{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T00:11:24Z","timestamp":1769127084762,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,16]],"date-time":"2019-02-16T00:00:00Z","timestamp":1550275200000},"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 accurate location of the mid-sagittal plane is fundamental for the assessment of craniofacial dysmorphisms and for a proper corrective surgery planning. To date, these elaborations are carried out by skilled operators within specific software environments. Since the whole procedure is based on the manual selection of specific landmarks, it is time-consuming, and the results depend on the operators\u2019 professional experience. This work aims to propose a new automatic and landmark-independent technique which is able to extract a reliable mid-sagittal plane from 3D CT images. The algorithm has been designed to perform a robust evaluation, also in the case of large defect areas. The presented method is an upgraded version of a mirroring-and registration technique for the automatic symmetry plane detection of 3D asymmetrically scanned human faces, previously published by the authors. With respect to the published algorithm, the improvements here introduced concern both the objective function formulation and the method used to minimize it. The automatic method here proposed has been verified in the analysis of real craniofacial skeletons also with large defects, and the results have been compared with other recent technologies.<\/jats:p>","DOI":"10.3390\/sym11020245","type":"journal-article","created":{"date-parts":[[2019,2,17]],"date-time":"2019-02-17T22:11:50Z","timestamp":1550441510000},"page":"245","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["A Robust and Automatic Method for the Best Symmetry Plane Detection of Craniofacial Skeletons"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5341-0500","authenticated-orcid":false,"given":"Luca","family":"Di Angelo","sequence":"first","affiliation":[{"name":"Department of Industrial and Information Engineering, and of Economics, University of L\u2019Aquila, via G. Gronchi 18, 67100 L\u2019Aquila, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5003-2084","authenticated-orcid":false,"given":"Paolo","family":"Di Stefano","sequence":"additional","affiliation":[{"name":"Department of Industrial and Information Engineering, and of Economics, University of L\u2019Aquila, via G. Gronchi 18, 67100 L\u2019Aquila, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7417-3487","authenticated-orcid":false,"given":"Lapo","family":"Governi","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, University of Florence, via di Santa Marta, 3, 50139 Firenze, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7842-4562","authenticated-orcid":false,"given":"Antonio","family":"Marzola","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, University of Florence, via di Santa Marta, 3, 50139 Firenze, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5668-1912","authenticated-orcid":false,"given":"Yary","family":"Volpe","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, University of Florence, via di Santa Marta, 3, 50139 Firenze, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0162-3095(94)90018-3","article-title":"Facial attractiveness, developmental stability, and fluctuating asymmetry","volume":"15","author":"Gangestad","year":"1994","journal-title":"Ethol. Sociobiol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1037\/0735-7036.108.3.233","article-title":"Human facial attractiveness and sexual selection: The role of averageness and symmetry","volume":"108","author":"Grammar","year":"1994","journal-title":"J. Comp. Psychol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1617","DOI":"10.1098\/rspb.2001.1703","article-title":"Symmetry, sexual dimorphism in facial proportions and male facial attractiveness","volume":"268","author":"Jones","year":"2001","journal-title":"Proc. Biol. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1037\/0022-3514.76.1.151","article-title":"Symmetry and perceived facial attractiveness: A monozygotic co-twin comparison","volume":"76","author":"Mealey","year":"1999","journal-title":"J. Pers. Soc. Psychol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"659","DOI":"10.3758\/BF03208842","article-title":"Facial symmetry and the perception of beauty","volume":"5","author":"Rhodes","year":"1998","journal-title":"Psychon. Bull. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1068\/p3123","article-title":"Attractiveness of facial averageness and symmetry in non-western cultures: In search of biologically based standards of beauty","volume":"30","author":"Rhodes","year":"2001","journal-title":"Perception"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1244","DOI":"10.1068\/p5712","article-title":"Perceived health contributes to the attractiveness of facial symmetry, averageness, and sexual dimorphism","volume":"36","author":"Rhodes","year":"2007","journal-title":"Perception"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/S1090-5138(00)00060-X","article-title":"Do facial averageness and symmetry signal health?","volume":"22","author":"Rhodes","year":"2001","journal-title":"Evol. Hum. Behav."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/BF02692201","article-title":"Human facial beauty: Averageness, symmetry, and parasite resistance","volume":"4","author":"Thornhill","year":"1993","journal-title":"Hum. Nat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/j.cub.2015.02.017","article-title":"Symmetry is less than meets the eye","volume":"25","author":"Apthorp","year":"2015","journal-title":"Curr. Biol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1038\/368239a0","article-title":"Facial shape and judgments of female attractiveness","volume":"368","author":"Perrett","year":"1994","journal-title":"Nature"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1016\/0002-9416(61)90062-8","article-title":"Dentofacial asymmetries and their clinical significance","volume":"47","author":"Cheney","year":"1961","journal-title":"Am. J. Orthod."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1093\/ejo\/12.4.408","article-title":"Relationship between craniofacial and condyle path asymmetry in unilateral cross-bite patients","volume":"12","author":"Pirttiniemi","year":"1990","journal-title":"Eur. J. Orthod."},{"key":"ref_14","first-page":"168.e1","article-title":"Impact of facial asymmetry in visual perception: A 3-dimensional data analysis","volume":"137","author":"Alpers","year":"2010","journal-title":"Am. J. Orthod. Dentofac. Orthop."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2080","DOI":"10.1016\/j.joms.2009.06.007","article-title":"Three-dimensional treatment planning of orthognathic surgery in the era of virtual imaging","volume":"67","author":"Swennen","year":"2009","journal-title":"J. Oral Maxillofac. Surg."},{"key":"ref_16","unstructured":"Proffit, W.R., and White, R.P. (1991). Dentofacial asymmetry. Surgical Orthodontic Treatment, Mosby Inc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1016\/j.ajodo.2005.02.021","article-title":"Maxillofacial 3-dimensional image analysis for the diagnosis of facial asymmetry","volume":"130","author":"Hwang","year":"2006","journal-title":"Am. J. Orthod. Dentofac. Orthop."},{"key":"ref_18","first-page":"729","article-title":"Three-dimensional evaluation of craniofacial asymmetry: An analysis using computed tomography","volume":"15","year":"2010","journal-title":"Clin. Oral Investig."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/j.jcms.2009.01.004","article-title":"A case report comparing 3-D evaluation in the diagnosis and treatment planning of hemimandibular hyperplasia with conventional radiography","volume":"37","author":"Tuncer","year":"2009","journal-title":"J. Craniomaxillofac. Surg."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1016\/j.tripleo.2007.03.002","article-title":"Skeletodental factors affecting chin point deviation in female patients with class III malocclusion and facial asymmetry: A three-dimensional analysis using computed tomography","volume":"104","author":"Baek","year":"2007","journal-title":"Oral Surg. Oral Med. Oral Pathol. Oral Radiol. Endod."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.ijom.2005.04.006","article-title":"A comparison of craniofacial morphology in patients with and without facial asymmetry\u2014A three-dimensional analysis with computed tomography","volume":"5","author":"Kwon","year":"2006","journal-title":"Int. J. Oral Maxillofa. Surg."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"103","DOI":"10.14733\/cadaps.2019.103-112","article-title":"Reverse engineering techniques for virtual reconstruction of defective skulls: An overview of existing approaches","volume":"16","author":"Buonamici","year":"2018","journal-title":"Comput-Aided Des. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1843","DOI":"10.1016\/j.medengphy.2013.06.006","article-title":"Development and evaluation of a semi-automatic technique for determining the bilateral symmetry plane of the facial skeleton","volume":"35","author":"Willing","year":"2013","journal-title":"Med. Eng. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/s00784-011-0512-4","article-title":"A three-dimensional comparison of a morphometric and conventional cephalometric midsagittal planes for craniofacial asymmetry","volume":"16","author":"Damstra","year":"2012","journal-title":"Clin. Oral Investig."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1016\/j.ijom.2006.01.028","article-title":"Automatic extraction of the mid-facial plane for cranio-maxillofacial surgery planning","volume":"35","author":"Chapuis","year":"2006","journal-title":"Int. J. Oral Maxillofac. Surg."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s00371-005-0352-9","article-title":"3D face authentication and recognition based on bilateral symmetry analysis","volume":"22","author":"Zhang","year":"2006","journal-title":"Vis. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bartalucci, C., Furferi, R., Governi, L., and Volpe, Y. (2018). A Survey of Methods for Symmetry Detection on 3D High Point Density Models in Biomedicine. Symmetry, 10.","DOI":"10.3390\/sym10070263"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1080\/16864360.2014.863487","article-title":"A Computational Method for Bilateral Symmetry Recognition in Asymmetrically Scanned Human Faces","volume":"11","year":"2014","journal-title":"Comput-Aided Des. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.ajodo.2016.06.027","article-title":"Precision of manual landmark identification between as-received and oriented volume-rendered cone-beam computed tomography images","volume":"151","author":"Gupta","year":"2017","journal-title":"Am. J. Orthod. Dentofac. Orthop."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"79","DOI":"10.5624\/isd.2011.41.2.79","article-title":"Determination of midsagittal plane for evaluation of facial asymmetry using three-dimensional computed tomography","volume":"41","author":"Kim","year":"2011","journal-title":"Imaging Sci. Dent."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1016\/j.ajodo.2016.12.025","article-title":"A simple and accurate craniofacial midsagittal plane definition","volume":"152","author":"Green","year":"2017","journal-title":"Am. J. Orthod. Dentofac. Orthop."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1038\/sj.dmfr.4600389","article-title":"Quantitative analysis of spiral computed tomography for craniofacial clinical applications","volume":"27","author":"Cavalcanti","year":"1998","journal-title":"Dentomaxillofac. Radiol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1737","DOI":"10.1007\/s11548-015-1173-6","article-title":"A knowledge-based algorithm for automatic detection of cephalometric landmarks on CBCT images","volume":"10","author":"Gupta","year":"2015","journal-title":"Int. J. Comput. Assisted Radiol. Surg."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"207","DOI":"10.5624\/isd.2014.44.3.207","article-title":"Deviation of landmarks in accordance with methods of establishing reference planes in three-dimensional facial CT evaluation","volume":"44","author":"Yoon","year":"2014","journal-title":"Imaging Sci. Dent."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1142\/S0219467805001884","article-title":"3D face recognition from range data","volume":"5","author":"Pan","year":"2005","journal-title":"Int. J. Image Graph."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1109\/34.121791","article-title":"A Method for Registration of 3-D Shapes","volume":"14","author":"Besl","year":"1992","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_37","first-page":"431","article-title":"An Algorithm for Least-Squares Estimation of Nonlinear Parameters","volume":"11","author":"Marquardt","year":"1963","journal-title":"J. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BF00054839","article-title":"Constructing simple stable descriptions for image partitioning","volume":"3","author":"Leclerc","year":"1989","journal-title":"Int. J. Comput. Vis."},{"key":"ref_39","unstructured":"Kennedy, J., and Eberhart, R.C. (December, January 27). Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks (ICNN), Perth, Australia."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/245\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:32:39Z","timestamp":1760185959000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/245"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,16]]},"references-count":39,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["sym11020245"],"URL":"https:\/\/doi.org\/10.3390\/sym11020245","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,16]]}}}