{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:25:59Z","timestamp":1760239559007,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T00:00:00Z","timestamp":1605830400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The literature has shown that the performance of the de-noising algorithm was greatly influenced by the dependencies between wavelet coefficients. In this paper, the bivariate probability density function (PDF) was proposed which was symmetric, and the dependencies between the coefficients were considered. The bivariate Cauchy distribution and the bivariate Student\u2019s distribution are special cases of the proposed bivariate PDF. One of the parameters in the probability density function gave the estimation method, and the other parameter can take any real number greater than 2. The algorithm adopted a maximum a posteriori estimator employing the dual-tree complex wavelet transform (DTCWT). Compared with the existing best results, the method is faster and more efficient than the previous numerical integration techniques. The bivariate shrinkage function of the proposed algorithm can be expressed explicitly. The proposed method is simple to implement.<\/jats:p>","DOI":"10.3390\/sym12111909","type":"journal-article","created":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T09:46:18Z","timestamp":1605865578000},"page":"1909","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Image Denoising Based on Bivariate Distribution"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1066-8608","authenticated-orcid":false,"given":"Ping","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Science, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xingyu","family":"Zhao","sequence":"additional","affiliation":[{"name":"CHELBI Engineering Consultants, Inc., Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chun","family":"Zhao","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics Science, Tianjin Normal University, Tianjin 300074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1532","DOI":"10.1109\/83.862633","article-title":"Adaptive wavelet thresholding for image denoising and compression","volume":"9","author":"Chang","year":"2000","journal-title":"IEEE Trans. Image Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1109\/97.803428","article-title":"Low-complexity image denoising based on statistical modeling of wavelet coefficients","volume":"6","author":"Kozintsev","year":"1999","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1338","DOI":"10.1109\/TIP.2003.818640","article-title":"Image denoising using scale mixtures of Gaussians in the wavelet domain","volume":"12","author":"Portilla","year":"2003","journal-title":"IEEE Trans. Image Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2346","DOI":"10.1049\/iet-ipr.2018.5292","article-title":"Denoising of ultrasound images affected by combined speckle and Gaussian noise","volume":"12","author":"Mafi","year":"2018","journal-title":"IET Image Process."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Deivalakshmi, S. (2017, January 23\u201324). Performance study of despeckling algorithm for different wavelet transforms. Proceedings of the 2017 International Conference on Inventive Computing and Informatics (ICICI), Coimbatore, India.","DOI":"10.1109\/ICICI.2017.8365280"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1109\/TIP.2005.864240","article-title":"Image denoising using a tight frame","volume":"15","author":"Shen","year":"2006","journal-title":"IEEE Trans. Image Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1847","DOI":"10.1007\/s00034-012-9412-3","article-title":"Four-channel tight wavelet frames design using Bernstein polynomial","volume":"31","author":"Zhao","year":"2012","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1109\/LSP.2002.806054","article-title":"Bivariate shrinkage with local variance estimation","volume":"9","author":"Sendur","year":"2002","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2744","DOI":"10.1109\/TSP.2002.804091","article-title":"Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency","volume":"50","author":"Sendur","year":"2002","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1109\/LSP.2004.839692","article-title":"Image denoising using bivariate \u03b1-stable distributions in the complex wavelet domain","volume":"12","author":"Achim","year":"2005","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1109\/TCSVT.2006.888020","article-title":"Spatially adaptive wavelet-based method using the Cauchy prior for denoising the SAR images","volume":"17","author":"Bhuiyan","year":"2007","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1049\/iet-ipr.2007.0096","article-title":"Spatially adaptive thresholding in wavelet domain for despeckling of ultrasound images","volume":"3","author":"Bhuiyan","year":"2009","journal-title":"IET Image Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1049\/iet-ipr:20070035","article-title":"Wavelet-based image denoising with the normal inverse Gaussian prior and linear MMSE estimator","volume":"2","author":"Bhuiyan","year":"2008","journal-title":"IET Image Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2723","DOI":"10.1109\/TGRS.2010.2041241","article-title":"Dual tree complex wavelet transform based despeckling using interscale dependency","volume":"48","author":"Ranjani","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"552","DOI":"10.1109\/LGRS.2010.2089780","article-title":"Generalized SAR Despeckling Based on DTCWT Exploiting Interscale and Intrascale Dependences","volume":"8","author":"Ranjani","year":"2011","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1049\/iet-ipr.2013.0863","article-title":"Bayesian denoising of ultrasound images using heavy-tailed Levy distribution","volume":"9","author":"Ranjani","year":"2015","journal-title":"IET Image Process."},{"key":"ref_17","first-page":"1","article-title":"Dual-Tree complex wavelet coefficient magnitude modeling using scale mixtures of Rayleigh distribution for image denoising","volume":"38","author":"Saeedzarandi","year":"2019","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2543","DOI":"10.1098\/rsta.1999.0447","article-title":"Image processing with complex wavelets","volume":"357","author":"Kingsbury","year":"1999","journal-title":"Philos. Trans. R. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1006\/acha.2000.0343","article-title":"Complex wavelets for shift invariant analysis and filtering of signals","volume":"10","author":"Kingsbury","year":"2001","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Dwivedy, P., Potnis, A., Soofi, S., and Mishra, M. (2017, January 25\u201327). Comparative study of MSVD, PCA, DCT, DTCWT, SWT and Laplacian pyramid based image fusion. Proceedings of the 2017 International Conference on Recent Innovations in Signal processing and Embedded Systems (RISE), Washington, DC, USA.","DOI":"10.1109\/RISE.2017.8378165"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2739","DOI":"10.1109\/TIP.2016.2552725","article-title":"Contrast sensitivity of the wavelet, dual tree complex wavelet, curvelet, and steerable pyramid transforms","volume":"25","author":"Hill","year":"2016","journal-title":"IEEE Trans. Image Process."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"HemaLatha, M., Varadarajan, S., and Babu, Y.M.M. (2017, January 16\u201318). Comparison of DWT, DWT-SWT, and DT-CWT for low resolution satellite images enhancement. Proceedings of the 2017 International Conference on Algorithms, Methodology, Models and Applications in Emerging Technologies (ICAMMAET), Chennai, India.","DOI":"10.1109\/ICAMMAET.2017.8186662"},{"key":"ref_23","unstructured":"Gradhsteyn, S.I., and Ryzhik, I.M. (1996). Table of Integrals, Series, and Products, Academic."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1093\/biomet\/81.3.425","article-title":"Ideal spatial adaptation by wavelet shrinkage","volume":"81","author":"Donoho","year":"1994","journal-title":"Biometrika"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"354","DOI":"10.2307\/3619777","article-title":"A new approach to solving the cubic: Cardan\u2019s solution revealed","volume":"77","author":"Nickalls","year":"1993","journal-title":"Math. Gaz."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1522","DOI":"10.1109\/83.862630","article-title":"Spatially adaptive wavelet thresholding with context modeling for image denoising","volume":"9","author":"Chang","year":"2000","journal-title":"IEEE Trans. Image Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1109\/78.668544","article-title":"Wavelet-based signal processing using hidden Markov models","volume":"46","author":"Crouse","year":"1998","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1647","DOI":"10.1109\/83.967393","article-title":"Information-theoretic analysis of interscale and intrascale dependencies between image wavelet coeffcients","volume":"10","author":"Liu","year":"2001","journal-title":"IEEE Trans. Image Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"4661","DOI":"10.1109\/TSP.2008.927461","article-title":"The shiftable complex directional pyramid-part II: Implementation and applications","volume":"56","author":"Nguyen","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.patcog.2008.03.017","article-title":"Multiscale facial structure representation for face recognition under varying illumination","volume":"42","author":"Zhang","year":"2009","journal-title":"Pattern Recognit."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1909\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:34:56Z","timestamp":1760178896000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1909"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,20]]},"references-count":30,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["sym12111909"],"URL":"https:\/\/doi.org\/10.3390\/sym12111909","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,11,20]]}}}