{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:24:47Z","timestamp":1757618687732,"version":"3.44.0"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T00:00:00Z","timestamp":1752105600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T00:00:00Z","timestamp":1752105600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000780","name":"European Commission","doi-asserted-by":"publisher","award":["RRF-2.3.1-21-2022-00004","RRF-2.3.1-21-2022-00004"],"award-info":[{"award-number":["RRF-2.3.1-21-2022-00004","RRF-2.3.1-21-2022-00004"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011019","name":"Nemzeti Kutat\u00e1si Fejleszt\u00e9si \u00e9s Innov\u00e1ci\u00f3s Hivatal","doi-asserted-by":"publisher","award":["TKP2021-NKTA-01","TKP2021-NKTA-01"],"award-info":[{"award-number":["TKP2021-NKTA-01","TKP2021-NKTA-01"]}],"id":[{"id":"10.13039\/501100011019","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009934","name":"E\u00f6tv\u00f6s Lor\u00e1nd University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100009934","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>The paper proposes a statistical learning approach to the problem of estimating missing pixels of images, crucial for image inpainting and super-resolution problems. One of the main novelties of the method is that it also provides uncertainty quantifications together with the estimated values. Our core assumption is that the underlying data-generating function comes from a reproducing kernel Hilbert space (RKHS). A special emphasis is put on band-limited functions, central to signal processing, which form Paley\u2013Wiener type RKHSs. The proposed method, which we call simultaneously guaranteed kernel interpolation (SGKI), is an extension and refinement of a recently developed kernel method. An advantage of SGKI is that it not only estimates the missing pixels, but also builds non-asymptotic confidence bands for the unobserved values, which are simultaneously guaranteed for all missing pixels. We also show how to compute these bands efficiently using Schur complements, we discuss a generalization to vector-valued functions, and we present a series of numerical experiments on various datasets containing synthetically generated and benchmark images, as well.<\/jats:p>","DOI":"10.1007\/s10994-025-06814-0","type":"journal-article","created":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T20:56:57Z","timestamp":1752181017000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Single image inpainting and super-resolution with simultaneous uncertainty guarantees by universal reproducing kernels"],"prefix":"10.1007","volume":"114","author":[{"given":"B\u00e1lint","family":"Horv\u00e1th","sequence":"first","affiliation":[]},{"given":"Bal\u00e1zs Csan\u00e1d","family":"Cs\u00e1ji","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,10]]},"reference":[{"key":"6814_CR1","doi-asserted-by":"publisher","DOI":"10.1002\/0471745790","volume-title":"Image processing: Principles and applications","author":"T Acharya","year":"2005","unstructured":"Acharya, T., & Ray, A. K. (2005). Image processing: Principles and applications. Wiley."},{"issue":"120","key":"6814_CR2","first-page":"1","volume":"17","author":"H Avron","year":"2016","unstructured":"Avron, H., Sindhwani, V., Yang, J., & Mahoney, M. W. (2016). Quasi-Monte Carlo feature maps for shift-invariant kernels. Journal of Machine Learning Research, 17(120), 1\u201338.","journal-title":"Journal of Machine Learning Research"},{"key":"6814_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9096-9","volume-title":"Reproducing kernel Hilbert spaces in probability and statistics","author":"A Berlinet","year":"2004","unstructured":"Berlinet, A., & Thomas-Agnan, C. (2004). Reproducing kernel Hilbert spaces in probability and statistics. Springer."},{"key":"6814_CR4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex optimization","author":"SP Boyd","year":"2004","unstructured":"Boyd, S. P., & Vandenberghe, L. (2004). Convex optimization. Cambridge University Press."},{"issue":"9","key":"6814_CR5","doi-asserted-by":"publisher","first-page":"1213","DOI":"10.1109\/TIP.2004.832922","volume":"13","author":"PL Combettes","year":"2004","unstructured":"Combettes, P. L., & Pesquet, J. C. (2004). Image restoration subject to a total variation constraint. IEEE Transactions on Image Processing, 13(9), 1213\u20131222.","journal-title":"IEEE Transactions on Image Processing"},{"key":"6814_CR6","doi-asserted-by":"publisher","first-page":"3355","DOI":"10.1109\/LCSYS.2022.3185143","volume":"6","author":"BC Cs\u00e1ji","year":"2022","unstructured":"Cs\u00e1ji, B. C., & Horv\u00e1th, B. (2022). Nonparametric, nonasymptotic confidence bands with Paley\u2013Wiener kernels for band-limited functions. IEEE Control Systems Letters, 6, 3355\u20133360.","journal-title":"IEEE Control Systems Letters"},{"key":"6814_CR7","doi-asserted-by":"crossref","unstructured":"Cs\u00e1ji, B. C., & Horv\u00e1th, B. (2023). Improving kernel-based nonasymptotic simultaneous confidence bands. In 23rd IFAC world congress, Yokohama, Japan 23rd IFAC World Congress, Yokohama, Japan (pp. 10357\u201310362). Elsevier.","DOI":"10.1016\/j.ifacol.2023.10.1047"},{"issue":"1","key":"6814_CR8","first-page":"3950312","volume":"2018","author":"SB Damelin","year":"2018","unstructured":"Damelin, S. B., & Hoang, N. (2018). On surface completion and image inpainting by biharmonic functions: Numerical aspects. International Journal of Mathematics and Mathematical Sciences, 2018(1), 3950312.","journal-title":"International Journal of Mathematics and Mathematical Sciences"},{"key":"6814_CR9","doi-asserted-by":"publisher","first-page":"2007","DOI":"10.1007\/s11063-019-10163-0","volume":"51","author":"O Elharrouss","year":"2020","unstructured":"Elharrouss, O., Almaadeed, N., Al-Maadeed, S., & Akbari, Y. (2020). Image inpainting: A review. Neural Processing Letters, 51, 2007\u20132028.","journal-title":"Neural Processing Letters"},{"issue":"10","key":"6814_CR10","first-page":"70","volume":"4","author":"S Fadnavis","year":"2014","unstructured":"Fadnavis, S. (2014). Image interpolation techniques in digital image processing: An overview. International Journal of Engineering Research and Applications, 4(10), 70\u201373.","journal-title":"International Journal of Engineering Research and Applications"},{"key":"6814_CR11","unstructured":"Horv\u00e1th, B., & Cs\u00e1ji, B. Cs. (2023). Nonparametric simultaneous confidence bands: The case of known input distributions. In 23rd European young statisticians meeting (EYSM), Ljubljana, Slovenia. 23rd European Young Statisticians Meeting (EYSM), Ljubljana, Slovenia."},{"key":"6814_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.media.2024.103223","volume":"97","author":"L Huang","year":"2024","unstructured":"Huang, L., Ruan, S., Xing, Y., & Feng, M. (2024). A review of uncertainty quantification in medical image analysis: Probabilistic and non-probabilistic methods. Medical Image Analysis, 97, Article 103223.","journal-title":"Medical Image Analysis"},{"issue":"6","key":"6814_CR13","doi-asserted-by":"publisher","first-page":"1153","DOI":"10.1109\/TASSP.1981.1163711","volume":"29","author":"R Keys","year":"1981","unstructured":"Keys, R. (1981). Cubic convolution interpolation for digital image processing. IEEE Transactions on Acoustics, Speech, and Signal Processing, 29(6), 1153\u20131160.","journal-title":"IEEE Transactions on Acoustics, Speech, and Signal Processing"},{"key":"6814_CR14","doi-asserted-by":"crossref","unstructured":"Ledig, C., Theis, L., Husz\u00e1r, F., Caballero, J., Cunningham, A., Acosta, A., & Shi, W. (2017). Photo-realistic single image super-resolution using a generative adversarial network. In Proceedings of the IEEE conference on computer vision and pattern recognition proceedings of the IEEE conference on computer vision and pattern recognition (pp. 4681\u20134690).","DOI":"10.1109\/CVPR.2017.19"},{"key":"6814_CR15","doi-asserted-by":"publisher","first-page":"1049","DOI":"10.1109\/42.816070","volume":"18","author":"TM Lehmann","year":"1999","unstructured":"Lehmann, T. M., Gonner, C., & Spitzer, K. (1999). Survey: Interpolation methods in medical image processing. IEEE Transactions on Medical Imaging, 18, 1049\u20131075.","journal-title":"IEEE Transactions on Medical Imaging"},{"key":"6814_CR16","doi-asserted-by":"crossref","unstructured":"Li, J., Fang, F., Mei, K., spsampsps Zhang, G. (2018). Multi-scale residual network for image super-resolution. In Proceedings of the European conference on computer vision (ECCV) Proceedings of the European conference on computer vision (ECCV) (pp. 517\u2013532).","DOI":"10.1007\/978-3-030-01237-3_32"},{"key":"6814_CR17","doi-asserted-by":"crossref","unstructured":"Li, Z., Yang, J., Liu, Z., Yang, X., Jeon, G., & Wu, W. (2019). Feedback network for image super-resolution. In Proceedings of the IEEE\/CVF conference on computer vision and pattern recognition proceedings of the IEEE\/CVF conference on computer vision and pattern recognition (pp. 3867\u20133876).","DOI":"10.1109\/CVPR.2019.00399"},{"key":"6814_CR18","doi-asserted-by":"crossref","unstructured":"Lim, B., Son, S., Kim, H., Nah, S., & Lee, K. M. (2017). Enhanced deep residual networks for single image super-resolution. In The IEEE conference on computer vision and pattern recognition (CVPR) workshops. The IEEE conference on computer vision and pattern recognition (CVPR) workshops.","DOI":"10.1109\/CVPRW.2017.151"},{"key":"6814_CR19","doi-asserted-by":"crossref","unstructured":"Madhukar, B., spsampsps Narendra, R. (2013). Lanczos resampling for the digital processing of remotely sensed images. In Proceedings of international conference on VLSI, communication, advanced devices, signals and systems and networking (VCASAN-2013) proceedings of international conference on VLSI, communication, advanced devices, signals and systems and networking (VCASAN-2013) (pp. 403\u2013411).","DOI":"10.1007\/978-81-322-1524-0_48"},{"issue":"12","key":"6814_CR20","first-page":"2651","volume":"7","author":"CA Micchelli","year":"2006","unstructured":"Micchelli, C. A., Xu, Y., & Zhang, H. (2006). Universal kernels. Journal of Machine Leing Research, 7(12), 2651\u20132667.","journal-title":"Journal of Machine Leing Research"},{"key":"6814_CR21","doi-asserted-by":"publisher","DOI":"10.1201\/9781439819319","volume-title":"Super-resolution imaging","author":"P Milanfar","year":"2017","unstructured":"Milanfar, P. (2017). Super-resolution imaging. CRC Press."},{"key":"6814_CR22","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781316219232","volume-title":"An introduction to the theory of reproducing kernel Hilbert spaces","author":"VI Paulsen","year":"2016","unstructured":"Paulsen, V. I., & Raghupathi, M. (2016). An introduction to the theory of reproducing kernel Hilbert spaces. Cambridge University Press."},{"key":"6814_CR23","doi-asserted-by":"publisher","DOI":"10.1002\/9781119994398","volume-title":"Image processing: The fundamentals","author":"MM Petrou","year":"2010","unstructured":"Petrou, M. M., & Petrou, C. (2010). Image processing: The fundamentals. Wiley."},{"issue":"5","key":"6814_CR24","doi-asserted-by":"publisher","first-page":"2848","DOI":"10.1109\/TAC.2022.3227907","volume":"68","author":"P Scharnhorst","year":"2022","unstructured":"Scharnhorst, P., Maddalena, E. T., Jiang, Y., & Jones, C. N. (2022). Robust uncertainty bounds in reproducing kernel Hilbert spaces: A convex optimization approach. IEEE Transactions on Automatic Control, 68(5), 2848\u20132861.","journal-title":"IEEE Transactions on Automatic Control"},{"key":"6814_CR25","first-page":"7","volume":"12","author":"BK Sriperumbudur","year":"2011","unstructured":"Sriperumbudur, B. K., Fukumizu, K., & Lanckriet, G. R. (2011). Universality, characteristic kernels and RKHS embedding of measures. Journal of Machine Learning Research, 12, 7.","journal-title":"Journal of Machine Learning Research"},{"issue":"5","key":"6814_CR26","doi-asserted-by":"publisher","first-page":"828","DOI":"10.1109\/TEVC.2019.2890858","volume":"23","author":"J Su","year":"2019","unstructured":"Su, J., Vargas, D. V., & Sakurai, K. (2019). One pixel attack for fooling deep neural networks. IEEE Transactions on Evolutionary Computation, 23(5), 828\u2013841.","journal-title":"IEEE Transactions on Evolutionary Computation"},{"key":"6814_CR27","doi-asserted-by":"crossref","unstructured":"Suvorov, R., Logacheva, E., Mashikhin, A., Remizova, A., Ashukha, A., Silvestrov, A., Kong, N., Goka, H., Park, K., & Lempitsky, V. (2022). Resolution-robust large mask inpainting with Fourier convolutions. In Proceedings of the IEEE\/CVF winter conference on applications of computer vision proceedings of the IEEE\/CVF winter conference on applications of computer vision (pp. 2149\u20132159).","DOI":"10.1109\/WACV51458.2022.00323"},{"key":"6814_CR28","unstructured":"Szegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D., Goodfellow, I., & Fergus, R. (2014). Intriguing properties of neural networks. In 2nd International conference on learning representations, Banff, AB, Canada, April 14\u2013162nd international conference on learning representations, Banff, AB, Canada, April 14\u201316."},{"key":"6814_CR29","doi-asserted-by":"crossref","unstructured":"Wang, Z., Simoncelli, E. P. & Bovik, A. C. (2003). Multiscale structural similarity for image quality assessment. In The thrity-seventh asilomar conference on signals, systems and computers, 2003 the thrity-seventh asilomar conference on signals, systems and computers (vol. 2, pp. 1398\u20131402).","DOI":"10.1109\/ACSSC.2003.1292216"},{"issue":"6","key":"6814_CR30","doi-asserted-by":"publisher","first-page":"1377","DOI":"10.1109\/TPAMI.2019.2895793","volume":"42","author":"W Yang","year":"2020","unstructured":"Yang, W., Tan, R. T., Feng, J., Guo, Z., Yan, S., & Liu, J. (2020). Joint rain detection and removal from a single image with contextualized deep networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(6), 1377\u20131393. https:\/\/doi.org\/10.1109\/TPAMI.2019.2895793","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"6814_CR31","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2021.3088914","author":"K Zhang","year":"2022","unstructured":"Zhang, K., Li, Y., Zuo, W., Zhang, L., Van Gool, L., & Timofte, R. (2022). Plug-and-play image restoration with deep denoiser prior. IEEE Transactions on Pattern Analysis and Machine Intelligence. https:\/\/doi.org\/10.1109\/TPAMI.2021.3088914","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"6814_CR32","doi-asserted-by":"publisher","first-page":"3142","DOI":"10.1109\/TIP.2017.2662206","volume":"26","author":"K Zhang","year":"2017","unstructured":"Zhang, K., Zuo, W., Chen, Y., Meng, D., & Zhang, L. (2017). Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising. IEEE Transactions on Image Processing, 26, 3142\u20133155.","journal-title":"IEEE Transactions on Image Processing"},{"issue":"7","key":"6814_CR33","doi-asserted-by":"publisher","first-page":"2480","DOI":"10.1109\/TPAMI.2020.2968521","volume":"43","author":"Y Zhang","year":"2021","unstructured":"Zhang, Y., Tian, Y., Kong, Y., Zhong, B., & Fu, Y. (2021). Residual dense network for image restoration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43(7), 2480\u20132495. https:\/\/doi.org\/10.1109\/TPAMI.2020.2968521","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06814-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-025-06814-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06814-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,7]],"date-time":"2025-09-07T04:44:09Z","timestamp":1757220249000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-025-06814-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,10]]},"references-count":33,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["6814"],"URL":"https:\/\/doi.org\/10.1007\/s10994-025-06814-0","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"type":"print","value":"0885-6125"},{"type":"electronic","value":"1573-0565"}],"subject":[],"published":{"date-parts":[[2025,7,10]]},"assertion":[{"value":"8 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 May 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 June 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 July 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"179"}}