Several numerical algorithms have been developed in the literature and employed for curves reconstruction. However, these techniques are developed within the discrete setting, namely the super-resolved image is defined on a finer grid than the observed images. Conversely, off-the-grid (or gridless) optimisation does not rely on a fine grid and offer a tractable theoretical and numerical framework. In this work, we present a gridless method accounting for the reconstruction of both open and closed curves, based on the latest theoretical development in off-the-grid curve reconstruction. This paper also shows (varGamma )-convergence results of the discretised surrogate functional towards the continuous energy we coined CROC.
{"title":"A $$varGamma $$ -Convergence Result and An Off-the-Grid Charge Algorithm for Curve Reconstruction in Inverse Problems","authors":"Bastien Laville, Laure Blanc-Féraud, Gilles Aubert","doi":"10.1007/s10851-024-01190-1","DOIUrl":"https://doi.org/10.1007/s10851-024-01190-1","url":null,"abstract":"<p>Several numerical algorithms have been developed in the literature and employed for curves reconstruction. However, these techniques are developed within the discrete setting, namely the super-resolved image is defined on a finer grid than the observed images. Conversely, off-the-grid (or gridless) optimisation does not rely on a fine grid and offer a tractable theoretical and numerical framework. In this work, we present a gridless method accounting for the reconstruction of both open and closed curves, based on the latest theoretical development in off-the-grid curve reconstruction. This paper also shows <span>(varGamma )</span>-convergence results of the discretised surrogate functional towards the continuous energy we coined CROC.\u0000</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939303","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 : 2024-04-29DOI: 10.1007/s10851-024-01186-x
Paolo Piras, Nicolas Guigui, Valerio Varano
Comparing the deformations of different beating hearts is a challenging operation. As in clinics the impaired condition is often recognized upon (local and global) deformation parameters, the particular nature of heart deformation during one beat can be compared among different individuals in the same ordination space more effectively if initial inter-individual form (shape + size) differences are filtered out. This is even more true if the shape of cardiac trajectory itself is under consideration. This need is satisfied by applying a geometric machinery named “parallel transport” in the field of differential geometry. In recent years several parallel transport methods have been applied to cardiological data acquired via echocardiography, CT scan or magnetic resonance. Concomitantly, some efforts were made for comparing different parallel transport algorithms applied to a variety of toy examples and real deformational data. Here we face the problem of comparing the heavily used LDDMM parallel transport with the recently proposed Riemannian “TPS space” in the context of the deformation of the right ventricle. Using local tensors diagnostics and global energy-based and shape distance-based parameters, we explored the maintenance of original deformations in transported data in four systo-diastolic deformations belonging to one healthy subject and three individuals affected by tetralogy of Fallot, atrial septal defect and pulmonary hypertension. We also do the same in a larger dataset relative to the left ventricle of 82 heathly subjects and 21 patients affected by hypertrophic cardiomyopathy. We also do the same in a larger dataset relative to the left ventricle of 82 heathly subjects and 21 patients affected by hypertrophic cardiomyopathy. In particular, we contrasted the TPS space with classic LDDMM and a modified LDDMM able to manage spherical differences. Our results point toward a neat superiority of TPS space over classic LDDMM. The modified LDDMM performs similarly as it maintains better the chosen diagnostics.
{"title":"Comparison of Different Parallel Transport Methods for the Study of Deformations in 3D Cardiac Data","authors":"Paolo Piras, Nicolas Guigui, Valerio Varano","doi":"10.1007/s10851-024-01186-x","DOIUrl":"https://doi.org/10.1007/s10851-024-01186-x","url":null,"abstract":"<p>Comparing the deformations of different beating hearts is a challenging operation. As in clinics the impaired condition is often recognized upon (local and global) deformation parameters, the particular nature of heart deformation during one beat can be compared among different individuals in the same ordination space more effectively if initial inter-individual form (shape + size) differences are filtered out. This is even more true if the shape of cardiac trajectory itself is under consideration. This need is satisfied by applying a geometric machinery named “parallel transport” in the field of differential geometry. In recent years several parallel transport methods have been applied to cardiological data acquired via echocardiography, CT scan or magnetic resonance. Concomitantly, some efforts were made for comparing different parallel transport algorithms applied to a variety of toy examples and real deformational data. Here we face the problem of comparing the heavily used LDDMM parallel transport with the recently proposed Riemannian “TPS space” in the context of the deformation of the right ventricle. Using local tensors diagnostics and global energy-based and shape distance-based parameters, we explored the maintenance of original deformations in transported data in four systo-diastolic deformations belonging to one healthy subject and three individuals affected by tetralogy of Fallot, atrial septal defect and pulmonary hypertension. We also do the same in a larger dataset relative to the left ventricle of 82 heathly subjects and 21 patients affected by hypertrophic cardiomyopathy. We also do the same in a larger dataset relative to the left ventricle of 82 heathly subjects and 21 patients affected by hypertrophic cardiomyopathy. In particular, we contrasted the TPS space with classic LDDMM and a modified LDDMM able to manage spherical differences. Our results point toward a neat superiority of TPS space over classic LDDMM. The modified LDDMM performs similarly as it maintains better the chosen diagnostics.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140835658","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 : 2024-04-25DOI: 10.1007/s10851-024-01189-8
Snehal Bhayani, Janne Heikkilä, Zuzana Kukelova
{"title":"Correction: Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix","authors":"Snehal Bhayani, Janne Heikkilä, Zuzana Kukelova","doi":"10.1007/s10851-024-01189-8","DOIUrl":"https://doi.org/10.1007/s10851-024-01189-8","url":null,"abstract":"","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140656159","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 : 2024-04-23DOI: 10.1007/s10851-024-01183-0
Ariane Fazeny, Daniel Tenbrinck, Kseniia Lukin, Martin Burger
The aim of this paper is to revisit the definition of differential operators on hypergraphs, which are a natural extension of graphs in systems based on interactions beyond pairs. In particular, we focus on the definition of Laplacian and p-Laplace operators for oriented and unoriented hypergraphs, their basic properties, variational structure, and their scale spaces. We illustrate that diffusion equations on hypergraphs are possible models for different applications such as information flow on social networks or image processing. Moreover, the spectral analysis and scale spaces induced by these operators provide a potential method to further analyze complex data and their multiscale structure. The quest for spectral analysis and suitable scale spaces on hypergraphs motivates in particular a definition of differential operators with trivial first eigenfunction and thus more interpretable second eigenfunctions. This property is not automatically satisfied in existing definitions of hypergraph p-Laplacians, and we hence provide a novel axiomatic approach that extends previous definitions and can be specialized to satisfy such (or other) desired properties.
{"title":"Hypergraph p-Laplacians and Scale Spaces","authors":"Ariane Fazeny, Daniel Tenbrinck, Kseniia Lukin, Martin Burger","doi":"10.1007/s10851-024-01183-0","DOIUrl":"https://doi.org/10.1007/s10851-024-01183-0","url":null,"abstract":"<p>The aim of this paper is to revisit the definition of differential operators on hypergraphs, which are a natural extension of graphs in systems based on interactions beyond pairs. In particular, we focus on the definition of Laplacian and <i>p</i>-Laplace operators for oriented and unoriented hypergraphs, their basic properties, variational structure, and their scale spaces. We illustrate that diffusion equations on hypergraphs are possible models for different applications such as information flow on social networks or image processing. Moreover, the spectral analysis and scale spaces induced by these operators provide a potential method to further analyze complex data and their multiscale structure. The quest for spectral analysis and suitable scale spaces on hypergraphs motivates in particular a definition of differential operators with trivial first eigenfunction and thus more interpretable second eigenfunctions. This property is not automatically satisfied in existing definitions of hypergraph <i>p</i>-Laplacians, and we hence provide a novel axiomatic approach that extends previous definitions and can be specialized to satisfy such (or other) desired properties.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799365","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}
The joint problem of reconstruction/feature extraction is a challenging task in image processing. It consists in performing, in a joint manner, the restoration of an image and the extraction of its features. In this work, we firstly propose a novel non-smooth and non-convex variational formulation of the problem. For this purpose, we introduce a versatile generalised Gaussian prior whose parameters, including its exponent, are space-variant. Secondly, we design an alternating proximal-based optimisation algorithm that efficiently exploits the structure of the proposed non-convex objective function. We also analyse the convergence of this algorithm. As shown in numerical experiments conducted on joint deblurring/segmentation tasks, the proposed method provides high-quality results.
{"title":"A Variational Approach for Joint Image Recovery and Feature Extraction Based on Spatially Varying Generalised Gaussian Models","authors":"Émilie Chouzenoux, Marie-Caroline Corbineau, Jean-Christophe Pesquet, Gabriele Scrivanti","doi":"10.1007/s10851-024-01184-z","DOIUrl":"https://doi.org/10.1007/s10851-024-01184-z","url":null,"abstract":"<p>The joint problem of reconstruction/feature extraction is a challenging task in image processing. It consists in performing, in a joint manner, the restoration of an image and the extraction of its features. In this work, we firstly propose a novel non-smooth and non-convex variational formulation of the problem. For this purpose, we introduce a versatile generalised Gaussian prior whose parameters, including its exponent, are space-variant. Secondly, we design an alternating proximal-based optimisation algorithm that efficiently exploits the structure of the proposed non-convex objective function. We also analyse the convergence of this algorithm. As shown in numerical experiments conducted on joint deblurring/segmentation tasks, the proposed method provides high-quality results.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140594632","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 : 2024-04-09DOI: 10.1007/s10851-024-01185-y
Patrick M. Jensen, J. Andreas Bærentzen, Anders B. Dahl, Vedrana A. Dahl
Dynamic 3D imaging is increasingly used to study evolving objects. We address the problem of detecting and tracking simple objects that merge or split in time. Common solutions involve detecting topological changes. Instead, we solve the problem in 4D by exploiting the observation that if objects only merge or only split, they appear as a single component in 4D. This allows us to initiate a topologically simple 3D hypersurface and deform it to fit the surface of all objects at all times. This gives an extremely compact representation of the objects’ evolution. We test our method on artificial 4D images and compare it to other segmentation methods. We also apply our method to a 4D X-ray data set to quantify evolving topology. Our method performs comparably to existing methods with better resource use and improved robustness.
动态三维成像技术越来越多地用于研究不断演变的物体。我们要解决的问题是检测和跟踪在时间上合并或分裂的简单物体。常见的解决方案包括检测拓扑变化。相反,我们利用物体只合并或只分裂时在 4D 中显示为单一成分这一观察结果,在 4D 中解决了这一问题。这样,我们就可以启动一个拓扑结构简单的三维超曲面,并对其进行变形,使其在任何时候都适合所有物体的表面。这样,物体的演变过程就得到了极其紧凑的呈现。我们在人工 4D 图像上测试了我们的方法,并将其与其他分割方法进行了比较。我们还将我们的方法应用于 4D X 射线数据集,以量化不断演变的拓扑结构。我们的方法与现有方法性能相当,资源利用率更高,鲁棒性更好。
{"title":"Finding Space-Time Boundaries with Deformable Hypersurfaces","authors":"Patrick M. Jensen, J. Andreas Bærentzen, Anders B. Dahl, Vedrana A. Dahl","doi":"10.1007/s10851-024-01185-y","DOIUrl":"https://doi.org/10.1007/s10851-024-01185-y","url":null,"abstract":"<p>Dynamic 3D imaging is increasingly used to study evolving objects. We address the problem of detecting and tracking simple objects that merge or split in time. Common solutions involve detecting topological changes. Instead, we solve the problem in 4D by exploiting the observation that if objects only merge or only split, they appear as a single component in 4D. This allows us to initiate a topologically simple 3D hypersurface and deform it to fit the surface of all objects at all times. This gives an extremely compact representation of the objects’ evolution. We test our method on artificial 4D images and compare it to other segmentation methods. We also apply our method to a 4D X-ray data set to quantify evolving topology. Our method performs comparably to existing methods with better resource use and improved robustness.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140594590","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 : 2024-04-01DOI: 10.1007/s10851-024-01175-0
Kristina Schaefer, Joachim Weickert
We introduce regularised diffusion–shock (RDS) inpainting as a modification of diffusion–shock inpainting from our SSVM 2023 conference paper. RDS inpainting combines two carefully chosen components: homogeneous diffusion and coherence-enhancing shock filtering. It benefits from the complementary synergy of its building blocks: The shock term propagates edge data with perfect sharpness and directional accuracy over large distances due to its high degree of anisotropy. Homogeneous diffusion fills large areas efficiently. The second order equation underlying RDS inpainting inherits a maximum–minimum principle from its components, which is also fulfilled in the discrete case, in contrast to competing anisotropic methods. The regularisation addresses the largest drawback of the original model: It allows a drastic reduction in model parameters without any loss in quality. Furthermore, we extend RDS inpainting to vector-valued data. Our experiments show a performance that is comparable to or better than many inpainting methods based on partial differential equations and related integrodifferential models, including anisotropic processes of second or fourth order.
{"title":"Regularised Diffusion–Shock Inpainting","authors":"Kristina Schaefer, Joachim Weickert","doi":"10.1007/s10851-024-01175-0","DOIUrl":"https://doi.org/10.1007/s10851-024-01175-0","url":null,"abstract":"<p>We introduce regularised diffusion–shock (RDS) inpainting as a modification of diffusion–shock inpainting from our SSVM 2023 conference paper. RDS inpainting combines two carefully chosen components: homogeneous diffusion and coherence-enhancing shock filtering. It benefits from the complementary synergy of its building blocks: The shock term propagates edge data with perfect sharpness and directional accuracy over large distances due to its high degree of anisotropy. Homogeneous diffusion fills large areas efficiently. The second order equation underlying RDS inpainting inherits a maximum–minimum principle from its components, which is also fulfilled in the discrete case, in contrast to competing anisotropic methods. The regularisation addresses the largest drawback of the original model: It allows a drastic reduction in model parameters without any loss in quality. Furthermore, we extend RDS inpainting to vector-valued data. Our experiments show a performance that is comparable to or better than many inpainting methods based on partial differential equations and related integrodifferential models, including anisotropic processes of second or fourth order.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140594594","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 : 2024-03-24DOI: 10.1007/s10851-024-01181-2
Abstract
Plug-and-Play (PnP) and Regularization-by-Denoising (RED) are recent paradigms for image reconstruction that leverage the power of modern denoisers for image regularization. In particular, they have been shown to deliver state-of-the-art reconstructions with CNN denoisers. Since the regularization is performed in an ad-hoc manner, understanding the convergence of PnP and RED has been an active research area. It was shown in recent works that iterate convergence can be guaranteed if the denoiser is averaged or nonexpansive. However, integrating nonexpansivity with gradient-based learning is challenging, the core issue being that testing nonexpansivity is intractable. Using numerical examples, we show that existing CNN denoisers tend to violate the nonexpansive property, which can cause PnP or RED to diverge. In fact, algorithms for training nonexpansive denoisers either cannot guarantee nonexpansivity or are computationally intensive. In this work, we construct contractive and averaged image denoisers by unfolding splitting-based optimization algorithms applied to wavelet denoising and demonstrate that their regularization capacity for PnP and RED can be matched with CNN denoisers. To our knowledge, this is the first work to propose a simple framework for training contractive denoisers using network unfolding.
摘要 即插即用(PnP)和去噪正则化(RED)是最近的图像重建范例,它们利用现代去噪器的强大功能进行图像正则化。特别是,它们已被证明能利用 CNN 去噪器实现最先进的重建。由于正则化是以临时方式进行的,因此了解 PnP 和 RED 的收敛性一直是一个活跃的研究领域。最近的研究表明,如果去噪器是平均的或非膨胀的,则可以保证迭代收敛。然而,将非膨胀性与基于梯度的学习结合起来具有挑战性,其核心问题是非膨胀性测试难以进行。我们利用数值示例表明,现有的 CNN 去噪器往往会违反非膨胀性特性,从而导致 PnP 或 RED 发散。事实上,训练非膨胀去噪器的算法要么不能保证非膨胀性,要么计算量很大。在这项工作中,我们通过应用于小波去噪的基于分裂的展开优化算法,构建了收缩和平均图像去噪器,并证明其对 PnP 和 RED 的正则化能力可与 CNN 去噪器相媲美。据我们所知,这是首次提出利用网络展开训练收缩去噪器的简单框架。
{"title":"Averaged Deep Denoisers for Image Regularization","authors":"","doi":"10.1007/s10851-024-01181-2","DOIUrl":"https://doi.org/10.1007/s10851-024-01181-2","url":null,"abstract":"<h3>Abstract</h3> <p>Plug-and-Play (PnP) and Regularization-by-Denoising (RED) are recent paradigms for image reconstruction that leverage the power of modern denoisers for image regularization. In particular, they have been shown to deliver state-of-the-art reconstructions with CNN denoisers. Since the regularization is performed in an ad-hoc manner, understanding the convergence of PnP and RED has been an active research area. It was shown in recent works that iterate convergence can be guaranteed if the denoiser is averaged or nonexpansive. However, integrating nonexpansivity with gradient-based learning is challenging, the core issue being that testing nonexpansivity is intractable. Using numerical examples, we show that existing CNN denoisers tend to violate the nonexpansive property, which can cause PnP or RED to diverge. In fact, algorithms for training nonexpansive denoisers either cannot guarantee nonexpansivity or are computationally intensive. In this work, we construct contractive and averaged image denoisers by unfolding splitting-based optimization algorithms applied to wavelet denoising and demonstrate that their regularization capacity for PnP and RED can be matched with CNN denoisers. To our knowledge, this is the first work to propose a simple framework for training contractive denoisers using network unfolding. </p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140199470","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 : 2024-03-23DOI: 10.1007/s10851-024-01182-1
Snehal Bhayani, Janne Heikkilä, Zuzana Kukelova
Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements. Minimal problems are usually formulated as complex systems of sparse polynomial equations. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants based on the Newton polytopes has not been used so often for generating efficient solvers, primarily because the polytopes do not respect the constraints amongst the coefficients. In an attempt to tackle this challenge, here we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via Schur complement computation. We show that for some camera geometry problems our resultant-based method leads to smaller and more stable solvers than the state-of-the-art Gröbner basis-based solvers, while being significantly smaller than the state-of-the-art resultant-based methods. The proposed method can be fully automated and incorporated into existing tools for the automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Gröbner basis-based methods for minimal problems in computer vision. Additionally, we study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically, we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers.
{"title":"Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix","authors":"Snehal Bhayani, Janne Heikkilä, Zuzana Kukelova","doi":"10.1007/s10851-024-01182-1","DOIUrl":"https://doi.org/10.1007/s10851-024-01182-1","url":null,"abstract":"<p>Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements. Minimal problems are usually formulated as complex systems of sparse polynomial equations. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants based on the Newton polytopes has not been used so often for generating efficient solvers, primarily because the polytopes do not respect the constraints amongst the coefficients. In an attempt to tackle this challenge, here we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via Schur complement computation. We show that for some camera geometry problems our resultant-based method leads to smaller and more stable solvers than the state-of-the-art Gröbner basis-based solvers, while being significantly smaller than the state-of-the-art resultant-based methods. The proposed method can be fully automated and incorporated into existing tools for the automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Gröbner basis-based methods for minimal problems in computer vision. Additionally, we study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically, we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers.\u0000</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198825","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 : 2024-03-19DOI: 10.1007/s10851-024-01179-w
Eyal Gofer, Guy Gilboa
Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore–Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore–Penrose axioms. We define the notion for general sets and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators.
{"title":"Generalized Inversion of Nonlinear Operators","authors":"Eyal Gofer, Guy Gilboa","doi":"10.1007/s10851-024-01179-w","DOIUrl":"https://doi.org/10.1007/s10851-024-01179-w","url":null,"abstract":"<p>Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore–Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore–Penrose axioms. We define the notion for general sets and then a refinement, termed <i>pseudo-inverse</i>, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140167632","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}