{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T01:39:29Z","timestamp":1768873169115,"version":"3.49.0"},"reference-count":54,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2018,5,22]],"date-time":"2018-05-22T00:00:00Z","timestamp":1526947200000},"content-version":"vor","delay-in-days":21,"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/http\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Computer Graphics Forum"],"published-print":{"date-parts":[[2018,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we consider the problem of information transfer across shapes and propose an extension to the widely used functional map representation. Our main observation is that in addition to the vector space structure of the functional spaces, which has been heavily exploited in the functional map framework, the functional<jats:italic>algebra<\/jats:italic>(i.e., the ability to take pointwise products of functions) can significantly extend the power of this framework. Equipped with this observation, we show how to improve one of the key applications of functional maps, namely transferring real\u2010valued functions without conversion to point\u2010to\u2010point correspondences. We demonstrate through extensive experiments that by decomposing a given function into a linear combination consisting not only of basis functions but also of their pointwise products, both the representation power and the quality of the function transfer can be improved significantly. Our modification, while computationally simple, allows us to achieve higher transfer accuracy while keeping the size of the basis and the functional map fixed. We also analyze the computational complexity of<jats:italic>optimally<\/jats:italic>representing functions through linear combinations of products in a given basis and prove NP\u2010completeness in some general cases. Finally, we argue that the use of function products can have a wide\u2010reaching effect in extending the power of functional maps in a variety of applications, in particular by enabling the transfer of high\u2010frequency functions without changing the representation size or complexity.<\/jats:p>","DOI":"10.1111\/cgf.13352","type":"journal-article","created":{"date-parts":[[2018,6,1]],"date-time":"2018-06-01T00:19:39Z","timestamp":1527812379000},"page":"179-190","update-policy":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Improved Functional Mappings via Product Preservation"],"prefix":"10.1111","volume":"37","author":[{"given":"D.","family":"Nogneng","sequence":"first","affiliation":[{"name":"LIX, \u00c9cole Polytechnique, UMR CNRS"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Melzi","sequence":"additional","affiliation":[{"name":"University of Verona"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"E.","family":"Rodol\u00e0","sequence":"additional","affiliation":[{"name":"Sapienza University of Rome"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"U.","family":"Castellani","sequence":"additional","affiliation":[{"name":"University of Verona"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Bronstein","sequence":"additional","affiliation":[{"name":"USI Lugano"},{"name":"Tel Aviv University"},{"name":"Intel Perceptual Computing"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Ovsjanikov","sequence":"additional","affiliation":[{"name":"LIX, \u00c9cole Polytechnique, UMR CNRS"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2018,5,22]]},"reference":[{"key":"e_1_2_16_2_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12174"},{"key":"e_1_2_16_3_2","doi-asserted-by":"publisher","DOI":"10.1137\/140977680"},{"key":"e_1_2_16_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073696"},{"key":"e_1_2_16_5_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1308708110"},{"key":"e_1_2_16_6_2","doi-asserted-by":"publisher","DOI":"10.1145\/2980179.2982412"},{"key":"e_1_2_16_7_2","doi-asserted-by":"publisher","DOI":"10.1145\/2766921"},{"key":"e_1_2_16_8_2","doi-asserted-by":"crossref","unstructured":"AubryM. SchlickeweiU. CremersD.: The wave kernel signature: A quantum mechanical approach to shape analysis. InICCV Workshops(2011) IEEE pp.1626\u20131633. 3 7","DOI":"10.1109\/ICCVW.2011.6130444"},{"key":"e_1_2_16_9_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0508601103"},{"key":"e_1_2_16_10_2","volume-title":"Numerical Geometry of Non\u2010Rigid Shapes","author":"Bronstein A. M.","year":"2008"},{"key":"e_1_2_16_11_2","doi-asserted-by":"crossref","unstructured":"BergA. C. BergT. L. MalikJ.: Shape matching and object recognition using low distortion correspondences. InComputer Vision and Pattern Recognition 2005. CVPR 2005. IEEE Computer Society Conference on(2005) Vol. 1 pp.26\u201333. 2","DOI":"10.1109\/CVPR.2005.320"},{"key":"e_1_2_16_12_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12734"},{"key":"e_1_2_16_13_2","doi-asserted-by":"crossref","unstructured":"BogoF. RomeroJ. LoperM. BlackM. J.: FAUST: Dataset and Evaluation for 3d Mesh Registration. InProc. CVPR(2014) pp.3794\u20133801. 1 7 9 10","DOI":"10.1109\/CVPR.2014.491"},{"key":"e_1_2_16_14_2","unstructured":"CuturiM.: Sinkhorn distances: Lightspeed computation of optimal transport. InAdvances in neural information processing systems(2013) pp.2292\u20132300. 2"},{"key":"e_1_2_16_15_2","doi-asserted-by":"crossref","unstructured":"HuangR. ChazalF. OvsjanikovM.: On the stability of functional maps and shape difference operators.Computer Graphics Forum(2017). 4","DOI":"10.1111\/cgf.13238"},{"key":"e_1_2_16_16_2","volume-title":"Algebras, rings and modules","author":"Hazewinkel M.","year":"2004"},{"key":"e_1_2_16_17_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01373.x"},{"key":"e_1_2_16_18_2","doi-asserted-by":"publisher","DOI":"10.1145\/2601097.2601111"},{"key":"e_1_2_16_19_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12064"},{"key":"e_1_2_16_20_2","doi-asserted-by":"crossref","unstructured":"KovnatskyA. GlashoffK. BronsteinM. M.: MADMM: a generic algorithm for non\u2010smooth optimization on manifolds. InProc. ECCV(2016) Springer pp.680\u2013696. 3","DOI":"10.1007\/978-3-319-46454-1_41"},{"key":"e_1_2_16_21_2","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964974"},{"key":"e_1_2_16_22_2","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276457"},{"key":"e_1_2_16_23_2","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531378"},{"key":"e_1_2_16_24_2","doi-asserted-by":"crossref","unstructured":"LeordeanuM. HebertM.: A spectral technique for correspondence problems using pairwise constraints. InProc. ICCV(2005) Vol. 2 IEEE pp.1482\u20131489. 2","DOI":"10.1109\/ICCV.2005.20"},{"key":"e_1_2_16_25_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12970"},{"key":"e_1_2_16_26_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13123"},{"key":"e_1_2_16_27_2","doi-asserted-by":"crossref","unstructured":"LitanyO. RemezT. Rodol\u00e0E. BronsteinA. M. BronsteinM. M.: Deep functional maps: Structured prediction for dense shape correspondence. InProc. ICCV(2017). 2","DOI":"10.1109\/ICCV.2017.603"},{"key":"e_1_2_16_28_2","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073671"},{"key":"e_1_2_16_29_2","unstructured":"M\u00e9moliF.: On the use of Gromov\u2010Hausdorff distances for shape comparison.Point\u2010Based Graphics(2007). 2"},{"key":"e_1_2_16_30_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-011-9093-5"},{"key":"e_1_2_16_31_2","doi-asserted-by":"crossref","unstructured":"MelziS. RodolaE. CastellaniU. BronsteinM.: Shape analysis with anisotropic windowed fourier transform. InInternational Conference on 3D Vision (3DV)(2016).","DOI":"10.1109\/3DV.2016.57"},{"key":"e_1_2_16_32_2","unstructured":"MelziS. Rodol\u00e0E. CastellaniU. BronsteinM. M.: Localized manifold harmonics for spectral shape analysis.arXiv:1707.02596(2017). 2 9"},{"key":"e_1_2_16_33_2","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539792240406"},{"key":"e_1_2_16_34_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13124"},{"key":"e_1_2_16_35_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12429"},{"key":"e_1_2_16_36_2","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185526"},{"key":"e_1_2_16_37_2","first-page":"51","article-title":"Computing and processing correspondences with functional maps","author":"Ovsianikov M.","year":"2017","journal-title":"ACM SIGGRAPH 2017 Courses"},{"key":"e_1_2_16_38_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2010.01764.x"},{"key":"e_1_2_16_39_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12167"},{"key":"e_1_2_16_40_2","unstructured":"Peyr\u00e9G.:Toolbox sparse optimization.https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/www.mathworks.com\/matlabcentral\/fileexchange\/16204. Accessed: 2017\u201010\u201010. 5 7"},{"key":"e_1_2_16_41_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12797"},{"key":"e_1_2_16_42_2","unstructured":"Rodol\u00e0E. MoellerM. CremersD.: Point\u2010wise map recovery and refinement from functional correspondence. InProc. Vision Modeling and Visualization (VMV)(2015). 2 3"},{"key":"e_1_2_16_43_2","doi-asserted-by":"publisher","DOI":"10.1145\/2766963"},{"key":"e_1_2_16_44_2","doi-asserted-by":"publisher","DOI":"10.3390\/axioms3030300"},{"key":"e_1_2_16_45_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cviu.2015.02.004"},{"key":"e_1_2_16_46_2","volume-title":"Composition Operators on Function Spaces","author":"Singh R. K.","year":"1993"},{"key":"e_1_2_16_47_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2012.03167.x"},{"key":"e_1_2_16_48_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2009.01515.x"},{"key":"e_1_2_16_49_2","doi-asserted-by":"publisher","DOI":"10.1145\/1015706.1015736"},{"key":"e_1_2_16_50_2","doi-asserted-by":"publisher","DOI":"10.1145\/2897824.2925903"},{"issue":"7","key":"e_1_2_16_51_2","first-page":"1199","article-title":"Registration of 3D point clouds and meshes: a survey from rigid to nonrigid","volume":"19","author":"Tam G. K.","year":"2013","journal-title":"IEEE TVCG"},{"key":"e_1_2_16_52_2","doi-asserted-by":"crossref","unstructured":"TombariF. SaltiS. Di StefanoL.: Unique signatures of histograms for local surface description. InInternational Conference on Computer Vision (ICCV)(2010) pp.356\u2013369. 7","DOI":"10.1007\/978-3-642-15558-1_26"},{"key":"e_1_2_16_53_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2011.01884.x"},{"key":"e_1_2_16_54_2","doi-asserted-by":"crossref","unstructured":"VestnerM. L\u00e1hnerZ. BoyarskiA. LitanyO. SlossbergR. RemezT. RodolaE. BronsteinA. BronsteinM. KimmelR. CremersD.: Efficient deformable shape correspondence via kernel matching. InProc. 3DV(2017). 2","DOI":"10.1109\/3DV.2017.00065"},{"key":"e_1_2_16_55_2","doi-asserted-by":"crossref","unstructured":"WangF. HuangQ. OvsjanikovM. GuibasL. J.: Unsupervised multi\u2010class joint image segmentation. InProc. CVPR(2014) pp.3142\u20133149. 2 3 4","DOI":"10.1109\/CVPR.2014.402"}],"container-title":["Computer Graphics Forum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fcgf.13352","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.13352","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T22:39:48Z","timestamp":1751668788000},"score":1,"resource":{"primary":{"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/onlinelibrary.wiley.com\/doi\/10.1111\/cgf.13352"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5]]},"references-count":54,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,5]]}},"alternative-id":["10.1111\/cgf.13352"],"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.1111\/cgf.13352","archive":["Portico"],"relation":{},"ISSN":["0167-7055","1467-8659"],"issn-type":[{"value":"0167-7055","type":"print"},{"value":"1467-8659","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,5]]},"assertion":[{"value":"2018-05-22","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}