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":"https://doi.org/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":1.5,"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":1.5,"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":"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}

{"title":"Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group","authors":"Michael Quellmalz, Léo Buecher, Gabriele Steidl","doi":"10.1007/s10851-024-01206-w","DOIUrl":"https://doi.org/10.1007/s10851-024-01206-w","url":null,"abstract":"<p>Sliced optimal transport, which is basically a Radon transform followed by one-dimensional optimal transport, became popular in various applications due to its efficient computation. In this paper, we deal with sliced optimal transport on the sphere <span>(mathbb {S}^{d-1})</span> and on the rotation group <span>(textrm{SO}(3))</span>. We propose a parallel slicing procedure of the sphere which requires again only optimal transforms on the line. We analyze the properties of the corresponding parallelly sliced optimal transport, which provides in particular a rotationally invariant metric on the spherical probability measures. For <span>(textrm{SO}(3))</span>, we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on <span>(textrm{SO}(3))</span>. As Wasserstein distances were extensively used in barycenter computations, we derive algorithms to compute the barycenters with respect to our new sliced Wasserstein distances and provide synthetic numerical examples on the 2-sphere that demonstrate their behavior for both the free- and fixed-support setting of discrete spherical measures. In terms of computational speed, they outperform the existing methods for semicircular slicing as well as the regularized Wasserstein barycenters.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"321 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777775","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":"An Edge-Preserving Regularization Model for the Demosaicing of Noisy Color Images","authors":"Antonio Boccuto, Ivan Gerace, Valentina Giorgetti, Francesca Martinelli, Anna Tonazzini","doi":"10.1007/s10851-024-01204-y","DOIUrl":"https://doi.org/10.1007/s10851-024-01204-y","url":null,"abstract":"<p>This paper proposes an edge-preserving regularization technique to solve the color image demosaicing problem in the realistic case of noisy data. We enforce intra-channel local smoothness of the intensity (low-frequency components) and inter-channel local similarity of the depth of object borders and textures (high-frequency components). Discontinuities of both the low-frequency and high-frequency components are accounted for implicitly, i.e., through suitable functions of the proper derivatives. For the treatment of even the finest image details, derivatives of first, second, and third orders are considered. The solution to the demosaicing problem is defined as the minimizer of an energy function, accounting for all these constraints plus a data fidelity term. This non-convex energy is minimized via an iterative deterministic algorithm, applied to a family of approximating functions, each implicitly referring to geometrically consistent image edges. Our method is general because it does not refer to any specific color filter array. However, to allow quantitative comparisons with other published results, we tested it in the case of the Bayer CFA and on the Kodak 24-image dataset, the McMaster (IMAX) 18-image dataset, the Microsoft Demosaicing Canon 57-image dataset, and the Microsoft Demosaicing Panasonic 500-image dataset. The comparisons with some of the most recent demosaicing algorithms show the good performance of our method in both the noiseless and noisy cases.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"20 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777776","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":"Re-initialization-Free Level Set Method via Molecular Beam Epitaxy Equation Regularization for Image Segmentation","authors":"Fanghui Song, Jiebao Sun, Shengzhu Shi, Zhichang Guo, Dazhi Zhang","doi":"10.1007/s10851-024-01205-x","DOIUrl":"https://doi.org/10.1007/s10851-024-01205-x","url":null,"abstract":"<p>Variational level set method has become a powerful tool in image segmentation due to its ability to handle complex topological changes and maintain continuity and smoothness in the process of evolution. However its evolution process can be unstable, which results in over flatted or over sharpened contours and segmentation failure. To improve the accuracy and stability of evolution, we propose a high-order level set variational segmentation method integrated with molecular beam epitaxy (MBE) equation regularization. This method uses the crystal growth in the MBE process to limit the evolution of the level set function. Thus can avoid the re-initialization in the evolution process and regulate the smoothness of the segmented curve and keep the segmentation results independent of the initial curve selection. It also works for noisy images with intensity inhomogeneity, which is a challenge in image segmentation. To solve the variational model, we derive the gradient flow and design a scalar auxiliary variable scheme, which can significantly improve the computational efficiency compared with the traditional semi-implicit and semi-explicit scheme. Numerical experiments show that the proposed method can generate smooth segmentation curves, preserve segmentation details and obtain robust segmentation results of small objects. Compared to existing level set methods, this model is state-of-the-art in both accuracy and efficiency.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"30 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531054","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}