[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85934-en":3,"doc-seo-85934-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85934,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Grassmannian Splatting I: Moving Rank-2 Spacetime Surfels for Dynamic Scene Rendering","Grassmannian splatting presents a dynamic scene representation using Gaussian primitives supported on 3-planes in spacetime R4, which generically become spatial rank-2 surfels under time conditioning. Each primitive is parameterized by a unit normal n and an unconstrained factor L, producing a 4D covariance whose projector depends on the Grassmannian Gr(3,4) embedding. Conditioning yields a translating disk surfel per frame without deformation fields, using a closed-form motion model and a standard 3D Gaussian splatting rasterizer interface. On 17 scenes from MonoDyGauBench, training is fastest versus baselines while ranking second in PSNR, MS-SSIM, and LPIPS.","arXiv :2607 . 10489v1 [ cs .CV] 11 Jul 2026  \nGRASSMANNIAN SPLATTING I:  \nMOVING RANK-2 SPACETIME SURFELS FOR DYNAMIC SCENE  \nRENDERING  \nAARON MAURICE BERMAN AND SHANTANU DAVE  \nAbstract . We introduce Grassmannian splatting, a dynamic scene representation whose primitives are Gaussians supported on 3-planes in spacetime R4 : generically, spatial 2-planes in uniform translation along their normals. Each primitive carries a unit normal n ∈ S3 /{±1}  Gr(3 , 4) and an unconstrained factor L ∈ R4×3, with covariance  \nΣ4D = (PnL)(PnL)T , Pn = I − nnT .  \nFor generic L and n  ±e0 , conditioning on time returns a rank-2 surfel at every frame. The normal of the disk and its velocity along that normal are read off from n; the disk shape and the tangential drift of its center are set by L. Existing native 4D Gaussian splatting methods [1, 2] slice full-rank spacetime covariances, so their per-frame primitive is a volumetric ellipsoid; since conditioning lowers rank by exactly one, a rank-2 surfel in the slice requires a rank-3 spacetime covariance, and the parameterization above realizes exactly these. The motion model is closed form, i.e. no deformation field is learned, and no custom CUDA is required: the conditioned disk feeds a standard 3DGS rasterizer through its precomputed-covariance interface. A soft clamp in the Schur denominator regularizes the static orientation and continuously bridges rank-3 static and rank-2 dynamic behavior, so static and moving primitives form a single continuous family. On the  \n17 HyperNeRF scenes of MonoDyGauBench, training is fastest among all compared methods (4.9 to 5.6 times faster than the strongest quality baselines), while ranking second in PSNR, MS-SSIM, and LPIPS. Code: [https://github.com/PaulCelanCoding/grassmannian-splatting](https://github.com/PaulCelanCoding/grassmannian-splatting)  \n1. Introduction  \nThe Grassmannian Gr(3 , 4) is a moduli space of constant-velocity plane motions: every 3-plane in spacetime R4 , with the single exception of the static slice [e0], is a spatial 2-plane in uniform translation along its normal (Lemma 2.2) . We take these motions as the primitives of dynamic scene rendering: each splat is a Gaussian supported on such a plane. Slicing it at any time yields arank-2 surfel whose normal and normal velocity are read off from the underlying point of Gr(3 , 4) . Concretely, we parameterize the plane by its unit normal n ∈ S3 /{±1} and an unconstrained factor L ∈ R4×3, and set  \n(1) Σ4D = (PnL)(PnL)T , Pn = I − nnT .  \nSince Σ4D depends on n only through the projector Pn = P−n, it is a well-defined function on Gr(3 , 4)  S3 /{±1}; the projector is the standard embedding of the Grassmannian into the symmetric matrices. The covariance is PSD of rank at most 3 with kernel direction n (Lemma 3.2), and for generic L and n  ±e0 , conditioning on time yields a spatial covariance of rank exactly 2: a disk in R3 , at every frame, with no per-frame flatness constraint and no learned deformation field. The primitive fills a specific cell of the design space. Static splatting offers a volumetric primitive, the rank-3 ellipsoid of 3DGS [3], and a planar one, the rank-2 surfel of 2DGS [4] . Dynamic splatting adds the temporal axis in three ways. Deformation- or tracking-based methods carry a primitive in a canonical frame and warp it into each frame [5 , 6]; among these, space-time 2D Gaussian splatting (ST-2DGS) [7] warps rank-2 surfels. Per-primitive trajectory methods attach an explicit  \nDate: July 2026 .  \n2 AARON MAURICE BERMAN AND SHANTANU DAVE  \nFigure 1 . Primitive types. Top (volumetric): (a) the 3D ellipsoid of 3DGS [3];  \n(b) its native 4D analogue [1 , 2], whose spacetime slice at t = t0 is again a rank-3 blob. Bottom (planar): (c) the rank-2 disk of 2DGS [4], with normal n; (d) ours, a Gaussian supported on a 3-plane in R4 with normal n ∈ S3 /{±1}  Gr(3 , 4), whose slice at t = t0 is a rank-2 surfel that translates along its spatial normal as t0 advances.  \nPanels (","cbCaie0l9aW7Bomp","https://ap.wps.com/l/cbCaie0l9aW7Bomp","pdf",1037529,1,15,"English","en",105,"# Introduction\n## Grassmannian primitives and surfel conditioning\n## Comparison to existing native and trajectory-based methods\n## Rank structure, parameterization, and redundancy\n## Motion model and rendering pipeline","[{\"question\":\"What are the core primitives used in Grassmannian splatting?\",\"answer\":\"The method represents scenes with Gaussian primitives supported on 3-planes in spacetime R4. Each primitive is associated with a unit normal n and a factor L that defines its 4D covariance.\"},{\"question\":\"How does conditioning on time produce the rank-2 surfel?\",\"answer\":\"Conditioning on a time coordinate applies a rank-reducing Schur identity so that a rank-3 spacetime covariance becomes a rank-2 object in the slice. For generic parameters, this yields a translating disk surfel at every frame.\"},{\"question\":\"How does this approach differ from existing native 4D Gaussian splatting methods?\",\"answer\":\"Native full-rank spacetime methods yield per-frame volumetric ellipsoids because their spacetime covariance remains higher rank after conditioning. Grassmannian splatting uses rank-deficient, native spacetime covariances whose slices are surfels, and it avoids learning deformation fields or requiring custom CUDA.\"}]",1784207253,38,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"grassmannian-splatting-i-moving-rank-2-spacetime-surfels-for-dynamic-scene-rendering","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/grassmannian-splatting-i-moving-rank-2-spacetime-surfels-for-dynamic-scene-rendering/85934/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are the core primitives used in Grassmannian splatting?","Question",{"text":75,"@type":76},"The method represents scenes with Gaussian primitives supported on 3-planes in spacetime R4. Each primitive is associated with a unit normal n and a factor L that defines its 4D covariance.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does conditioning on time produce the rank-2 surfel?",{"text":80,"@type":76},"Conditioning on a time coordinate applies a rank-reducing Schur identity so that a rank-3 spacetime covariance becomes a rank-2 object in the slice. For generic parameters, this yields a translating disk surfel at every frame.",{"name":82,"@type":73,"acceptedAnswer":83},"How does this approach differ from existing native 4D Gaussian splatting methods?",{"text":84,"@type":76},"Native full-rank spacetime methods yield per-frame volumetric ellipsoids because their spacetime covariance remains higher rank after conditioning. Grassmannian splatting uses rank-deficient, native spacetime covariances whose slices are surfels, and it avoids learning deformation fields or requiring custom CUDA.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]