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Zhang R, Cao Z, Yang S, Si L, Sun H, Xu L, Sun F. Cognition-Driven Structural Prior for Instance-Dependent Label Transition Matrix Estimation. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2025; 36:3730-3743. [PMID: 38190682 DOI: 10.1109/tnnls.2023.3347633] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/10/2024]
Abstract
The label transition matrix has emerged as a widely accepted method for mitigating label noise in machine learning. In recent years, numerous studies have centered on leveraging deep neural networks to estimate the label transition matrix for individual instances within the context of instance-dependent noise. However, these methods suffer from low search efficiency due to the large space of feasible solutions. Behind this drawback, we have explored that the real murderer lies in the invalid class transitions, that is, the actual transition probability between certain classes is zero but is estimated to have a certain value. To mask the invalid class transitions, we introduced a human-cognition-assisted method with structural information from human cognition. Specifically, we introduce a structured transition matrix network (STMN) designed with an adversarial learning process to balance instance features and prior information from human cognition. The proposed method offers two advantages: 1) better estimation effectiveness is obtained by sparing the transition matrix and 2) better estimation accuracy is obtained with the assistance of human cognition. By exploiting these two advantages, our method parametrically estimates a sparse label transition matrix, effectively converting noisy labels into true labels. The efficiency and superiority of our proposed method are substantiated through comprehensive comparisons with state-of-the-art methods on three synthetic datasets and a real-world dataset. Our code will be available at https://github.com/WheatCao/STMN-Pytorch.
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Song H, Kim M, Park D, Shin Y, Lee JG. Learning From Noisy Labels With Deep Neural Networks: A Survey. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023; 34:8135-8153. [PMID: 35254993 DOI: 10.1109/tnnls.2022.3152527] [Citation(s) in RCA: 76] [Impact Index Per Article: 38.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
Deep learning has achieved remarkable success in numerous domains with help from large amounts of big data. However, the quality of data labels is a concern because of the lack of high-quality labels in many real-world scenarios. As noisy labels severely degrade the generalization performance of deep neural networks, learning from noisy labels (robust training) is becoming an important task in modern deep learning applications. In this survey, we first describe the problem of learning with label noise from a supervised learning perspective. Next, we provide a comprehensive review of 62 state-of-the-art robust training methods, all of which are categorized into five groups according to their methodological difference, followed by a systematic comparison of six properties used to evaluate their superiority. Subsequently, we perform an in-depth analysis of noise rate estimation and summarize the typically used evaluation methodology, including public noisy datasets and evaluation metrics. Finally, we present several promising research directions that can serve as a guideline for future studies.
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Ma F, Wu Y, Yu X, Yang Y. Learning With Noisy Labels via Self-Reweighting From Class Centroids. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2022; 33:6275-6285. [PMID: 33961567 DOI: 10.1109/tnnls.2021.3073248] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Although deep neural networks have been proved effective in many applications, they are data hungry, and training deep models often requires laboriously labeled data. However, when labeled data contain erroneous labels, they often lead to model performance degradation. A common solution is to assign each sample with a dynamic weight during optimization, and the weight is adjusted in accordance with the loss. However, those weights are usually unreliable since they are measured by the losses of corrupted labels. Thus, this scheme might impede the discriminative ability of neural networks trained on noisy data. To address this issue, we propose a novel reweighting method, dubbed self-reweighting from class centroids (SRCC), by assigning sample weights based on the similarities between the samples and our online learned class centroids. Since we exploit statistical class centers in the image feature space to reweight data samples in learning, our method is robust to noise caused by corrupted labels. In addition, even after reweighting the noisy data, the decision boundaries might still suffer distortions. Thus, we leverage mixed inputs that are generated by linearly interpolating two random images and their labels to further regularize the boundaries. We employ the learned class centroids to evaluate the confidence of our generated mixed data via measuring feature similarities. During the network optimization, the class centroids are updated as more discriminative feature representations of original images are learned. In doing so, SRCC will generate more robust weighting coefficients for noisy and mixed data and facilitates our feature representation learning in return. Extensive experiments on both the synthetic and real image recognition tasks demonstrate that our method SRCC outperforms the state of the art on learning with noisy data.
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Xia S, Chen B, Wang G, Zheng Y, Gao X, Giem E, Chen Z. mCRF and mRD: Two Classification Methods Based on a Novel Multiclass Label Noise Filtering Learning Framework. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2022; 33:2916-2930. [PMID: 33428577 DOI: 10.1109/tnnls.2020.3047046] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Mitigating label noise is a crucial problem in classification. Noise filtering is an effective method of dealing with label noise which does not need to estimate the noise rate or rely on any loss function. However, most filtering methods focus mainly on binary classification, leaving the more difficult counterpart problem of multiclass classification relatively unexplored. To remedy this deficit, we present a definition for label noise in a multiclass setting and propose a general framework for a novel label noise filtering learning method for multiclass classification. Two examples of noise filtering methods for multiclass classification, multiclass complete random forest (mCRF) and multiclass relative density, are derived from their binary counterparts using our proposed framework. In addition, to optimize the NI_threshold hyperparameter in mCRF, we propose two new optimization methods: a new voting cross-validation method and an adaptive method that employs a 2-means clustering algorithm. Furthermore, we incorporate SMOTE into our label noise filtering learning framework to handle the ubiquitous problem of imbalanced data in multiclass classification. We report experiments on both synthetic data sets and UCI benchmarks to demonstrate our proposed methods are highly robust to label noise in comparison with state-of-the-art baselines. All code and data results are available at https://github.com/syxiaa/Multiclass-Label-Noise-Filtering-Learning.
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Gao M, Feng X, Geng M, Jiang Z, Zhu L, Meng X, Zhou C, Ren Q, Lu Y. Bayesian statistics-guided label refurbishment mechanism: Mitigating label noise in medical image classification. Med Phys 2022; 49:5899-5913. [PMID: 35678232 DOI: 10.1002/mp.15799] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2022] [Revised: 03/26/2022] [Accepted: 05/31/2022] [Indexed: 11/12/2022] Open
Abstract
PURPOSE Deep neural networks (DNNs) have been widely applied in medical image classification, benefiting from its powerful mapping capability among medical images. However, these existing deep learning-based methods depend on an enormous amount of carefully labeled images. Meanwhile, noise is inevitably introduced in the labeling process, degrading the performance of models. Hence, it is significant to devise robust training strategies to mitigate label noise in the medical image classification tasks. METHODS In this work, we propose a novel Bayesian statistics-guided label refurbishment mechanism (BLRM) for DNNs to prevent overfitting noisy images. BLRM utilizes maximum a posteriori probability in the Bayesian statistics and the exponentially time-weighted technique to selectively correct the labels of noisy images. The training images are purified gradually with the training epochs when BLRM is activated, further improving classification performance. RESULTS Comprehensive experiments on both synthetic noisy images (public OCT & Messidor datasets) and real-world noisy images (ANIMAL-10N) demonstrate that BLRM refurbishes the noisy labels selectively, curbing the adverse effects of noisy data. Also, the anti-noise BLRMs integrated with DNNs are effective at different noise ratio and are independent of backbone DNN architectures. In addition, BLRM is superior to state-of-the-art comparative methods of anti-noise. CONCLUSIONS These investigations indicate that the proposed BLRM is well capable of mitigating label noise in medical image classification tasks.
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Affiliation(s)
- Mengdi Gao
- Department of Biomedical Engineering, College of Future Technology, Peking University, Beijing, China.,Institute of Medical Technology, Peking University Health Science Center, Peking University, Beijing, China.,Institute of Biomedical Engineering, Peking University Shenzhen Graduate School, Shenzhen, China.,Shenzhen Bay Laboratory 5F, Institute of Biomedical Engineering, Shenzhen, China
| | - Ximeng Feng
- Department of Biomedical Engineering, College of Future Technology, Peking University, Beijing, China.,Institute of Medical Technology, Peking University Health Science Center, Peking University, Beijing, China.,Institute of Biomedical Engineering, Peking University Shenzhen Graduate School, Shenzhen, China.,Shenzhen Bay Laboratory 5F, Institute of Biomedical Engineering, Shenzhen, China
| | - Mufeng Geng
- Department of Biomedical Engineering, College of Future Technology, Peking University, Beijing, China.,Institute of Medical Technology, Peking University Health Science Center, Peking University, Beijing, China.,Institute of Biomedical Engineering, Peking University Shenzhen Graduate School, Shenzhen, China.,Shenzhen Bay Laboratory 5F, Institute of Biomedical Engineering, Shenzhen, China
| | - Zhe Jiang
- Department of Biomedical Engineering, College of Future Technology, Peking University, Beijing, China.,Institute of Medical Technology, Peking University Health Science Center, Peking University, Beijing, China.,Institute of Biomedical Engineering, Peking University Shenzhen Graduate School, Shenzhen, China.,Shenzhen Bay Laboratory 5F, Institute of Biomedical Engineering, Shenzhen, China
| | - Lei Zhu
- Department of Biomedical Engineering, College of Future Technology, Peking University, Beijing, China.,Institute of Medical Technology, Peking University Health Science Center, Peking University, Beijing, China.,Institute of Biomedical Engineering, Peking University Shenzhen Graduate School, Shenzhen, China.,Shenzhen Bay Laboratory 5F, Institute of Biomedical Engineering, Shenzhen, China
| | - Xiangxi Meng
- Key Laboratory of Carcinogenesis and Translational Research (Ministry of Education), Key Laboratory for Research and Evaluation of Radiopharmaceuticals (National Medical Products Administration), Department of Nuclear Medicine, Beijing Cancer Hospital & Institute, Beijing, China
| | - Chuanqing Zhou
- Shenzhen Bay Laboratory 5F, Institute of Biomedical Engineering, Shenzhen, China
| | - Qiushi Ren
- Department of Biomedical Engineering, College of Future Technology, Peking University, Beijing, China.,Institute of Medical Technology, Peking University Health Science Center, Peking University, Beijing, China.,Institute of Biomedical Engineering, Peking University Shenzhen Graduate School, Shenzhen, China.,Shenzhen Bay Laboratory 5F, Institute of Biomedical Engineering, Shenzhen, China
| | - Yanye Lu
- Institute of Medical Technology, Peking University Health Science Center, Peking University, Beijing, China.,Institute of Biomedical Engineering, Peking University Shenzhen Graduate School, Shenzhen, China
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Deconfounded classification by an intervention approach. INT J MACH LEARN CYB 2022. [DOI: 10.1007/s13042-021-01486-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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7
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TEMImageNet training library and AtomSegNet deep-learning models for high-precision atom segmentation, localization, denoising, and deblurring of atomic-resolution images. Sci Rep 2021; 11:5386. [PMID: 33686158 PMCID: PMC7940611 DOI: 10.1038/s41598-021-84499-w] [Citation(s) in RCA: 36] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2020] [Accepted: 02/10/2021] [Indexed: 02/07/2023] Open
Abstract
Atom segmentation and localization, noise reduction and deblurring of atomic-resolution scanning transmission electron microscopy (STEM) images with high precision and robustness is a challenging task. Although several conventional algorithms, such has thresholding, edge detection and clustering, can achieve reasonable performance in some predefined sceneries, they tend to fail when interferences from the background are strong and unpredictable. Particularly, for atomic-resolution STEM images, so far there is no well-established algorithm that is robust enough to segment or detect all atomic columns when there is large thickness variation in a recorded image. Herein, we report the development of a training library and a deep learning method that can perform robust and precise atom segmentation, localization, denoising, and super-resolution processing of experimental images. Despite using simulated images as training datasets, the deep-learning model can self-adapt to experimental STEM images and shows outstanding performance in atom detection and localization in challenging contrast conditions and the precision consistently outperforms the state-of-the-art two-dimensional Gaussian fit method. Taking a step further, we have deployed our deep-learning models to a desktop app with a graphical user interface and the app is free and open-source. We have also built a TEM ImageNet project website for easy browsing and downloading of the training data.
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Algan G, Ulusoy I. Image classification with deep learning in the presence of noisy labels: A survey. Knowl Based Syst 2021. [DOI: 10.1016/j.knosys.2021.106771] [Citation(s) in RCA: 62] [Impact Index Per Article: 15.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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9
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Robust multiclass least squares support vector classifier with optimal error distribution. Knowl Based Syst 2021. [DOI: 10.1016/j.knosys.2020.106652] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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Zhang CB, Jiang PT, Hou Q, Wei Y, Han Q, Li Z, Cheng MM. Delving Deep Into Label Smoothing. IEEE TRANSACTIONS ON IMAGE PROCESSING : A PUBLICATION OF THE IEEE SIGNAL PROCESSING SOCIETY 2021; 30:5984-5996. [PMID: 34166191 DOI: 10.1109/tip.2021.3089942] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/14/2023]
Abstract
Label smoothing is an effective regularization tool for deep neural networks (DNNs), which generates soft labels by applying a weighted average between the uniform distribution and the hard label. It is often used to reduce the overfitting problem of training DNNs and further improve classification performance. In this paper, we aim to investigate how to generate more reliable soft labels. We present an Online Label Smoothing (OLS) strategy, which generates soft labels based on the statistics of the model prediction for the target category. The proposed OLS constructs a more reasonable probability distribution between the target categories and non-target categories to supervise DNNs. Experiments demonstrate that based on the same classification models, the proposed approach can effectively improve the classification performance on CIFAR-100, ImageNet, and fine-grained datasets. Additionally, the proposed method can significantly improve the robustness of DNN models to noisy labels compared to current label smoothing approaches. The source code is available at our project page: https://mmcheng.net/ols/.
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Wei Y, Gong C, Chen S, Liu T, Yang J, Tao D. Harnessing Side Information for Classification Under Label Noise. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2020; 31:3178-3192. [PMID: 31562108 DOI: 10.1109/tnnls.2019.2938782] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Practical data sets often contain the label noise caused by various human factors or measurement errors, which means that a fraction of training examples might be mistakenly labeled. Such noisy labels will mislead the classifier training and severely decrease the classification performance. Existing approaches to handle this problem are usually developed through various surrogate loss functions under the framework of empirical risk minimization. However, they are only suitable for binary classification and also require strong prior knowledge. Therefore, this article treats the example features as side information and formulates the noisy label removal problem as a matrix recovery problem. We denote our proposed method as "label noise handling via side information" (LNSI). Specifically, the observed label matrix is decomposed as the sum of two parts, in which the first part reveals the true labels and can be obtained by conducting a low-rank mapping on the side information; and the second part captures the incorrect labels and is modeled by a row-sparse matrix. The merits of such formulation lie in three aspects: 1) the strong recovery ability of this strategy has been sufficiently demonstrated by intensive theoretical works on side information; 2) multi-class situations can be directly handled with the aid of learned projection matrix; and 3) only very weak assumptions are required for model design, making LNSI applicable to a wide range of practical problems. Moreover, we theoretically derive the generalization bound of LNSI and show that the expected classification error of LNSI is upper bounded. The experimental results on a variety of data sets including UCI benchmark data sets and practical data sets confirm the superiority of LNSI to state-of-the-art approaches on label noise handling.
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Chen Z, Duan J, Yang C, Kang L, Qiu G. SMLBoost-adopting a soft-margin like strategy in boosting. Knowl Based Syst 2020. [DOI: 10.1016/j.knosys.2020.105705] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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13
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Han B, Tsang IW, Chen L, Zhou JT, Yu CP. Beyond Majority Voting: A Coarse-to-Fine Label Filtration for Heavily Noisy Labels. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2019; 30:3774-3787. [PMID: 30892236 DOI: 10.1109/tnnls.2019.2899045] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
Crowdsourcing has become the most appealing way to provide a plethora of labels at a low cost. Nevertheless, labels from amateur workers are often noisy, which inevitably degenerates the robustness of subsequent learning models. To improve the label quality for subsequent use, majority voting (MV) is widely leveraged to aggregate crowdsourced labels due to its simplicity and scalability. However, when crowdsourced labels are "heavily" noisy (e.g., 40% of noisy labels), MV may not work well because of the fact "garbage (heavily noisy labels) in, garbage (full aggregated labels) out." This issue inspires us to think: if the ultimate target is to learn a robust model using noisy labels, why not provide partial aggregated labels and ensure that these labels are reliable enough for learning models? To solve this challenge by improving MV, we propose a coarse-to-fine label filtration model called double filter machine (DFM), which consists of a (majority) voting filter and a sparse filter serially. Specifically, the DFM refines crowdsourced labels from coarse filtering to fine filtering. In the stage of coarse filtering, the DFM aggregates crowdsourced labels by voting filter, which yields (quality-acceptable) full aggregated labels. In the stage of fine filtering, DFM further digs out a set of high-quality labels from full aggregated labels by sparse filter, since this filter can identify high-quality labels by the methodology of support selection. Based on the insight of compressed sensing, DFM recovers a ground-truth signal from heavily noisy data under a restricted isometry property. To sum up, the primary benefits of DFM are to keep the scalability by voting filter, while improve the robustness by sparse filter. We also derive theoretical guarantees for the convergence and recovery of DFM and reveal its complexity. We conduct comprehensive experiments on both the UCI simulated and the AMT crowdsourced datasets. Empirical results show that partial aggregated labels provided by DFM effectively improve the robustness of learning models.
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Tao D, Cheng J, Yu Z, Yue K, Wang L. Domain-Weighted Majority Voting for Crowdsourcing. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2019; 30:163-174. [PMID: 29994339 DOI: 10.1109/tnnls.2018.2836969] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Crowdsourcing labeling systems provide an efficient way to generate multiple inaccurate labels for given observations. If the competence level or the "reputation," which can be explained as the probabilities of annotating the right label, for each crowdsourcing annotators is equal and biased to annotate the right label, majority voting (MV) is the optimal decision rule for merging the multiple labels into a single reliable one. However, in practice, the competence levels of annotators employed by the crowdsourcing labeling systems are often diverse very much. In these cases, weighted MV is more preferred. The weights should be determined by the competence levels. However, since the annotators are anonymous and the ground-truth labels are usually unknown, it is hard to compute the competence levels of the annotators directly. In this paper, we propose to learn the weights for weighted MV by exploiting the expertise of annotators. Specifically, we model the domain knowledge of different annotators with different distributions and treat the crowdsourcing problem as a domain adaptation problem. The annotators provide labels to the source domains and the target domain is assumed to be associated with the ground-truth labels. The weights are obtained by matching the source domains with the target domain. Although the target-domain labels are unknown, we prove that they could be estimated under mild conditions. Both theoretical and empirical analyses verify the effectiveness of the proposed method. Large performance gains are shown for specific data sets.
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Yu X, Liu T, Gong M, Batmanghelich K, Tao D. An Efficient and Provable Approach for Mixture Proportion Estimation Using Linear Independence Assumption. CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION WORKSHOPS. IEEE COMPUTER SOCIETY CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION. WORKSHOPS 2018; 2018:4480-4489. [PMID: 32089968 DOI: 10.1109/cvpr.2018.00471] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
In this paper, we study the mixture proportion estimation (MPE) problem in a new setting: given samples from the mixture and the component distributions, we identify the proportions of the components in the mixture distribution. To address this problem, we make use of a linear independence assumption, i.e., the component distributions are independent from each other, which is much weaker than assumptions exploited in the previous MPE methods. Based on this assumption, we propose a method (1) that uniquely identifies the mixture proportions, (2) whose output provably converges to the optimal solution, and (3) that is computationally efficient. We show the superiority of the proposed method over the state-of-the-art methods in two applications including learning with label noise and semi-supervised learning on both synthetic and real-world datasets.
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Affiliation(s)
- Xiyu Yu
- UBTECH Sydney AI Centre, SIT, FEIT, The University of Sydney, Australia
| | - Tongliang Liu
- UBTECH Sydney AI Centre, SIT, FEIT, The University of Sydney, Australia
| | - Mingming Gong
- Department of Biomedical Informatics, University of Pittsburgh.,Department of Philosophy, Carnegie Mellon University
| | | | - Dacheng Tao
- UBTECH Sydney AI Centre, SIT, FEIT, The University of Sydney, Australia
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Fast Image Segmentation Using Two-Dimensional Otsu Based on Estimation of Distribution Algorithm. JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING 2017. [DOI: 10.1155/2017/1735176] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
Traditional two-dimensional Otsu algorithm has several drawbacks; that is, the sum of probabilities of target and background is approximate to 1 inaccurately, the details of neighborhood image are not obvious, and the computational cost is high. In order to address these problems, a method of fast image segmentation using two-dimensional Otsu based on estimation of distribution algorithm is proposed. Firstly, in order to enhance the performance of image segmentation, the guided filtering is employed to improve neighborhood image template instead of mean filtering. Additionally, the probabilities of target and background in two-dimensional histogram are exactly calculated to get more accurate threshold. Finally, the trace of the interclass dispersion matrix is taken as the fitness function of estimation of distributed algorithm, and the optimal threshold is obtained by constructing and sampling the probability model. Extensive experimental results demonstrate that our method can effectively preserve details of the target, improve the segmentation precision, and reduce the running time of algorithms.
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