Uncertainty quantification for semi-supervised multi-class classification in image processing and ego-motion analysis of body-worn videos

Qiao, Yi-Ling; Shi, Chang Zheng; Wang, Chenjian; Li, Hao; Haberland, Matt; Luo, Xiaoqiang; Stuart, Andrew M.; Bertozzi, Andrea L.

doi:10.2352/issn.2470-1173.2019.11.ipas-264

Cited by 11 publications

(11 citation statements)

References 16 publications

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“…We will seek to extend both our theoretical and numerical framework to multiclass graph‐based classification, as considered for example in [22, 28, 37]. The groundwork for this extension was laid by the authors in [14, Section 6].…”

Section: Resultsmentioning

confidence: 99%

“…Note In each of these examples we took as reference data a separate reference data image. However, our algorithm does not require this, and one could take a subset of the pixels of a single image to be the reference data, and thereby investigate the impact of the relative size of the reference data on the segmentation, which is beyond the scope of this paper but is explored for the [32] MBO segmentation algorithm and related methods in [37, Figure 4].…”

Section: Applications In Image Processingmentioning

confidence: 99%

“…This is of course not a rigorous Bayesian uncertainty quantification, as for instance is given in [7, 37] for graph‐based learning. However the use of stochastic algorithms for inference tasks, and the use of their output as a method of uncertainty quantification, has for instance been motivated by [31].…”

Section: Applications In Image Processingmentioning

confidence: 99%

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Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

2021

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This paper introduces a semi‐discrete implicit Euler (SDIE) scheme for the Allen‐Cahn equation (ACE) with fidelity forcing on graphs. The continuous‐in‐time version of this differential equation was pioneered by Bertozzi and Flenner in 2012 as a method for graph classification problems, such as semi‐supervised learning and image segmentation. In 2013, Merkurjev et. al. used a Merriman‐Bence‐Osher (MBO) scheme with fidelity forcing instead, as heuristically it was expected to give similar results to the ACE. The current paper rigorously establishes the graph MBO scheme with fidelity forcing as a special case of an SDIE scheme for the graph ACE with fidelity forcing. This connection requires the use of the double‐obstacle potential in the ACE, as was already demonstrated by Budd and Van Gennip in 2020 in the context of ACE without a fidelity forcing term. We also prove that solutions of the SDIE scheme converge to solutions of the graph ACE with fidelity forcing as the discrete time step converges to zero. In the second part of the paper we develop the SDIE scheme as a classification algorithm. We also introduce some innovations into the algorithms for the SDIE and MBO schemes. For large graphs, we use a QR decomposition method to compute an eigendecomposition from a Nyström extension, which outperforms the method used by, for example, Bertozzi and Flenner in 2012, in accuracy, stability, and speed. Moreover, we replace the Euler discretization for the scheme's diffusion step by a computation based on the Strang formula for matrix exponentials. We apply this algorithm to a number of image segmentation problems, and compare the performance with that of the graph MBO scheme with fidelity forcing. We find that while the general SDIE scheme does not perform better than the MBO special case at this task, our other innovations lead to a significantly better segmentation than that from previous literature. We also empirically quantify the uncertainty that this segmentation inherits from the randomness in the Nyström extension.

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Section: Resultsmentioning

confidence: 99%

Section: Applications In Image Processingmentioning

confidence: 99%

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Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

2021

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“…Bayesian Interpretation of SSL Problems. These graph-based SSL objective functions lend themselves to a Bayesian probabilistic interpretation, as discussed in prior literature [42,7,15,18,29,32,31]. In the binary case, (2.1) is equivalent to finding the maximum a posteriori (MAP) estimate of a posterior probability distribution whose density P(u|y) relates to the objective function via P(u|y) ∝ exp (−J (u; y)) = e − 1 2 u,Lτ u e − j∈L (u j ,y j ) ∝ µ(u)e −Φ (u;y) , (2.4) where the prior µ(u) follows a Gaussian prior, N (0, L −1 τ ), and the likelihood, q(y|u) ∝ exp(−Φ (u; y)), is defined by the likelihood potential Φ (u; y) := j∈L (u j , y j ).…”

Section: This Workmentioning

confidence: 99%

Model-Change Active Learning in Graph-Based Semi-Supervised Learning

Miller,

Bertozzi

2021

Preprint

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Active learning in semi-supervised classification involves introducing additional labels for unlabelled data to improve the accuracy of the underlying classifier. A challenge is to identify which points to label to best improve performance while limiting the number of new labels. "Model-change" active learning quantifies the resulting change incurred in the classifier by introducing the additional label(s). We pair this idea with graph-based semi-supervised learning methods, that use the spectrum of the graph Laplacian matrix, which can be truncated to avoid prohibitively large computational and storage costs. We consider a family of convex loss functions for which the acquisition function can be efficiently approximated using the Laplace approximation of the posterior distribution. We show a variety of multiclass examples that illustrate improved performance over prior state-of-art.

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“…In each of these examples we took as reference data a separate reference data image. However, our algorithm does not require this, and one could take a subset of the pixels of a single image to be the reference data, and thereby investigate the impact of the relative size of the reference data on the segmentation, which is beyond the scope of this paper but is explored for the [2] MBO segmentation algorithm and related methods in [35,Fig. 4].…”

Section: Data Image Imagementioning

confidence: 99%

Classification and image processing with a semi-discrete scheme for fidelity forced Allen--Cahn on graphs

Budd,

van Gennip,

Latz

2020

Preprint

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This paper introduces a semi-discrete implicit Euler (SDIE) scheme for the Allen-Cahn equation (ACE) with fidelity forcing on graphs. Bertozzi and Flenner (2012) pioneered the use of this differential equation as a method for graph classification problems, such as semi-supervised learning and image segmentation. In Merkurjev, Kostić, and Bertozzi (2013), a Merriman-Bence-Osher (MBO) scheme with fidelity forcing was used instead, as the MBO scheme is heuristically similar to the ACE. This paper rigorously establishes the graph MBO scheme with fidelity forcing as a special case of an SDIE scheme for the graph ACE with fidelity forcing. This connection requires using the double-obstacle potential in the ACE, as was shown in Budd and Van Gennip (2020) for ACE without fidelity forcing. We also prove that solutions of the SDIE scheme converge to solutions of the graph ACE with fidelity forcing as the SDIE time step tends to zero.Next, we develop the SDIE scheme as a classification algorithm. We also introduce some innovations into the algorithms for the SDIE and MBO schemes. For large graphs, we use a QR decomposition method to compute an eigendecomposition from a Nyström extension, which outperforms the method used in e.g. Bertozzi and Flenner (2012) in accuracy, stability, and speed. Moreover, we replace the Euler discretisation for the scheme's diffusion step by a computation based on the Strang formula for matrix exponentials. We apply this algorithm to a number of image segmentation problems, and compare the performance of the SDIE and MBO schemes. We find that whilst the general SDIE scheme does not perform better than the MBO special case at this task, our other innovations lead to a significantly better segmentation than that from previous literature. We also empirically quantify the uncertainty that this segmentation inherits from the randomness in the Nyström extension.

show abstract

Uncertainty quantification for semi-supervised multi-class classification in image processing and ego-motion analysis of body-worn videos

Cited by 11 publications

References 16 publications

Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

Model-Change Active Learning in Graph-Based Semi-Supervised Learning

Classification and image processing with a semi-discrete scheme for fidelity forced Allen--Cahn on graphs

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