numerical evaluations on simulated datasets demonstrate that TIENO achieves high-accuracy phase recovery and maintains stable inference performance at unseen spatial resolutions, while experimental validation on biological cell data further confirms its applicability to practical QPI and shows that it outperforms representative learning-based models and conventional TIE solvers. These results highlight the potential of TIENO for accurate and resolution-flexible TIE-based quantitative phase imaging.<\/jats:p>","DOI":"10.1364\/oe.612314","type":"journal-article","created":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T06:00:08Z","timestamp":1789711208000},"page":"37576","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["TIENO: a neural operator for transport-of-intensity-equation-based quantitative phase imaging"],"prefix":"10.1364","volume":"34","author":[{"given":"Yutong","family":"Zhang","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0051rme32","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinjie","family":"Zhao","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0051rme32","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hui","family":"Chen","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0051rme32","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chunxi","family":"Hao","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0051rme32","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rui","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0051rme32","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4156-4016","authenticated-orcid":true,"given":"Wenyi","family":"Ren","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0051rme32","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ying","family":"Li","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0051rme32","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"285","published-online":{"date-parts":[[2026,9,25]]},"reference":[{"key":"oe-34-20-37576-R1","doi-asserted-by":"publisher","first-page":"578","DOI":"10.1038\/s41566-018-0253-x","type":"journal-article","volume":"12","author":"Park","year":"2018","journal-title":"Nat. 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