Journal of Mathematical Imaging and Vision最新文献

{"title":"Diffusion-Shock PDEs for Deep Learning on Position-Orientation Space.","authors":"Finn M Sherry, Kristina Schaefer, Remco Duits","doi":"10.1007/s10851-026-01291-z","DOIUrl":"10.1007/s10851-026-01291-z","url":null,"abstract":"<p><p>We extend regularised diffusion-shock (RDS) filtering from Euclidean space <math> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </math> (Schaefer and Weickert in J Math Imaging Vis 66:447-463, 2024. 10.1007/s10851-024-01175-0) to position-orientation space <math> <mrow><msub><mi>M</mi> <mn>2</mn></msub> <mo>≅</mo> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> <mo>×</mo> <msup><mi>S</mi> <mn>1</mn></msup> </mrow> </math> . This has numerous advantages, e.g. making it possible to enhance and inpaint crossing structures, since they become disentangled when lifted to <math><msub><mi>M</mi> <mn>2</mn></msub> </math> . We create a version of the algorithm using gauge frames to mitigate issues caused by lifting to a finite number of orientations. This leads us to study generalisations of diffusion, since the gauge frame diffusion is not generated by the Laplace-Beltrami operator. RDS filtering compares favourably to existing techniques such as total roto-translational variation (TR-TV) flow (Smets et al. in J Math Imaging Vis 63:237-262, 2021. 10.1007/s10851-020-00991-4; Chambolle and Pock in Numer Math 142:611-666, 2019. 10.1007/s00211-019-01026-w), NLM (Buades et al. in Image Process On Line 1:208-212, 2011. 10.5201/ipol.2011.bcm_nlm), and BM3D (Dabov et al. in Trans Image Process 16:2080-2095, 2007. 10.1109/TIP.2007.901238) when denoising images with crossing structures, particularly if they are segmented. Furthermore, we see that <math><msub><mi>M</mi> <mn>2</mn></msub> </math> RDS inpainting is indeed able to restore crossing structures, unlike <math> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </math> RDS inpainting. In addition to the contributions of our SSVM submission (Sherry et al. in: Bubba, Gaburro, Gazzola, Papafitsoros, Pereyra, Schönlieb (eds) 10th International Conference on Scale Space and Variational Methods in Computer Vision II (SSVM), vol. 15668, pp. 205-217. Springer, Cham, 2025. 10.1007/978-3-031-92369-2_16), in this extended work we provide new theorical results and automate RDS filtering by integrating it into a geometric deep learning framework. Regarding our theoretical contributions, we prove that our generalised diffusions are still well posed, smoothing, and analytic. We developed an RDS filtering PDE layer for the PDE-CNN and PDE-G-CNN deep learning frameworks, using a novel gating mechanism. We show that these new RDS PDE layers can be beneficial in various impainting and denoising tasks.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"68 3","pages":"17"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13144200/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147838994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Connected Components on Lie Groups and Applications to Multi-Orientation Image Analysis.","authors":"Nicky J van den Berg, Olga Mula, Leanne Vis, Remco Duits","doi":"10.1007/s10851-026-01287-9","DOIUrl":"10.1007/s10851-026-01287-9","url":null,"abstract":"<p><p>We develop and analyze a new algorithm to find the connected components of a compact set <i>I</i> from a Lie group <i>G</i> endowed with a left-invariant Riemannian distance. For a given <math><mrow><mi>δ</mi> <mo>></mo> <mn>0</mn></mrow> </math> , the algorithm finds the largest cover of <i>I</i> such that all sets in the cover are separated by at least distance <math><mi>δ</mi></math> . We call the sets in the cover the <math><mi>δ</mi></math> -connected components of I (closely related to <math><mover><mtext>C</mtext> <mo>ˇ</mo></mover> </math> ech complexes of radius <math><mrow><mi>δ</mi> <mo>/</mo> <mn>2</mn></mrow> </math> ). The grouping relies on an iterative procedure involving morphological dilations with Hamilton-Jacobi-Bellman kernels on <i>G</i> and notions of <math><mi>δ</mi></math> -thickened sets. We prove that the algorithm converges in finitely many iteration steps. We find the optimal value for <math><mi>δ</mi></math> using persistence diagrams. We also propose to use specific affinity matrices. This allows for grouping of <math><mi>δ</mi></math> -connected components based on their local proximity and alignment. Among the many different applications of the algorithm, in this article, we focus on illustrating that the method can efficiently identify (possibly overlapping) branches in complex vascular trees on retinal images. This is done by applying an orientation score transform to the images that allows us to view them as functions from <math> <mrow><msub><mi>L</mi> <mn>2</mn></msub> <mrow><mo>(</mo> <mi>G</mi> <mo>)</mo></mrow> </mrow> </math> where <math><mrow><mi>G</mi> <mo>=</mo> <mi>S</mi> <mi>E</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo></mrow> </math> , the Lie group of roto-translations. By applying our algorithm in this Lie group, we illustrate that we obtain <math><mi>δ</mi></math> -connected components that differentiate between crossing structures and that group well-aligned, nearby structures. This contrasts standard connected component algorithms in <math> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </math> .</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"68 2","pages":"11"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13021705/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147574293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"MG-SpaIR: Multi-Grade Sparse-Guided Implicit Representation for Training-Data-Free Image Restoration.","authors":"Jianmin Liao, Lei Huang, Ronglong Fang, Ashley Prater-Bennette, Lixin Shen, Yuesheng Xu","doi":"10.1007/s10851-026-01329-2","DOIUrl":"10.1007/s10851-026-01329-2","url":null,"abstract":"<p><p>MG-SpaIR is a training-data-free framework for restoring a clean image from a single observation corrupted by a mixture of blur, downsampling, noise, and missing pixels. Building on implicit neural representations (INRs), we introduce a multi-grade residual hierarchy that progressively refines the reconstruction from low to high spatial frequencies across grades, improving representational fidelity and mitigating spectral limitations. To stabilize reconstruction optimization and suppress INR-induced artifacts, we further propose an explicit sparse proximal regularization (e.g., <math><msub><mi>ℓ</mi> <mn>0</mn></msub> </math> type) applied directly in the high-resolution image domain, which discourages spurious high-frequency patterns while preserving sharp structures. The resulting optimization is solved efficiently via a multi-grade proximal alternating scheme, and we establish convergence guarantees for the associated updates under standard regularity conditions. Experiments on mixed-degradation benchmarks demonstrate that MG-SpaIR consistently outperforms strong training-data-free baselines such as Deep Image Prior, providing a stable, interpretable, and data-efficient alternative to conventional learning-based restoration methods.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"68 4","pages":"48"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13423958/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148653736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Product-of-Gaussian-mixture diffusion models for joint nonlinear MRI reconstruction.","authors":"Laurenz Nagler, Martin Zach, Thomas Pock","doi":"10.1007/s10851-026-01297-7","DOIUrl":"10.1007/s10851-026-01297-7","url":null,"abstract":"<p><p>Recently, diffusion models have attracted considerable attention for magnetic resonance image reconstruction due to their high sample quality. However, most existing methods rely on large networks with opaque time conditioning mechanisms and require offline coil sensitivity estimation. This results in limited interpretability of the reconstruction process and reduced flexibility in the acquisition setup. To address these limitations, we jointly reconstruct the image and the coil sensitivities by combining the parameter-efficient product-of-Gaussian-mixture diffusion model as an image prior with a classical smoothness prior on the coil sensitivities. The proposed method is fast and robust to both contrast and anatomical distribution shifts as well as changing k-space trajectories. Finally, we propose a more expressive parameterization of the image prior which improves results in denoising and magnetic resonance image reconstruction.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"68 3","pages":"30"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13241435/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148199201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Linear Optimal Transport Subspaces for Point Set Classification.","authors":"Mohammad Shifat-E-Rabbi, Naqib Sad Pathan, Shiying Li, Yan Zhuang, Abu Hasnat Mohammad Rubaiyat, Gustavo K Rohde","doi":"10.1007/s10851-025-01261-x","DOIUrl":"10.1007/s10851-025-01261-x","url":null,"abstract":"<p><p>Learning from point sets is an essential component in many computer vision and machine learning applications. Native, unordered, and permutation-invariant set structure space is challenging to model, particularly for point set classification under spatial deformations. Here, we propose a framework for classifying point sets experiencing certain types of spatial deformations, with a particular emphasis on datasets featuring affine deformations. Our approach employs the linear optimal transport (LOT) transform to obtain a linear embedding of set-structured data. Utilizing the mathematical properties of the LOT transform, we demonstrate its capacity to accommodate variations in point sets by constructing a convex data space, effectively simplifying point set classification problems. Our method, which employs a nearest-subspace algorithm in the LOT space, demonstrates label efficiency, non-iterative behavior, and requires no hyperparameter tuning. It achieves competitive accuracies compared to state-of-the-art methods across various point set classification tasks. Furthermore, our approach exhibits robustness in out-of-distribution scenarios where training and test distributions vary in terms of deformation magnitudes.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"67 4","pages":""},"PeriodicalIF":1.8,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13229578/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148156908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"CoRRECT: A Deep Unfolding Framework for Motion-Corrected Quantitative R2* Mapping.","authors":"Xiaojian Xu, Weijie Gan, Satya V V N Kothapalli, Dmitriy A Yablonskiy, Ulugbek S Kamilov","doi":"10.1007/s10851-025-01236-y","DOIUrl":"10.1007/s10851-025-01236-y","url":null,"abstract":"<p><p>Quantitative MRI (qMRI) refers to a class of MRI methods for quantifying the spatial distribution of biological tissue parameters. Traditional qMRI methods usually deal separately with artifacts arising from accelerated data acquisition, involuntary physical motion, and magnetic field inhomogeneities, leading to sub-optimal end-to-end performance. This paper presents CoRRECT, a unified deep unfolding (DU) framework for qMRI consisting of a model-based end-to-end neural network, a method for motion artifact reduction, and a self-supervised learning scheme. The network is trained to produce R2* maps whose k-space data matches the real data by also accounting for motion and field inhomogeneities. When deployed, CoRRECT only uses the k-space data without any pre-computed parameters for motion or inhomogeneity correction. Our results on experimentally collected multi-gradient recalled echo (mGRE) MRI data show that CoRRECT recovers motion and inhomogeneity artifact-free R2* maps in highly accelerated acquisition settings. This work opens the door to DU methods that can integrate physical measurement models, biophysical signal models, and learned prior models for high-quality qMRI.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"67 2","pages":""},"PeriodicalIF":1.5,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12369581/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144957223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mathematical Morphology on Directional Data","authors":"Konstantin Hauch, Claudia Redenbach","doi":"10.1007/s10851-024-01210-0","DOIUrl":"https://doi.org/10.1007/s10851-024-01210-0","url":null,"abstract":"<p>We define morphological operators and filters for directional images whose pixel values are unit vectors. This requires an ordering relation for unit vectors which is obtained by using depth functions. They provide a centre-outward ordering with respect to a specified centre vector. We apply our operators on synthetic directional images and compare them with classical morphological operators for grey-scale images. As application examples, we enhance the fault region in a compressed glass foam and segment misaligned fibre regions of glass fibre-reinforced polymers.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"59 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mixing Support Detection-Based Alternating Direction Method of Multipliers for Sparse Hyperspectral Image Unmixing","authors":"Jie Huang, Shuang Liang, Liang-Jian Deng","doi":"10.1007/s10851-024-01208-8","DOIUrl":"https://doi.org/10.1007/s10851-024-01208-8","url":null,"abstract":"<p>Spectral unmixing is important in analyzing and processing hyperspectral images (HSIs). With the availability of large spectral signature libraries, the main task of spectral unmixing is to estimate corresponding proportions called <i>abundances</i> of pure spectral signatures called <i>endmembers</i> in mixed pixels. In this vein, only a few endmembers participate in the formation of mixed pixels in the scene and so we call them active endmembers. A plethora of sparse unmixing algorithms exploit spectral and spatial information in HSIs to enhance abundance estimation results. Many algorithms, however, treat the abundances corresponding to active and nonactive endmembers in the scene equivalently. In this article, we propose a framework named <i>mixing support detection</i> (MSD) for the spectral unmixing problem. The main idea is first to detect the active and nonactive endmembers at each iteration and then to treat the corresponding abundances differently. It follows that we only focus on the estimation of active abundances with the assumption of zero abundances corresponding to nonactive endmembers. It can be expected to reduce the computational cost, avoid the perturbations in nonactive abundances, and enhance the sparsity of the abundances. We embed the MSD framework in classic <i>alternating direction method of multipliers</i> (ADMM) updates and obtain an ADMM-MSD algorithm. In particular, five ADMM-MSD-based unmixing algorithms are provided. The residual and objective convergence results of the proposed algorithm are given under certain assumptions. Both simulated and real-data experiments demonstrate the efficacy and superiority of the proposed algorithm compared with some state-of-the-art algorithms.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"122 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Inferring Object Boundaries and Their Roughness with Uncertainty Quantification","authors":"Babak Maboudi Afkham, Nicolai André Brogaard Riis, Yiqiu Dong, Per Christian Hansen","doi":"10.1007/s10851-024-01207-9","DOIUrl":"https://doi.org/10.1007/s10851-024-01207-9","url":null,"abstract":"<p>This work describes a Bayesian framework for reconstructing the boundaries that represent targeted features in an image, as well as the regularity (i.e., roughness vs. smoothness) of these boundaries. This regularity often carries crucial information in many inverse problem applications, e.g., for identifying malignant tissues in medical imaging. We represent the boundary as a radial function and characterize the regularity of this function by means of its fractional differentiability. We propose a hierarchical Bayesian formulation which, simultaneously, estimates the function and its regularity, and in addition we quantify the uncertainties in the estimates. Numerical results suggest that the proposed method is a reliable approach for estimating and characterizing object boundaries in imaging applications, as illustrated with examples from high-intensity X-ray CT and image inpainting with Gaussian and Laplace additive noise models. We also show that our method can quantify uncertainties for these noise types, various noise levels, and incomplete data scenarios.\u0000</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"276 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Graph Multi-separator Problem for Image Segmentation","authors":"Jannik Irmai, Shengxian Zhao, Mark Schöne, Jannik Presberger, Bjoern Andres","doi":"10.1007/s10851-024-01201-1","DOIUrl":"https://doi.org/10.1007/s10851-024-01201-1","url":null,"abstract":"<p>We propose a novel abstraction of the image segmentation task in the form of a combinatorial optimization problem that we call the <i>multi-separator problem</i>. Feasible solutions indicate for every pixel whether it belongs to a segment or a segment separator, and indicate for pairs of pixels whether or not the pixels belong to the same segment. This is in contrast to the closely related lifted multicut problem, where every pixel is associated with a segment and no pixel explicitly represents a separating structure. While the multi-separator problem is <span>np</span>-hard, we identify two special cases for which it can be solved efficiently. Moreover, we define two local search algorithms for the general case and demonstrate their effectiveness in segmenting simulated volume images of foam cells and filaments.\u0000</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"32 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}