Pub Date : 2024-09-06DOI: 10.1007/s10851-024-01210-0
Konstantin Hauch, Claudia Redenbach
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.
{"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}
Pub Date : 2024-08-16DOI: 10.1007/s10851-024-01208-8
Jie Huang, Shuang Liang, Liang-Jian Deng
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 abundances of pure spectral signatures called endmembers 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 mixing support detection (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 alternating direction method of multipliers (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.
{"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}
Pub Date : 2024-08-13DOI: 10.1007/s10851-024-01207-9
Babak Maboudi Afkham, Nicolai André Brogaard Riis, Yiqiu Dong, Per Christian Hansen
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.
这项工作描述了一个贝叶斯框架,用于重建图像中代表目标特征的边界,以及这些边界的规则性(即粗糙度与平滑度)。在许多逆问题应用中,例如在医学成像中识别恶性组织时,这种规则性往往蕴含着至关重要的信息。我们将边界表示为一个径向函数,并通过其分数可微分性来表征该函数的规则性。我们提出了一种分层贝叶斯公式,可以同时估计函数及其规律性,此外,我们还量化了估计值的不确定性。数值结果表明,所提出的方法是在成像应用中估计和描述物体边界的可靠方法,高强度 X 射线 CT 和使用高斯和拉普拉斯加性噪声模型的图像绘制就是例证。我们还表明,我们的方法可以量化这些噪声类型、各种噪声水平和不完整数据情况下的不确定性。
{"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}
Pub Date : 2024-08-12DOI: 10.1007/s10851-024-01201-1
Jannik Irmai, Shengxian Zhao, Mark Schöne, Jannik Presberger, Bjoern Andres
We propose a novel abstraction of the image segmentation task in the form of a combinatorial optimization problem that we call the multi-separator problem. 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 np-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.
{"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}
Pub Date : 2024-07-26DOI: 10.1007/s10851-024-01206-w
Michael Quellmalz, Léo Buecher, Gabriele Steidl
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 (mathbb {S}^{d-1}) and on the rotation group (textrm{SO}(3)). 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 (textrm{SO}(3)), we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on (textrm{SO}(3)). 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.
{"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}
Pub Date : 2024-07-26DOI: 10.1007/s10851-024-01204-y
Antonio Boccuto, Ivan Gerace, Valentina Giorgetti, Francesca Martinelli, Anna Tonazzini
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.
本文提出了一种边缘保留正则化技术,用于解决现实中存在噪声数据的彩色图像去马赛克问题。我们对强度(低频成分)实施通道内局部平滑,对物体边界和纹理深度(高频成分)实施通道间局部相似。低频分量和高频分量的不连续性通过适当的正导数函数隐含地加以考虑。为了处理最精细的图像细节,还考虑了一阶、二阶和三阶导数。去马赛克问题的解决方案被定义为能量函数的最小化,它考虑了所有这些约束条件和数据保真度项。这种非凸能量是通过迭代确定性算法最小化的,该算法适用于一系列近似函数,每个函数都隐含着几何上一致的图像边缘。我们的方法是通用的,因为它不涉及任何特定的滤色器阵列。不过,为了与其他已发表的结果进行定量比较,我们在拜耳 CFA 的情况下,并在柯达 24 幅图像数据集、麦克马斯特(IMAX)18 幅图像数据集、微软去马赛克佳能 57 幅图像数据集和微软去马赛克松下 500 幅图像数据集上进行了测试。与一些最新的去马赛克算法的比较结果表明,我们的方法在无噪声和有噪声的情况下都有良好的表现。
{"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}
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.
{"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}
Pub Date : 2024-06-24DOI: 10.1007/s10851-024-01203-z
Petra Wiederhold
To study convexity properties of digital planar objects, the minimum perimeter polygon (MPP) was defined in the 1970 s in articles by Sklansky, Chazin, Hansen, Kibler, and Kim, where pixels were identified with polygonal tiles in mosaics, and two algorithms (1972, 1976) were proposed to determine the MPP vertices. These algorithms are based on constructing and iteratively restricting visibility cones, the MPP vertices result as special vertices of the tiles. The present paper proposes a novel MPP algorithm for objects given as regular complexes in rectangular mosaics, which are edge-adjacency-connected sets of tiles that have neither end tiles nor holes and whose boundaries not necessarily are simple. The new algorithm takes as input the canonical boundary path, we also propose a boundary tracing algorithm to obtain this path. We review the two classic MPP algorithms for rectangular tiles and a simplified adaptation for square tiles that is recommended in widely used modern textbooks on digital image analysis (2018, 2020) to produce approximations of simple digital 4-contours. We show that all these algorithms fail and that their mathematical basis is flawed, we correct the errors to develop the new MPP algorithm. Our MPP algorithm is illustrated using examples and its correctness is proved. Under our assumptions, the MPP coincides with the relative convex hull of a set A with respect to a polygon (Bsupset A), where A is not necessarily a polygon, not even connected.
为了研究数字平面对象的凹凸特性,20 世纪 70 年代,Sklansky、Chazin、Hansen、Kibler 和 Kim 在文章中定义了最小周长多边形(MPP),将像素与马赛克中的多边形瓷砖进行识别,并提出了两种算法(1972 年和 1976 年)来确定 MPP 的顶点。这些算法都是基于能见度锥的构建和迭代限制,而 MPP 顶点则是瓦片的特殊顶点。本文提出了一种新的 MPP 算法,适用于矩形马赛克中作为规则复合物给出的对象,这些规则复合物是边缘相接的瓦片集,既没有末端瓦片,也没有孔洞,其边界不一定是简单的。新算法将典型边界路径作为输入,我们还提出了一种边界追踪算法来获取该路径。我们回顾了用于矩形瓷砖的两种经典 MPP 算法,以及广泛使用的现代数字图像分析教科书(2018 年,2020 年)中推荐的用于正方形瓷砖的简化改编算法,以生成简单数字 4 轮廓的近似值。我们证明了所有这些算法都是失败的,它们的数学基础存在缺陷,我们纠正了这些错误,开发了新的 MPP 算法。我们用实例说明了我们的 MPP 算法,并证明了其正确性。根据我们的假设,MPP 与多边形 (Bsupset A) 的集合 A 的相对凸壳重合,其中 A 不一定是多边形,甚至不一定是连通的。
{"title":"Computing the Minimal Perimeter Polygon for Sets of Rectangular Tiles based on Visibility Cones","authors":"Petra Wiederhold","doi":"10.1007/s10851-024-01203-z","DOIUrl":"https://doi.org/10.1007/s10851-024-01203-z","url":null,"abstract":"<p>To study convexity properties of digital planar objects, the minimum perimeter polygon (MPP) was defined in the 1970 s in articles by Sklansky, Chazin, Hansen, Kibler, and Kim, where pixels were identified with polygonal tiles in mosaics, and two algorithms (1972, 1976) were proposed to determine the MPP vertices. These algorithms are based on constructing and iteratively restricting visibility cones, the MPP vertices result as special vertices of the tiles. The present paper proposes a novel MPP algorithm for objects given as regular complexes in rectangular mosaics, which are edge-adjacency-connected sets of tiles that have neither end tiles nor holes and whose boundaries not necessarily are simple. The new algorithm takes as input the canonical boundary path, we also propose a boundary tracing algorithm to obtain this path. We review the two classic MPP algorithms for rectangular tiles and a simplified adaptation for square tiles that is recommended in widely used modern textbooks on digital image analysis (2018, 2020) to produce approximations of simple digital 4-contours. We show that all these algorithms fail and that their mathematical basis is flawed, we correct the errors to develop the new MPP algorithm. Our MPP algorithm is illustrated using examples and its correctness is proved. Under our assumptions, the MPP coincides with the relative convex hull of a set <i>A</i> with respect to a polygon <span>(Bsupset A)</span>, where <i>A</i> is not necessarily a polygon, not even connected.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"28 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509057","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}
Pub Date : 2024-06-17DOI: 10.1007/s10851-024-01196-9
Tony Lindeberg
This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and the Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways of discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region, to aim at suppressing some of the severe artefacts of sampled Gaussian kernels and sampled Gaussian derivatives at very fine scales, or (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data.
We study the properties of these three main discretization methods both theoretically and experimentally and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and the sampled Gaussian derivatives as well as the integrated Gaussian kernels and the integrated Gaussian derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in most of the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing. Below a standard deviation of about 0.75, the derivative estimates obtained from convolutions with the sampled Gaussian derivative kernels are, however, not numerically accurate or consistent, while the results obtained from the discrete analogue of the Gaussian kernel, with its associated central difference operators applied to the spatially smoothed image data, are then a much better choice.
{"title":"Discrete Approximations of Gaussian Smoothing and Gaussian Derivatives","authors":"Tony Lindeberg","doi":"10.1007/s10851-024-01196-9","DOIUrl":"https://doi.org/10.1007/s10851-024-01196-9","url":null,"abstract":"<p>This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and the Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways of discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region, to aim at suppressing some of the severe artefacts of sampled Gaussian kernels and sampled Gaussian derivatives at very fine scales, or (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data.</p><p>We study the properties of these three main discretization methods both theoretically and experimentally and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and the sampled Gaussian derivatives as well as the integrated Gaussian kernels and the integrated Gaussian derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in most of the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing. Below a standard deviation of about 0.75, the derivative estimates obtained from convolutions with the sampled Gaussian derivative kernels are, however, not numerically accurate or consistent, while the results obtained from the discrete analogue of the Gaussian kernel, with its associated central difference operators applied to the spatially smoothed image data, are then a much better choice.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"45 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141500559","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}
Pub Date : 2024-06-17DOI: 10.1007/s10851-024-01202-0
Pascal Peter
Diffusion probabilistic models excel at sampling new images from learned distributions. Originally motivated by drift-diffusion concepts from physics, they apply image perturbations such as noise and blur in a forward process that results in a tractable probability distribution. A corresponding learned reverse process generates images and can be conditioned on side information, which leads to a wide variety of practical applications. Most of the research focus currently lies on practice-oriented extensions. In contrast, the theoretical background remains largely unexplored, in particular the relations to drift-diffusion. In order to shed light on these connections to classical image filtering, we propose a generalised scale-space theory for diffusion probabilistic models. Moreover, we show conceptual and empirical connections to diffusion and osmosis filters.
{"title":"Generalised Diffusion Probabilistic Scale-Spaces","authors":"Pascal Peter","doi":"10.1007/s10851-024-01202-0","DOIUrl":"https://doi.org/10.1007/s10851-024-01202-0","url":null,"abstract":"<p>Diffusion probabilistic models excel at sampling new images from learned distributions. Originally motivated by drift-diffusion concepts from physics, they apply image perturbations such as noise and blur in a forward process that results in a tractable probability distribution. A corresponding learned reverse process generates images and can be conditioned on side information, which leads to a wide variety of practical applications. Most of the research focus currently lies on practice-oriented extensions. In contrast, the theoretical background remains largely unexplored, in particular the relations to drift-diffusion. In order to shed light on these connections to classical image filtering, we propose a generalised scale-space theory for diffusion probabilistic models. Moreover, we show conceptual and empirical connections to diffusion and osmosis filters.\u0000</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":"15 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509058","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}