{"url":"/method/m-s-structure","slug":"m-s-structure","name":"M-S structure","full_name":"Motion-Separable backbone structure","full_name_withheld":false,"description_markdown":"Based on the theoretical analyses in RAN paper, a novel multi-scale backbone structure is designed in the paper. This structure enables the network to efficiently predict motion patterns with larger separable upper bounds by using optimized dilation convolution on high-resolution feature maps, while maintaining a capturable range of motion with low computational complexity.\r\n\r\nTo quantify the network's capacity for large deformation capturing, the accessible motion capture range is defined as:\r\n\r\n**Definition 1: Accessible Motion Range**\r\n\r\nThe radius of capture range of the $k^{\\text{th}}$-level registration by the registration module$\\mathcal{R}_k$ is defined as the smallest upper bound of its accessible Deformation Displacement Field:\r\n\r\n$$\r\na_k := \\min_{\\mathbf{x}}(\\{\\sup(\\|\\varphi_{k}[\\mathbf{x}]\\|_{\\infty})\\})\r\n$$\r\n\r\nwhere $\\|\\cdot\\|_{\\infty}$ denotes the L-$\\infty$ norm of a vector, $\\sup(\\cdot)$ denotes the supremum or the maximum value of a given function with varying inputs and trainable weights of networks, and $\\mathbf{x}$ denotes one coordinate entry of the images or Deformation Displacement Fields.\r\n\r\n\r\nTo quantify the Degree-of-Freedom limitation in the discontinuity of the estimated Deformation Displacement Field, we define the separability of the predicted motion:\r\n\r\n\r\n**Definition 2: Separability Bottleneck of Predicted Motion**\r\n\r\nThe motion separability bottleneck is defined as the minimum value of the upper bound of the Chebyshev difference of a network's predicted DDF $\\phi$ between two locations $\\mathbf{x}, \\mathbf{y} \\in \\mathbb{Z}^d$ with the specific Chebyshev distance $p \\in \\mathbb{Z}^d$:\r\n\r\n$$\r\n\\Delta_\\infty(p) := \\min_{\\mathbf{x}, \\mathbf{y}}\\left\\{\\sup(\\|\\phi[\\mathbf{x}] - \\phi[\\mathbf{y}]\\|_{\\infty}) : \\|\\mathbf{x} - \\mathbf{y}\\|_{\\infty} = p\\right\\}\r\n$$\r\n\r\nwhere $p$ denotes the L-$\\infty$ distance between the two pixels.\r\n\r\n\r\n**Theorem: Regional Dependency** \r\n\r\nThe upper boundary of motion difference is related to $a_k$ and $p_k$:\r\n\r\n$$\r\n\\begin{align*}\r\n\\forall \\mathbf{x}, \\mathbf{y} \\in \\mathbb{Z}^d, \\|\\mathbf{x} - \\mathbf{y}\\|_\\infty \\geq p_{k''} + 2\\sum_{k'=k''+1}^{k} a_{k'}, &\\quad \\sup(\\|\\phi_{k}[\\mathbf{x}] - \\phi_{k}[\\mathbf{y}]\\|_\\infty) \\geq 2\\sum_{k'=k''}^{k} a_{k'}; \\\\\r\n\\exists \\mathbf{x}, \\mathbf{y} \\in \\mathbb{Z}^d, \\|\\mathbf{x} - \\mathbf{y}\\|_\\infty < p_{k''-1} + 2\\sum_{k'=k''}^{k} a_{k'}, &\\quad \\sup(\\|\\phi_{k}[\\mathbf{x}] - \\phi_{k}[\\mathbf{y}]\\|_\\infty) = 2\\sum_{k'=k''}^{k} a_{k'};\r\n\\end{align*}\r\n$$\r\n\r\nwhere $k'', k,$ denote two recursive numbers satisfying $0 \\leq k'' < k$, and $\\mathbf{x}, \\mathbf{y}$ denote two coordinate entries of images or DDFs.\r\n\r\n\r\nThus a **Motion-Separable structure** is designed with the upsampled feature maps processed by the corresponding atrous convolution layers.","description_state":"present","introduced_year":null,"introduced_by":{"title":"Residual Aligner-based Network (RAN): Motion-separable structure for coarse-to-fine discontinuous deformable registration","paper":"/paper/residual-aligner-based-network-ran-motion","first_author":"Jian-Qing Zheng","n_authors":5,"url_abs":null,"archive_paper_url":"https://paperswithcode.com/paper/residual-aligner-based-network-ran-motion"},"source":{"url":"https://doi.org/10.1016/j.media.2023.103038","title":"Residual Aligner-based Network (RAN): Motion-separable structure for coarse-to-fine discontinuous deformable registration","url_on_a_paper_host":true},"code_snippet_url":null,"code_snippet_url_on_a_code_host":false,"categories":[{"area":"Computer Vision","area_id":"computer-vision","collection":"Backbone Architectures","url":"/methods/category/backbone-architectures","pwc_aliases":[]}],"n_papers_tagged":1,"archive_num_papers":1,"papers_newest_first":[{"paper":"/paper/residual-aligner-based-network-ran-motion","title":"Residual Aligner-based Network (RAN): Motion-separable structure for coarse-to-fine discontinuous deformable registration","date":"2023-11-21","arxiv_id":null,"n_code_links":1,"syntology":null}],"papers_shown":1,"tasks":[{"task":"/task/computed-tomography-ct","name":"Computed Tomography (CT)","papers":1},{"task":"/task/deformable-medical-image-registration","name":"Deformable Medical Image Registration","papers":1},{"task":"/task/image-registration","name":"Image Registration","papers":1},{"task":"/task/medical-image-registration","name":"Medical Image Registration","papers":1},{"task":"/task/unsupervised-image-registration","name":"Unsupervised Image Registration","papers":1}],"tasks_shown":5,"n_tasks":5,"usage_by_year":[{"year":"2023","papers":1}],"row_source":"methods_table","archive":{"source":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","archive_url":"https://paperswithcode.com/method/m-s-structure"},"syntology_read_at":"2026-09-24T18:15:14+00:00"}