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However, the MS model, which operates at the pixel level of the images, faces challenges when dealing with medical images with low contrast or unclear edges. In this paper, we begin by using a feature extractor to capture high-dimensional deep features that contain more comprehensive semantic information than pixel-level data alone. Inspired by the MS model, we develop a variational model that incorporates threshold dynamics (TD) regularization for segmenting each feature. We obtain the final segmentation result for the original image by assembling segmentation results of all the features. This process results in MS-MGNet, a lightweight trainable segmentation network with a similar architecture to many encoder\u2013decoder networks. The intermediate layers of MS-MGNet are designed by unrolling the numerical scheme based on the multigrid method for solving the variational model. We provide interpretability for the encoder\u2013decoder architecture by elucidating the roles of each layer and offering explanations of the underlying mathematical models. By incorporating the TD regularizer, we integrate spatial priors from the variational models into the network architecture, resulting in better segmentation results with smoother edges and a certain robustness to noise. Compared to some relevant methods, experimental results on the selected data sets with low contrast or unclear edges show that the proposed method can achieve better segmentation performance with fewer parameters, even when trained on smaller data sets.<\/jats:p>","DOI":"10.1137\/23m1577663","type":"journal-article","created":{"date-parts":[[2024,5,22]],"date-time":"2024-05-22T06:33:31Z","timestamp":1716359611000},"page":"1007-1039","source":"Crossref","is-referenced-by-count":5,"title":["Assembling a Learnable Mumford\u2013Shah Type Model with Multigrid Technique for Image Segmentation"],"prefix":"10.1137","volume":"17","author":[{"given":"Junying","family":"Meng","sequence":"first","affiliation":[{"name":"Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875 China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5796-3527","authenticated-orcid":true,"given":"Weihong","family":"Guo","sequence":"additional","affiliation":[{"name":"Co-corresponding author.\u00a0Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8697-3089","authenticated-orcid":true,"given":"Jun","family":"Liu","sequence":"additional","affiliation":[{"name":"Co-corresponding author.\u00a0Laboratory of Mathematics and Complex Systems Ministry of Education of China, School of Mathematical Sciences, Beijing Normal University, Beijing 100875 China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mingrui","family":"Yang","sequence":"additional","affiliation":[{"name":"Lerner Research Institute, Cleveland Clinic, Cleveland, OH 44195 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,5,22]]},"reference":[{"key":"ref1","doi-asserted-by":"crossref","unstructured":"M. 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