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Kang S, Oh HS. Probabilistic Principal Curves on Riemannian Manifolds. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2024; 46:4843-4849. [PMID: 38265902 DOI: 10.1109/tpami.2024.3357801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/26/2024]
Abstract
This paper studies a new curve-fitting approach to data on Riemannian manifolds. We define a principal curve based on a mixture model for observations and unobserved latent variables and propose a new algorithm to estimate the principal curve for given data points on Riemannian manifolds.
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Peng T, Gu Y, Zhang J, Dong Y, DI G, Wang W, Zhao J, Cai J. A Robust and Explainable Structure-Based Algorithm for Detecting the Organ Boundary From Ultrasound Multi-Datasets. J Digit Imaging 2023; 36:1515-1532. [PMID: 37231289 PMCID: PMC10406792 DOI: 10.1007/s10278-023-00839-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2023] [Revised: 04/19/2023] [Accepted: 04/20/2023] [Indexed: 05/27/2023] Open
Abstract
Detecting the organ boundary in an ultrasound image is challenging because of the poor contrast of ultrasound images and the existence of imaging artifacts. In this study, we developed a coarse-to-refinement architecture for multi-organ ultrasound segmentation. First, we integrated the principal curve-based projection stage into an improved neutrosophic mean shift-based algorithm to acquire the data sequence, for which we utilized a limited amount of prior seed point information as the approximate initialization. Second, a distribution-based evolution technique was designed to aid in the identification of a suitable learning network. Then, utilizing the data sequence as the input of the learning network, we achieved the optimal learning network after learning network training. Finally, a scaled exponential linear unit-based interpretable mathematical model of the organ boundary was expressed via the parameters of a fraction-based learning network. The experimental outcomes indicated that our algorithm 1) achieved more satisfactory segmentation outcomes than state-of-the-art algorithms, with a Dice score coefficient value of 96.68 ± 2.2%, a Jaccard index value of 95.65 ± 2.16%, and an accuracy of 96.54 ± 1.82% and 2) discovered missing or blurry areas.
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Affiliation(s)
- Tao Peng
- School of Future Science and Engineering, Soochow University, Suzhou, China
- Department of Health Technology and Informatics, Hong Kong Polytechnic University, Hong Kong, China
- Department of Radiation Oncology, UT Southwestern Medical Center, Dallas, TX USA
| | - Yidong Gu
- School of Future Science and Engineering, Soochow University, Suzhou, China
- Department of Medical Ultrasound, the Affiliated Suzhou Hospital of Nanjing Medical University, Suzhou Municipal Hospital, Suzhou, Jiangsu China
| | - Ji Zhang
- Department of Radiology, The Affiliated Taizhou People’s Hospital of Nanjing Medical University, Taizhou, Jiangsu Province, China
| | - Yan Dong
- Department of Ultrasonography, The First Affiliated Hospital of Soochow University, Suzhou, Jiangsu Province, China
| | - Gongye DI
- Department of Ultrasonic, The Affiliated Taizhou People’s Hospital of Nanjing Medical University, Taizhou, Jiangsu Province, China
| | - Wenjie Wang
- Department of Radio-Oncology, The Affiliated Suzhou Hospital of Nanjing Medical University, Suzhou Municipal Hospital, Suzhou, Jiangsu China
| | - Jing Zhao
- Department of Ultrasound, Tsinghua University Affiliated Beijing Tsinghua Changgung Hospital, Beijing, China
| | - Jing Cai
- Department of Health Technology and Informatics, Hong Kong Polytechnic University, Hong Kong, China
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Beinert R, Bërdëllima A, Gräf M, Steidl G. On the dynamical system of principal curves in. COMMUN STAT-SIMUL C 2022. [DOI: 10.1080/03610918.2022.2092139] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Affiliation(s)
- Robert Beinert
- Institute of Mathematics, Technische Universität Berlin, Berlin, Germany
| | - Arian Bërdëllima
- Institute of Mathematics, Technische Universität Berlin, Berlin, Germany
| | - Manuel Gräf
- Institute of Mathematics, Technische Universität Berlin, Berlin, Germany
| | - Gabriele Steidl
- Institute of Mathematics, Technische Universität Berlin, Berlin, Germany
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Fan X, Yang CH, Vemuri BC. Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design. PROCEEDINGS. IEEE COMPUTER SOCIETY CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION 2022; 2022:356-365. [PMID: 36911245 PMCID: PMC9997089 DOI: 10.1109/cvpr52688.2022.00045] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
Abstract
Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group which is a semi-Riemannian manifold, i.e. a manifold equipped with an indefinite metric. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) followed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower-dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transformations, which are the natural isometric transformations in hyperbolic spaces. This projection is computationally efficient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it allows for weight sharing. The nested hyperbolic space representation is the core component of our network and therefore, we first compare this representation - independent of the network - with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projection. Finally, we present experiments demonstrating comparative performance of our network on several publicly available data sets.
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Affiliation(s)
- Xiran Fan
- University of Florida, Department of Statistics
| | - Chun-Hao Yang
- National Taiwan University, Institute of Applied Mathematical Sciences
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Lee J, Kim JH, Oh HS. Spherical Principal Curves. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2021; 43:2165-2171. [PMID: 32956037 DOI: 10.1109/tpami.2020.3025327] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
This paper presents a new approach for dimension reduction of data observed on spherical surfaces. Several dimension reduction techniques have been developed in recent years for non-euclidean data analysis. As a pioneer work, (Hauberg 2016) attempted to implement principal curves on Riemannian manifolds. However, this approach uses approximations to process data on Riemannian manifolds, resulting in distorted results. This study proposes a new approach to project data onto a continuous curve to construct principal curves on spherical surfaces. Our approach lies in the same line of (Hastie and Stuetzle et al. 1989) that proposed principal curves for data on euclidean space. We further investigate the stationarity of the proposed principal curves that satisfy the self-consistency on spherical surfaces. The results on the real data analysis and simulation examples show promising empirical characteristics of the proposed approach.
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Meng K, Eloyan A. Principal Manifold Estimation via Model Complexity Selection. J R Stat Soc Series B Stat Methodol 2021; 83:369-394. [PMID: 35813449 PMCID: PMC9264480 DOI: 10.1111/rssb.12416] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
We propose a framework of principal manifolds to model high-dimensional data. This framework is based on Sobolev spaces and designed to model data of any intrinsic dimension. It includes principal component analysis and principal curve algorithm as special cases. We propose a novel method for model complexity selection to avoid overfitting, eliminate the effects of outliers, and improve the computation speed. Additionally, we propose a method for identifying the interiors of circle-like curves and cylinder/ball-like surfaces. The proposed approach is compared to existing methods by simulations and applied to estimate tumor surfaces and interiors in a lung cancer study.
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Affiliation(s)
- Kun Meng
- Department of Biostatistics, Brown University School of Public Health, Providence, RI 02903, USA
| | - Ani Eloyan
- Department of Biostatistics, Brown University School of Public Health, Providence, RI 02903, USA
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Affiliation(s)
- Zhigang Yao
- Department of Statistics and Applied Probability, National University of Singapore, Singapore
| | - Zhenyue Zhang
- School of Mathematical Science, Zhejiang University, Hangzhou, China
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Intelligent active fault-tolerant system for multi-source integrated navigation system based on deep neural network. Neural Comput Appl 2019. [DOI: 10.1007/s00521-018-03975-z] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Laplacian Eigenmaps Feature Conversion and Particle Swarm Optimization-Based Deep Neural Network for Machine Condition Monitoring. APPLIED SCIENCES-BASEL 2018. [DOI: 10.3390/app8122611] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
This work reports a novel method by fusing Laplacian Eigenmaps feature conversion and deep neural network (DNN) for machine condition assessment. Laplacian Eigenmaps is adopted to transform data features from original high dimension space to projected lower dimensional space, the DNN is optimized by the particle swarm optimization algorithm, and the machine run-to-failure experiment were investigated for validation studies. Through a series of comparative experiments with the original features, two other effective space transformation techniques, Principal Component Analysis (PCA) and Isometric map (Isomap), and two other artificial intelligence methods, hidden Markov model (HMM) as well as back-propagation neural network (BPNN), the present method in this paper proved to be more effective for machine operation condition assessment.
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