[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81913-en":3,"doc-seo-81913-105":31,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81913,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Towards Fully Dynamic Omnitrees: Moment-Conserving Anisotropic Compression With Wavelets","Omnitrees are a flexible space partitioning tree combining strengths of octrees and k-d trees for efficient encoding of anisotropic refinements, targeted to anisotropic and high-dimensional applications. The paper introduces new operations on omnitree encodings that enable coarsening in addition to refinement, yielding fully adaptive compression. By integrating wavelets, it conserves moments by construction and uses wavelet coefficients as local refinement priorities. Algorithms for coarsening and downsplit guide adaptation, validated on 3D shape datasets and a cloud density field versus OpenVDB, showing major storage savings and low visual loss under lossy compression.","arXiv :2607 .0488 1v 1 [ cs .DS] 6 Jul 2026  \nTowards Fully Dynamic Omnitrees: Moment-Conserving Anisotropic Compression  \nWith Wavelets  \nTheresa Pollinger, Masado Ishii, Jens Domke RIKEN Center for Computational Science, Kobe, Japan  \nAbstract  \nRecently, omnitrees were introduced as a flexible space partitioning tree that improves upon the benefits of both octrees and k-d trees: Omnitrees’ efficient encoding of anisotropic refinements holds particular interest for applications with anisotropic features and high dimensionality. These include, but are not limited to, computer graphics, databases, machine learning, and physics simulations. The present paper defines new operations on the omnitree encoding that extend its capabilities from the existing refinement to also include coarsening and therefore fully adaptive compression. It demonstrates natural integration of omnitrees with wavelets, which conserves moments of the stored function by design. For omnitrees, the wavelet coefficients can be interpreted as local refinement priorities, which can be used to guide the adaptation process. We derive algorithms for coarsening and downsplit that are guided by wavelet coefficients, and show their application to a large dataset of 3D shapes, as well as the continuous-valued density field of a cloud. The comparison to OpenVDB, a widely-used data structure for sparse volumetric data in computer graphics, enables a demonstration of the practical benefits of omnitrees even for moderately anisotropic three-dimensional data. Compared to OpenVDB, objects can be stored using up to 28 × less space, and asymptotically show savings that exceed theoretical expectations. Using lossy compression, the cloud dataset can be compressed by ≈ 5 × compared to OpenVDB, with negligible loss of visual quality. This demonstrates the potential of omnitrees for efficient storage and processing, and motivates further research into their applications in various domains.  \n1 Introduction  \nOmnitrees are introduced by [29] as a type of flexible space partitioning tree that improves upon the benefits of both octrees and bintrees (k-d trees) . This is achieved by efficiently encoding anisotropic refinements. Whereas octrees bisect all dimensions of a volume at once, and bintrees bisect one dimension at a time,  \nomnitrees bisect a locally optimal subset of dimensions on any level. It is shown in [29] that, for the same error threshold, omnitree discretizations of anisotropic problems are smaller than their octree counterparts by a power law, Nomni ∼(Noct )γ , γ \u003C 1, where N is the number of cells in the discretization. That is, Nis improved by faster than a constant factor. Omnitrees also generalize bintrees to allow any dimension to be split on any level without a regular period. The ability for multiple dimensions to be split at once further distills the tree into a shallower encoding with shorter traversal depth.  \nThe present paper defines new operations on this omnitree encoding that enable not only top-down refinement but also fully adaptive compression. The compression scheme integrates naturally with wavelet bases: Wavelet coefficients below a threshold indicate that coarsening is possible, and this paper describes how the updated tree and wavelet coefficients can be computed. For the purposes of this work, we select Haar wavelets as one of the simplest possible multiscale bases. Haar wavelet compression conserves the mass of the function, and higher orders of conservation (momentum, energy, ...) are possible with a choice of higher-order wavelets. By comparing with OpenVDB—a state-ofthe-art adaptive hierarchical storage format—this paper provides a validation of increased compression and even approximation rate for three-dimensional objects.  \nThese coarsening operations represent a major step towards a fully dynamic omnitree data structure that can be used for adaptive storage and processing of anisotropic data in various domains, including computer graph","cbCaicYlsjAIxKQN","https://ap.wps.com/l/cbCaicYlsjAIxKQN","pdf",1237668,4,1,29,"English","en",105,"# Abstract\n# Introduction\n# Related Work","[{\"question\":\"What are omnitrees and why are they useful for anisotropic data?\",\"answer\":\"Omnitrees are a flexible space partitioning tree that efficiently encodes anisotropic refinements, improving upon octrees and k-d trees. Their design supports anisotropic features and high dimensionality.\"},{\"question\":\"How does the method achieve fully adaptive compression in omnitrees?\",\"answer\":\"The paper defines new operations on omnitree encodings that add coarsening alongside refinement. This allows the structure to adapt across scales using wavelet-driven decisions.\"},{\"question\":\"How are wavelets used to guide the adaptation process and what conservation properties are preserved?\",\"answer\":\"Wavelet coefficients below a threshold indicate coarsening is possible and are interpreted as local refinement priorities. The integration conserves mass by design, with higher-order wavelets enabling higher moment conservation.\"}]","Towards Fully Dynamic Omnitrees: Moment-Conserving Anisotropic Compression With Wavelets | PDF",1784177013,73,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"towards-fully-dynamic-omnitrees-moment-conserving-anisotropic-compression-with-wavelets","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":20},"https://docshare.wps.com/document/towards-fully-dynamic-omnitrees-moment-conserving-anisotropic-compression-with-wavelets/81913/",{"url":53,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What are omnitrees and why are they useful for anisotropic data?","Question",{"text":76,"@type":77},"Omnitrees are a flexible space partitioning tree that efficiently encodes anisotropic refinements, improving upon octrees and k-d trees. Their design supports anisotropic features and high dimensionality.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the method achieve fully adaptive compression in omnitrees?",{"text":81,"@type":77},"The paper defines new operations on omnitree encodings that add coarsening alongside refinement. This allows the structure to adapt across scales using wavelet-driven decisions.",{"name":83,"@type":74,"acceptedAnswer":84},"How are wavelets used to guide the adaptation process and what conservation properties are preserved?",{"text":85,"@type":77},"Wavelet coefficients below a threshold indicate coarsening is possible and are interpreted as local refinement priorities. The integration conserves mass by design, with higher-order wavelets enabling higher moment conservation.","https://schema.org",{"og:url":53,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]