Journal of Mathematical Imaging and Vision最新文献_第7页

Analysis of (sub-)Riemannian PDE-G-CNNs (亚)黎曼pde - g - cnn的分析

4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2023-04-16 DOI: 10.1007/s10851-023-01147-w

Gijs Bellaard, Daan L. J. Bon, Gautam Pai, Bart M. N. Smets, Remco Duits

{"title":"Analysis of (sub-)Riemannian PDE-G-CNNs","authors":"Gijs Bellaard, Daan L. J. Bon, Gautam Pai, Bart M. N. Smets, Remco Duits","doi":"10.1007/s10851-023-01147-w","DOIUrl":"https://doi.org/10.1007/s10851-023-01147-w","url":null,"abstract":"Abstract Group equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. Typically, G-CNNs have the advantage over CNNs that they do not waste network capacity on training symmetries that should have been hard-coded in the network. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalizes G-CNNs. PDE-G-CNNs have the core advantages that they simultaneously (1) reduce network complexity, (2) increase classification performance, and (3) provide geometric interpretability. Their implementations primarily consist of linear and morphological convolutions with kernels. In this paper, we show that the previously suggested approximative morphological kernels do not always accurately approximate the exact kernels accurately. More specifically, depending on the spatial anisotropy of the Riemannian metric, we argue that one must resort to sub-Riemannian approximations. We solve this problem by providing a new approximative kernel that works regardless of the anisotropy. We provide new theorems with better error estimates of the approximative kernels, and prove that they all carry the same reflectional symmetries as the exact ones. We test the effectiveness of multiple approximative kernels within the PDE-G-CNN framework on two datasets, and observe an improvement with the new approximative kernels. We report that the PDE-G-CNNs again allow for a considerable reduction of network complexity while having comparable or better performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have the advantage of better geometric interpretability over G-CNNs, as the morphological kernels are related to association fields from neurogeometry.","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136243274","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}

引用次数: 1

Appreciation to Journal of Mathematical Imaging and Vision Reviewers 感谢《数学成像与视觉评论杂志》

IF 2 4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2023-04-01 DOI: 10.1007/s10851-023-01141-2

引用次数: 0

A Generalisation of Flat Morphology, II: Main Properties, Duality and Hybrid Operators 平面形态学的一个推广，Ⅱ：主要性质、对偶性和混合算子

IF 2 4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2023-03-28 DOI: 10.1007/s10851-023-01145-y

C. Ronse

引用次数: 0

A Physically Admissible Stokes Vector Reconstruction in Linear Polarimetric Imaging 线性偏振成像中物理上可接受的Stokes矢量重建

IF 2 4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2023-01-11 DOI: 10.1007/s10851-022-01139-2

Carole Le Guyader, Samia Ainouz, S. Canu

引用次数: 0

Properties of Morphological Dilation in Max-Plus and Plus-Prod Algebra in Connection with the Fourier Transformation Max-Plus和Plus-Prod代数中形态扩张的性质及傅立叶变换

IF 2 4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2023-01-09 DOI: 10.1007/s10851-022-01138-3

Marvin Kahra, M. Breuß

引用次数: 1

Guest Editorial JMIV Special Issue SSVM’21 客座编辑JMIV特刊SSVM ' 21

IF 2 4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2023-01-01 DOI: 10.1007/s10851-023-01140-3

M. Fadili, A. Moataz, Loïc Simon, J. Rabin, Y. Quéau

引用次数: 0

Connections Between Numerical Algorithms for PDEs and Neural Networks. PDE 数值算法与神经网络之间的联系。

IF 1.3 4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2023-01-01 Epub Date: 2022-06-24 DOI: 10.1007/s10851-022-01106-x

Tobias Alt, Karl Schrader, Matthias Augustin, Pascal Peter, Joachim Weickert

{"title":"Connections Between Numerical Algorithms for PDEs and Neural Networks.","authors":"Tobias Alt, Karl Schrader, Matthias Augustin, Pascal Peter, Joachim Weickert","doi":"10.1007/s10851-022-01106-x","DOIUrl":"10.1007/s10851-022-01106-x","url":null,"abstract":"<p><p>We investigate numerous structural connections between numerical algorithms for partial differential equations (PDEs) and neural architectures. Our goal is to transfer the rich set of mathematical foundations from the world of PDEs to neural networks. Besides structural insights, we provide concrete examples and experimental evaluations of the resulting architectures. Using the example of generalised nonlinear diffusion in 1D, we consider explicit schemes, acceleration strategies thereof, implicit schemes, and multigrid approaches. We connect these concepts to residual networks, recurrent neural networks, and U-net architectures. Our findings inspire a symmetric residual network design with provable stability guarantees and justify the effectiveness of skip connections in neural networks from a numerical perspective. Moreover, we present U-net architectures that implement multigrid techniques for learning efficient solutions of partial differential equation models, and motivate uncommon design choices such as trainable nonmonotone activation functions. Experimental evaluations show that the proposed architectures save half of the trainable parameters and can thus outperform standard ones with the same model complexity. Our considerations serve as a basis for explaining the success of popular neural architectures and provide a blueprint for developing new mathematically well-founded neural building blocks.</p>","PeriodicalId":16196,"journal":{"name":"Journal of Mathematical Imaging and Vision","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9883332/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10607081","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}

引用次数: 0

Analysis of Joint Shape Variation from Multi-Object Complexes 多目标综合体关节形状变化分析

IF 2 4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2022-12-17 DOI: 10.1007/s10851-022-01136-5

Zhiyuan Liu, Jörn Schulz, Mohsen Taheri, M. Styner, J. Damon, S. Pizer, J. S. Marron

引用次数: 3

Local Binary Patterns of Segments of a Binary Object for Shape Analysis 用于形状分析的二值对象的局部二值模式

IF 2 4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2022-11-24 DOI: 10.1007/s10851-022-01130-x

Ratnesh Kumar, Kalyani Mali

引用次数: 1

On a Variational Problem with a Nonstandard Growth Functional and Its Applications to Image Processing 非标准生长泛函的变分问题及其在图像处理中的应用

IF 2 4区数学

Journal of Mathematical Imaging and Vision Pub Date : 2022-11-16 DOI: 10.1007/s10851-022-01131-w

C. D'apice, P. Kogut, O. Kupenko, R. Manzo

引用次数: 5