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In this paper, we present an approach that blends state\u2010space modeling with Wasserstein geometry, enabling a more fluid and coherent representation of dynamic scenes. We introduce a State Consistency Filter that merges prior predictions with the current observations, enabling Gaussians to maintain coherent trajectories over time. We also employ Wasserstein Consistency Constraint to ensure smooth, consistent updates of Gaussian parameters, reducing motion artifacts. Lastly, we leverage Wasserstein geometry to capture both translational motion and shape deformations, creating a more geometrically consistent model for dynamic scenes. Our approach models the evolution of Gaussians along geodesics on the manifold of Gaussian distributions, achieving smoother, more realistic motion and stronger temporal coherence. Experimental results show consistent improvements in rendering quality and efficiency.<\/jats:p>\n                  <jats:p>\n                    (see\n                    <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/www.acm.org\/publications\/class-2012\">https:\/\/www.acm.org\/publications\/class-2012<\/jats:ext-link>\n                    )\n                  <\/jats:p>","DOI":"10.1111\/cgf.70271","type":"journal-article","created":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:10:18Z","timestamp":1760184618000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Gaussians on their Way: Wasserstein\u2010Constrained 4D Gaussian Splatting with State\u2010Space Modeling"],"prefix":"10.1111","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8294-1274","authenticated-orcid":false,"given":"J.","family":"Deng","sequence":"first","affiliation":[{"name":"Communication University of China  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-7744-6246","authenticated-orcid":false,"given":"P.","family":"Shi","sequence":"additional","affiliation":[{"name":"Communication University of China  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Y.","family":"Luo","sequence":"additional","affiliation":[{"name":"Imperial College London  London UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-1995-8395","authenticated-orcid":false,"given":"Q.","family":"Li","sequence":"additional","affiliation":[{"name":"Communication University of China  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2025,10,11]]},"reference":[{"key":"e_1_2_7_2_2","volume-title":"Gradient flows: in metric spaces and in the space of probability measures","author":"Ambrosio L.","year":"2008"},{"key":"e_1_2_7_3_2","doi-asserted-by":"crossref","unstructured":"AttalB. 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