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Regularization techniques and neighborhood filters have been widely used, but degraded spatial resolution and bias are\u00a0concerns.<\/jats:p><\/jats:sec><jats:sec><jats:title>Purpose<\/jats:title><jats:p>This paper proposes likelihood\u2010based bilateral filters that can be applied to pre\u2010estimated basis sinograms to reduce the noise while minimally affecting spatial resolution and\u00a0accuracy.<\/jats:p><\/jats:sec><jats:sec><jats:title>Methods<\/jats:title><jats:p>The proposed method needs system models (e.g., incident spectrum, detector response) to calculate the likelihood. First, it performs maximum likelihood (ML)\u2010based estimation in the projection domain to obtain basis sinograms. The estimated basis sinograms suffer from severe noise but are asymptotically unbiased without degrading spatial resolution. Then it calculates the neighborhood likelihoods for a given measurement at the center pixel using the neighborhood estimates and designs the weights based on the distance of likelihoods. It is also analyzed in terms of statistical inference, and then two variations of the filter are introduced: one that requires the significance level instead of the empirical hyperparameter. The other is a measurement\u2010based filter, which can be applied when accurate estimates are given without the system models. The proposed methods were validated by analyzing the local property of noise and spatial resolution and the global trend of noise and bias using numerical thorax and abdominal phantoms for a two\u2010material decomposition (water and bone). They were compared to the conventional neighborhood filters and the model\u2010based iterative reconstruction with an edge\u2010preserving penalty applied in the basis\u00a0images.<\/jats:p><\/jats:sec><jats:sec><jats:title>Results<\/jats:title><jats:p>The proposed method showed comparable or superior performance for the local and global properties to conventional methods in many cases. The thorax phantom: The full width at half maximum (FWHM) decreased by \u22122%\u201331% (\u22122 indicates that it increased by 2% compared to the best performance from conventional methods), and the global bias was reduced by 2%\u201319% compared to other methods for similar noise levels (local: 51% of the ML, global: 49%) in the water basis image. The FWHM decreased by 8%\u201331%, and the global bias was reduced by 9%\u201344% for similar noise levels (local: 44% of the ML, global: 36%) in the CT image at 65\u2009keV. The abdominal phantom: The FWHM decreased by 10%\u201332%, and the global bias was reduced by 3%\u201335% compared to other methods for similar noise levels (local: 66% of the ML, global: 67%) in the water basis image. The FWHM decreased by up to \u221211%\u201347%, and the global bias was reduced by 13%\u201335% for similar noise levels (local: 71% of the ML, global: 70%) in the CT image at 60\u2009keV.<\/jats:p><\/jats:sec><jats:sec><jats:title>Conclusions<\/jats:title><jats:p>This paper introduced the likelihood\u2010based bilateral filters as a post\u2010processing method applied to the ML\u2010based estimates of basis sinograms. The proposed filters effectively reduced the noise in the basis images and the synthesized monochromatic CT images. It showed the potential of using likelihood\u2010based filters in the projection domain as a substitute for conventional regularization or filtering\u00a0methods.<\/jats:p><\/jats:sec>","DOI":"10.1002\/mp.16251","type":"journal-article","created":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T10:58:30Z","timestamp":1674817110000},"page":"2733-2758","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Likelihood\u2010based bilateral filters for pre\u2010estimated basis sinograms using photon\u2010counting CT"],"prefix":"10.1002","volume":"50","author":[{"given":"Okkyun","family":"Lee","sequence":"first","affiliation":[{"name":"Department of Robotics and Mechatronics Engineering Daegu Gyeongbuk Institute of Science and Technology (DGIST) Daegu Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2023,2,5]]},"reference":[{"key":"e_1_2_9_2_1","doi-asserted-by":"publisher","DOI":"10.1118\/1.4820371"},{"key":"e_1_2_9_3_1","doi-asserted-by":"publisher","DOI":"10.1109\/TRPMS.2020.3020212"},{"key":"e_1_2_9_4_1","doi-asserted-by":"publisher","DOI":"10.1118\/1.4894724"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2019.2914370"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1002\/mp.15621"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.1118\/1.4922654"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.1148\/radiol.10092473"},{"key":"e_1_2_9_9_1","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6560\/ac0cc1"},{"key":"e_1_2_9_10_1","volume-title":"Statistical Inference","author":"Casella G","year":"2001"},{"key":"e_1_2_9_11_1","doi-asserted-by":"publisher","DOI":"10.1088\/0031-9155\/52\/15\/020"},{"key":"e_1_2_9_12_1","doi-asserted-by":"publisher","DOI":"10.1088\/0031-9155\/53\/15\/002"},{"key":"e_1_2_9_13_1","doi-asserted-by":"publisher","DOI":"10.1088\/0031-9155\/54\/5\/014"},{"key":"e_1_2_9_14_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2016.2621821"},{"key":"e_1_2_9_15_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2017.2746269"},{"key":"e_1_2_9_16_1","doi-asserted-by":"publisher","DOI":"10.1201\/9780429486111-19"},{"key":"e_1_2_9_17_1","doi-asserted-by":"publisher","DOI":"10.1002\/mp.14920"},{"key":"e_1_2_9_18_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2013.2250991"},{"key":"e_1_2_9_19_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2014.2320284"},{"key":"e_1_2_9_20_1","doi-asserted-by":"publisher","DOI":"10.1088\/0031-9155\/61\/10\/3784"},{"key":"e_1_2_9_21_1","doi-asserted-by":"publisher","DOI":"10.1109\/TRPMS.2020.3007380"},{"key":"e_1_2_9_22_1","doi-asserted-by":"publisher","DOI":"10.1109\/83.535846"},{"key":"e_1_2_9_23_1","doi-asserted-by":"publisher","DOI":"10.1109\/83.491322"},{"key":"e_1_2_9_24_1","first-page":"1","article-title":"Multi\u2010energy CT decomposition using convolutional neural networks. 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