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A sparse control\u2010point parameterization provides a compact, smooth deformation representation while reducing memory and improving\u00a0stability.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Purpose<\/jats:title>\n                    <jats:p>This work investigates the required control points for learning\u2010based registration network development. In particular, as sparse as  control points are configured and compared with alternative approaches, including those using scattered control points and displacements sampled at every voxel, that is, dense displacement\u00a0fields.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Method<\/jats:title>\n                    <jats:p>We present GridReg, a learning\u2010based registration framework that replaces dense voxel\u2010wise decoding with displacement predictions at a sparse grid of control points. This design substantially cuts the parameter count and memory while retaining registration accuracy. Multiscale 3D encoder feature maps are flattened into a 1D token sequence with positional encoding to retain spatial context. The model then predicts a sparse gridded deformation field using a cross\u2010attention module: Each control point attends to encoder tokens within its local grid neighborhood to estimate its displacement, which is subsequently interpolated to a dense field. We further introduce grid\u2010adaptive training, enabling an adaptive model to operate at multiple grid sizes at inference without\u00a0retraining.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Results<\/jats:title>\n                    <jats:p>This work quantitatively demonstrates the benefits of using sparse grids. Using three data sets for registering prostate gland, pelvic organs and neurological structures, the experimental results suggest a much improved computational efficiency, due to the prediction of sparse\u2010grid\u2010sampled displacements. Alternatively, the superior registration performance was obtained using the proposed approach, with the similiar or less compute cost, compared with existing algorithms that predict DDFs (e.g., VoxelMorph\/TransMorph) or displacements sampled on scattered key points (KeyMorph).<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Conclusion<\/jats:title>\n                    <jats:p>\n                      We conclude that predicting sparsely gridded displacements provides reduced computational cost and\/or improved performance, independent of the encoder architecture, and can be readily implemented. Therefore, GridReg should potentially be considered for many registration tasks with adaptive grid sizes. The code is available via\n                      <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/git@github.com:yanwenCi\/GridReg.git\">git@github.com:yanwenCi\/GridReg.git<\/jats:ext-link>\n                      .\n                    <\/jats:p>\n                  <\/jats:sec>","DOI":"10.1002\/mp.70343","type":"journal-article","created":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T16:30:37Z","timestamp":1771518637000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the degrees of freedom of gridded control points in learning\u2010based medical image registration"],"prefix":"10.1002","volume":"53","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3962-5994","authenticated-orcid":false,"given":"Wen","family":"Yan","sequence":"first","affiliation":[{"name":"Department of Medical Physics and Biomedical Engineering UCL Hawkes Institute, University College London London UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4401-5311","authenticated-orcid":false,"given":"Qianye","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Medical Physics and Biomedical Engineering UCL Hawkes Institute, University College London London UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9589-7177","authenticated-orcid":false,"given":"Yipei","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Medical Physics and Biomedical Engineering UCL Hawkes Institute, University College London London UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1014-0870","authenticated-orcid":false,"given":"Shonit","family":"Punwani","sequence":"additional","affiliation":[{"name":"Centre for Medical Imaging, Division of Medicine University College London London UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4230-0338","authenticated-orcid":false,"given":"Mark","family":"Emberton","sequence":"additional","affiliation":[{"name":"Division of Surgery and Interventional Science University College London London UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0011-9792","authenticated-orcid":false,"given":"Vasilis","family":"Stavrinides","sequence":"additional","affiliation":[{"name":"Cancer Institute, Urology Department, UCL Hospital University College London London UK"},{"name":"Radiology Department Imperial College Healthcare London UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4902-0486","authenticated-orcid":false,"given":"Yipeng","family":"Hu","sequence":"additional","affiliation":[{"name":"Department of Medical Physics and Biomedical Engineering UCL Hawkes Institute, University College London London UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2916-655X","authenticated-orcid":false,"given":"Dean","family":"Barratt","sequence":"additional","affiliation":[{"name":"Department of Medical Physics and Biomedical Engineering UCL Hawkes Institute, University College London London UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2026,2,19]]},"reference":[{"key":"e_1_2_11_2_1","doi-asserted-by":"publisher","DOI":"10.1088\/0031-9155\/46\/3\/201"},{"key":"e_1_2_11_3_1","first-page":"447","volume-title":"Handbook of Medical Imaging","author":"Fitzpatrick JM","year":"2000"},{"key":"e_1_2_11_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2019.2897538"},{"key":"e_1_2_11_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.media.2022.102615"},{"key":"e_1_2_11_6_1","volume-title":"Medical Imaging With Deep Learning","author":"Evan MY","year":"2022"},{"key":"e_1_2_11_7_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2021.3116879"},{"key":"e_1_2_11_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.media.2024.103385"},{"key":"e_1_2_11_9_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2022.3187873"},{"key":"e_1_2_11_10_1","unstructured":"elastix team.Elastix the Manual. 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