{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T18:14:37Z","timestamp":1772475277073,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,8,16]],"date-time":"2019-08-16T00:00:00Z","timestamp":1565913600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Imaging"],"abstract":"<jats:p>A type-2 fuzzy edge detection method is presented in this paper. The general process consists of first obtaining the image gradients in the four directions\u2014horizontal, vertical, and the two diagonals\u2014and this technique is known as the morphological gradient. After that, the general type-2 fuzzy Sugeno integral (GT2 FSI) is used to integrate the four image gradients. In this second step, the GT2 FSI establishes criteria to determine at which level the obtained image gradient belongs to an edge during the process; this is calculated assigning different general type-2 fuzzy densities, and these fuzzy gradients are aggregated using the meet and join operators. The gradient integration using the GT2 FSI provides a methodology for achieving more robust edge detection, even more if we are working with blurry images. The experimental evaluations are performed on synthetic and real images, and the accuracy is quantified using Pratt\u2019s Figure of Merit. The results values demonstrate that the proposed edge detection method outperforms other existing algorithms.<\/jats:p>","DOI":"10.3390\/jimaging5080071","type":"journal-article","created":{"date-parts":[[2019,8,19]],"date-time":"2019-08-19T06:10:14Z","timestamp":1566195014000},"page":"71","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["General Type-2 Fuzzy Sugeno Integral for Edge Detection"],"prefix":"10.3390","volume":"5","author":[{"given":"Gabriela E.","family":"Mart\u00ednez","sequence":"first","affiliation":[{"name":"Division of Graduate Studies and Research, Tijuana Institute of Technology, Tijuana 22414, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1631-033X","authenticated-orcid":false,"given":"Claudia I.","family":"Gonzalez","sequence":"additional","affiliation":[{"name":"Division of Graduate Studies and Research, Tijuana Institute of Technology, Tijuana 22414, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Olivia","family":"Mendoza","sequence":"additional","affiliation":[{"name":"School of Engineering, Autonomous University of Baja California, Tijuana 22390, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5798-1426","authenticated-orcid":false,"given":"Patricia","family":"Melin","sequence":"additional","affiliation":[{"name":"Division of Graduate Studies and Research, Tijuana Institute of Technology, Tijuana 22414, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3175","DOI":"10.1016\/j.fss.2009.04.009","article-title":"Toward extended fuzzy logic\u2014A first step","volume":"160","author":"Zadeh","year":"2009","journal-title":"Fuzzy Sets Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1109\/34.494638","article-title":"Image filtering, edge detection, and edge tracing using fuzzy reasoning","volume":"18","author":"Law","year":"1996","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1102","DOI":"10.1109\/19.746564","article-title":"Edge Detection in Noisy Images Using Fuzzy Reasoning","volume":"47","author":"Russo","year":"1998","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/S0165-0114(98)00226-7","article-title":"FIRE operators for image processing","volume":"103","author":"Russo","year":"1999","journal-title":"Fuzzy Sets Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/S1568-4946(03)00008-5","article-title":"Competitive fuzzy edge detection","volume":"3","author":"Liang","year":"2003","journal-title":"Appl. Soft Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1109\/LSP.2006.888087","article-title":"The Fast Multilevel Fuzzy Edge Detection of Blurry Images","volume":"14","author":"Wu","year":"2007","journal-title":"IEEE Signal. Process. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1819","DOI":"10.1016\/j.fss.2008.08.005","article-title":"Interval-valued fuzzy sets constructed from matrices: Application to edge detection","volume":"160","author":"Bustince","year":"2009","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1109\/TFUZZ.2011.2146260","article-title":"Construction of Interval-Valued Fuzzy Relations with Application to the Generation of the Fuzzy Edge Images","volume":"19","author":"Barrenechea","year":"2011","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1002\/int.20378","article-title":"Interval type-2 fuzzy logic for edge detection in digital images","volume":"24","author":"Mendoza","year":"2009","journal-title":"Int. J. Intell. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"8527","DOI":"10.1016\/j.eswa.2010.05.023","article-title":"An improved method for edge detection based on interval type-2 fuzzy logic","volume":"37","author":"Melin","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1571","DOI":"10.1016\/j.cviu.2011.07.003","article-title":"Generating fuzzy edge images from gradient magnitudes","volume":"115","author":"Molina","year":"2011","journal-title":"Comput. Vis. Image Underst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1515","DOI":"10.1109\/TFUZZ.2013.2297159","article-title":"Edge-Detection Method for Image Processing Based on Generalized Type-2 Fuzzy Logic","volume":"22","author":"Melin","year":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Gonzalez, C.I., Melin, P., Castro, J.R., and Castillo, O. (2017). Edge Detection Methods Based on Generalized Type-2 Fuzzy Logic, Springer International Publishing.","DOI":"10.1007\/978-3-319-53994-2"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Melin, P., and Martinez, G.E. (2019). Extension of the Fuzzy Sugeno Integral Based on Generalized Type-2 Fuzzy Logic, Springer.","DOI":"10.1007\/978-3-030-16416-4"},{"key":"ref_15","unstructured":"Sugeno, M. (1974). Theory of Fuzzy Integrals and Its Applications. [Ph.D. Thesis, Tokyo Institute of Technology]."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1109\/TSMCA.2010.2104318","article-title":"Face Recognition with an Improved Interval Type-2 Fuzzy Logic Sugeno Integral and Modular Neural Networks","volume":"41","author":"Melin","year":"2011","journal-title":"IEEE Trans. Syst. Man Cybern. Part A Syst. Hum."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1016\/j.asoc.2006.01.009","article-title":"A hybrid modular neural network architecture with fuzzy Sugeno integration for time series forecasting","volume":"7","author":"Melin","year":"2007","journal-title":"Appl. Soft Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1109\/TFUZZ.2010.2045386","article-title":"Toward general type-2 fuzzy logic systems based on zSlices","volume":"18","author":"Wagner","year":"2010","journal-title":"Fuzzy Syst. IEEE Trans."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1109\/TFUZZ.2009.2024411","article-title":"\u03b1-Plane Representation for Type-2 Fuzzy Sets","volume":"17","author":"Mendel","year":"2009","journal-title":"Theory Appl. IEEE Trans. Fuzzy Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1162","DOI":"10.1109\/TFUZZ.2013.2286414","article-title":"General Type-2 Fuzzy Logic Systems Made Simple: A Tutorial","volume":"22","author":"Mendel","year":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5904","DOI":"10.1016\/j.eswa.2015.03.024","article-title":"Generalized Type-2 Fuzzy Systems for controlling a mobile robot and a performance comparison with Interval Type-2 and Type-1 Fuzzy Systems","volume":"42","author":"Sanchez","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1080\/16168658.2018.1509519","article-title":"A New Method for Parameterization of General Type-2 Fuzzy Sets","volume":"10","author":"Castro","year":"2018","journal-title":"Fuzzy Inf. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3887","DOI":"10.1007\/s00500-018-3503-4","article-title":"An approach for parameterized shadowed type-2 fuzzy membership functions applied in control applications","volume":"23","author":"Melin","year":"2019","journal-title":"Soft Comput."},{"key":"ref_24","unstructured":"Sobel, I. (1970). Camera Models and Perception. [Ph.D. Thesis, Stanford University]."},{"key":"ref_25","unstructured":"Prewitt, J.M.S. (1970). Object enhancement and extraction. Picture Analysis and Psychopictorics, Academic Press."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1454","DOI":"10.1109\/TIP.2005.864164","article-title":"A morphological gradient approach to color edge detection","volume":"15","author":"Evans","year":"2006","journal-title":"IEEE Trans. Image Process."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Becerikli, Y., and Karan, T.M. (2005). A New Fuzzy Approach for Edge Detection. Computational Intelligence and Bioinspired Systems, Springer Verlag. LNCS.","DOI":"10.1007\/11494669_116"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1109\/TFUZZ.2007.902024","article-title":"Minimum classification error training for Choquet integrals with applications to landmine detection","volume":"16","author":"Gader","year":"2008","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/int.20254","article-title":"A knowledge-based approach to adversarial decision making","volume":"23","author":"Yager","year":"2008","journal-title":"Int. J. Intell. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Klir, G. (2005). Uncertainty and Information, Wiley.","DOI":"10.1002\/0471755575"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Huang, K.K., Shieh, J., Lee, K.J., and Wu, S.N. (2010, January 11\u201314). Applying a generalized Choquet integral with signed fuzzy measure based on the complexity to evaluate the overall satisfaction of the patients. Proceedings of the International Conference on Machine Learning and Cybernetics, Qingdao, China.","DOI":"10.1109\/ICMLC.2010.5580669"},{"key":"ref_32","unstructured":"Bezdek, J.C., Keller, J., and Pal, N.R. (2005). Fuzzy Models and Algorithms for Pattern Recognition and Image Processing, Springer."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Torra, V., and Narukawa, Y. (2007). Modeling Decisions, Information Fusion and Aggregation Operators, Springer.","DOI":"10.1007\/978-3-540-68791-7"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1016\/S0167-8655(99)00012-4","article-title":"Soft combination of neural classifiers: A comparative study","volume":"20","author":"Verikas","year":"1999","journal-title":"Pattern Recognit."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.fss.2012.04.014","article-title":"Robust optimization of the Choquet integral","volume":"213","author":"Timonin","year":"2013","journal-title":"Fuzzy Sets Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"898","DOI":"10.1109\/TPAMI.2010.161","article-title":"Contour Detection and Hierarchical Image Segmentation","volume":"33","author":"Arbelaez","year":"2011","journal-title":"IEEE TPAMI"},{"key":"ref_37","unstructured":"Pratt, W. (1978). Digital Image Processing, Wiley."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1109\/PROC.1979.11325","article-title":"Quantitative design and evaluation of enhancement\/thresholding edge detectors","volume":"67","author":"Abdou","year":"1979","journal-title":"Proc. IEEE"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Perez-Ornelas, F., Mendoza, O., Melin, P., Castro, J.R., Rodriguez-Diaz, A., and Castillo, O. (2015). Fuzzy Index to Evaluate Edge Detection in Digital Images. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0131161"}],"container-title":["Journal of Imaging"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2313-433X\/5\/8\/71\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:11:36Z","timestamp":1760188296000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2313-433X\/5\/8\/71"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,16]]},"references-count":39,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2019,8]]}},"alternative-id":["jimaging5080071"],"URL":"https:\/\/doi.org\/10.3390\/jimaging5080071","relation":{},"ISSN":["2313-433X"],"issn-type":[{"value":"2313-433X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8,16]]}}}