{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:41:00Z","timestamp":1760244060584,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2009,9,28]],"date-time":"2009-09-28T00:00:00Z","timestamp":1254096000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In geometric models with high-valence vertices, current subdivision wavelets may not deal with the special cases well for good visual effect of multiresolution surfaces. In this paper, we present the novel biorthogonal polar subdivision wavelets, which can efficiently perform wavelet analysis to the control nets with polar structures. The polar subdivision can generate more natural subdivision surfaces around the high-valence vertices and avoid the ripples and saddle points where Catmull-Clark subdivision may produce. Based on polar subdivision, our wavelet scheme supports special operations on the polar structures, especially suitable to models with many facets joining. For seamless fusing with Catmull-Clark subdivision wavelet, we construct the wavelets in circular and radial layers of polar structures, so can combine the subdivision wavelets smoothly for composite models formed by quadrilaterals and polar structures. The computations of wavelet analysis and synthesis are highly efficient and fully in-place. The experimental results have confirmed the stability of our proposed approach.<\/jats:p>","DOI":"10.3390\/a2041263","type":"journal-article","created":{"date-parts":[[2009,9,28]],"date-time":"2009-09-28T13:03:56Z","timestamp":1254143036000},"page":"1263-1280","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Compound Biorthogonal Wavelets on Quadrilaterals and Polar Structures"],"prefix":"10.3390","volume":"2","author":[{"given":"Chong","family":"Zhao","sequence":"first","affiliation":[{"name":"Department of Computer Science Engineering, The Chinese University of Hong Kong, The Chinese University of Hong Kong, Shatin N.T., Hong Kong"}]},{"given":"Hanqiu","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Computer Science Engineering, The Chinese University of Hong Kong, The Chinese University of Hong Kong, Shatin N.T., Hong Kong"}]},{"given":"Huawei","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of MEEM, The City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong"}]},{"given":"Kaihuai","family":"Qin","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Technology, East Main Building, Tsinghua University, Beijing 100084, China"}]}],"member":"1968","published-online":{"date-parts":[[2009,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1145\/237748.237750","article-title":"Multiresolution analysis for surfaces of arbitrary topological type","volume":"16","author":"Lounsbery","year":"1997","journal-title":"ACM Trans. Graph."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1109\/TVCG.2004.1260763","article-title":"Wavelet-based multiresolution analysis of irregular surface meshes","volume":"10","author":"Valette","year":"2004","journal-title":"IEEE Trans. Visual. Comput. Graph."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1109\/TVCG.2004.1260764","article-title":"Wavelet-based progressive compression scheme for triangle meshes: wavemesh","volume":"10","author":"Valette","year":"2004","journal-title":"IEEE Trans. Visual. Comput. Graph."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1111\/1467-8659.00361","article-title":"Multiresolution curve and surface representation: reversing subdivision rules by least-squares data fitting","volume":"18","author":"Samavati","year":"1999","journal-title":"Comput. Graph. Forum"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1111\/1467-8659.00572","article-title":"Multiresolution surfaces having arbitrary topologies by a reverse Doo subdivision method","volume":"21","author":"Samavati","year":"2002","journal-title":"Computer Graphics Forum"},{"key":"ref_6","unstructured":"Khodakovsky, A., Schr\u00f6der, P., and Swelddens, W. (, January July,). Progressive geometry compression. Proceedings of the 27th annual conference on Computer graphics and interactive techniques, New York, NY, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1006\/acha.1996.0015","article-title":"The lifting scheme: A custom-design construction of biorthogonal wavelets","volume":"3","author":"Sweldens","year":"1996","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Schr\u00f6der, P., and Sweldens, W. (,  1995). Spherical wavelets: efficiently representing functions on the sphere. Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, New York, NY, USA.","DOI":"10.1145\/218380.218439"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1109\/TVCG.2004.1272731","article-title":"Generalized B-spline subdivision-surface wavelets for geometry compression","volume":"10","author":"Bertram","year":"2004","journal-title":"IEEE Trans. Visual. Comput. Graph."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bertram, M., Duchaineau, M., Hamann, B., and Joy, K. (2000, January October). Bicubic subdivision-surface wavelets for large-scale isosurface representation and visualization. Proceedings of the conference on Visualization \u201900, Salt Lake City, UT, USA.","DOI":"10.1109\/VISUAL.2000.885720"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s00607-003-0044-0","article-title":"Biorthogonal loop-subdivision wavelets","volume":"72","author":"Bertram","year":"2004","journal-title":"Computing"},{"key":"ref_12","unstructured":"Li, D., Qin, K., and Sun, H. (2004, January October). Unlifted Loop Subdivision Wavelets. Proceedings of 12th Pacific Conference on the Computer Graphics and Applications, Seoul, Korea."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1007\/s00371-006-0074-7","article-title":"Efficient wavelet construction with Catmull-Clark subdivision","volume":"22","author":"Wang","year":"2006","journal-title":"Vis. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1109\/TVCG.2007.1031","article-title":"3\n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t        -Subdivision-based biorthogonal wavelets","volume":"13","author":"Wang","year":"2007","journal-title":"IEEE Trans. Visual. Comput. Graph."},{"key":"ref_15","unstructured":"Kar\u010diauskas, K., Myles, A., and Peters, J. (2006, January June). A C2 polar jet subdivision. Proceedings of the 4th Eurographics symposium on Geometry processing, Cagliari, Sardinia, Italy."},{"key":"ref_16","first-page":"1","article-title":"Bicubic polar subdivision","volume":"26","author":"Peters","year":"2007","journal-title":"ACM Trans. Graph."},{"key":"ref_17","unstructured":"Myles, A., Karciauskas, K., and Peters, J. (2,, January October). Extending catmull-clark subdivision and PCCM with polar structures. Proceedings of the 15th Pacific Conference on Computer Graphics and Applications, Maui, Hawaii, USA."},{"key":"ref_18","unstructured":"Myles, A., and Peters, J. (2009). SIGGRAPH 2009 Papers, ACM."},{"key":"ref_19","unstructured":"Khodakovsky, A., Schroder, P., and Sweldens, W. (, January July,). Progressive geometry compression. Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, New York, NY, USA."},{"key":"ref_20","unstructured":"Zhao, C., Wang, H., Sun, H., and Qin, K. (2008, January July). Polar subdivision based wavelet analysis. Proceedings of the 2008 International Conference on Computer Graphics and Virtual Reality (Computer Graphics International 2008), Las Vegas, NV, USA. 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