Optimal Transport in Biomedical Imaging

Description

In this tutorial we will present a set of image analysis tools based on the mathematics of optimal transport, which can be used for a variety of biomedical imaging applications. In addition to providing basic theory relating to optimal mass transport, we will describe the recent development of image transforms with well-defined forward and inverse operations that have demonstrable advantages over other image transforms (e.g. Fourier, wavelet, Radon).

The final part of the tutorial will focus on applications of optimal transport relevant to biomedical imaging: image modeling, statistical analysis, classification, and inverse problems. The tutorial will feature live code demonstrations, and we will provide attendees with software that can be used to aid their own research.

Presentations

Code

Outline

The tutorial is divided into three parts as follows:

Part I: Introduction to optimal transport

Monge and Kantorovich formulations

• S.Kolouri, S.R. Park, M. Thorpe, D. Slepcev, and G.K. Rohde. Optimal mass transport: Signal processing and machine learning applications. IEEE Signal Processing Magazine, 34:43–59, 2017.
• C. Villani. Optimal Transport: Old and new. Springer Science and Business Media, 2008.

Riemmanian geometry and optimal transport

• S. Kolouri, A.B. Tosun, J.A. Ozolek, and G.K. Rohde. A continuous linear optimal transport approach for pattern analysis in image datasets. Pattern Recognition, 51:453–462, 2016.
• W. Wang, D. Slepcev, S. Basu, J.A. Ozolek, and G.K. Rohde. A linear optimal transportation framework for quantifying and visualizing variations in sets of images. International Journal of Computer Vision, 101:254–269, 2013.
• C. Villani. Optimal Transport: Old and new. Springer Science and Business Media, 2008.

Optimal transport in 1D and ND, and optimization methods

• S.Kolouri, S.R. Park, M. Thorpe, D. Slepcev, and G.K. Rohde. Optimal mass transport: Signal processing and machine learning applications. IEEE Signal Processing Magazine, 34:43–59, 2017.
• S.R. Park, S. Kolouri, S. Kundu, and G.K. Rohde. The cumulative distribution transform and linear pattern classification. Applied and Computational Harmonic Analysis, 2017.
• S. Kolouri, A.B. Tosun, J.A. Ozolek, and G.K. Rohde. A continuous linear optimal transport approach for pattern analysis in image datasets. Pattern Recognition, 51:453–462, 2016.

Part II: Transport-related transforms

Cumulative distribution transform (CDT) and Radon-CDT

• S.R. Park, S. Kolouri, S. Kundu, and G.K. Rohde. The cumulative distribution transform and linearpattern classification. Applied and Computational Harmonic Analysis, 2017.
• S. Kolouri, S.R. Park, and G.K. Rohde. The Radon cumulative distribution transform and its application to image classfication. IEEE Transactions on Image Processing, 25(2):920–934, 2016.

Linear optimal transport

• S. Kolouri, A.B. Tosun, J.A. Ozolek, and G.K. Rohde. A continuous linear optimal transport approach for pattern analysis in image datasets. Pattern Recognition, 51:453–462, 2016.
• S. Basu, S. Kolouri, and G.K. Rohde. Detecting and visualizing cell phenotype differences frommicroscopy images using transport-based morphometry. Proceedings of the National Academy of Sciences, 111(9):3448–3453, 2014.
• W. Wang, D. Slepcev, S. Basu, J.A. Ozolek, and G.K. Rohde. A linear optimal transportation framework for quantifying and visualizing variations in sets of images. International Journal of Computer Vision, 101:254–269, 2013.

Transport-based kernel methods

• S. Kolouri, Y. Zou, and G.K. Rohde. Sliced wasserstein kernels for probability distributions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 5258–5267, 2016.

Part III: Applications in biomedical image analysis

Image-based modeling

• S. Kolouri, A.B. Tosun, J.A. Ozolek, and G.K. Rohde. A continuous linear optimal transport approach for pattern analysis in image datasets. Pattern Recognition, 51:453–462, 2016.
• S. Basu, S. Kolouri, and G.K. Rohde. Detecting and visualizing cell phenotype differences frommicroscopy images using transport-based morphometry. Proceedings of the National Academy of Sciences, 111(9):3448–3453, 2014.
• W. Wang, D. Slepcev, S. Basu, J.A. Ozolek, and G.K. Rohde. A linear optimal transportation framework for quantifying and visualizing variations in sets of images. International Journal of Computer Vision, 101:254–269, 2013.
• W. Wang, J.A. Ozolek, D. Slepcev, A.B. Lee, C. Chen, and G.K. Rohde. An optimal transportation approach for nuclear structure-based pathology. IEEE Transactions on Medical Imaging, 30(3):621–631, 2011.

Statistical analysis

• S. Kundu, S. Kolouri, K.I. Erickson, A.F. Kramer, E. McAuley, and G.K. Rohde. Discovery and visualization of structural biomarkers from MRI using transport-based morphometry. NeuroImage, 167:256-275, 2018

Image reconstruction and super-resolution

• S. Kolouri, and G.K. Rohde. Transport-based single frame super resolution of very low resolution face images. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2015.

Data classification

• S.R. Park, S. Kolouri, S. Kundu, and G.K. Rohde. The cumulative distribution transform and linearpattern classification. Applied and Computational Harmonic Analysis, 2017.
• S. Kolouri, S.R. Park, and G.K. Rohde. The Radon cumulative distribution transform and its application to image classfication. IEEE Transactions on Image Processing, 25(2):920–934, 2016.

Additional References

• S. Kolouri, G.K. Rohde, and H. Hoffmann. Sliced Wasserstein Distance for Learning Gaussian Mixture Models. To appear in Conference on Computer Vision and Pattern Recognition, 2018.
• S. Kolouri, C.E. Martin, and G.K. Rohde. Sliced Wasserstein Auto-Encoders: An embarrassingly simple generative model. arXiv, 2018

Biography

Liam Cattell earned his M.Eng degree in Engineering Science from the University of Oxford, UK, in 2012. He also received his D.Phil in Biomedical Image Analysis from the University of Oxford, UK, in 2016, where his research was focused on combining image registration and machine learning methods to help aid the diagnosis of Alzheimer's disease. He is currently a post-doctoral researcher in the Imaging and Data Science Lab at the University of Virginia, Charlottesville, USA, where he conducts research on biomedical image analysis problems using the mathematics of optimal transport.

Soheil Kolouri is a research scientist at HRL Laboratories, Malibu, CA. His research interests include machine learning, computer vision, and statistical signal processing. He received his B.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2010, and his M.Sc. degree in electrical engineering in 2012 from Colorado State University, Fort Collins, Colorado. He received his doctorate degree in biomedical engineering from Carnegie Mellon University in 2015. He was the recipient of the Bertucci fellowship award in 2014 and his thesis, titled, “Transport-based pattern recognition and image modeling”, won the outstanding thesis award from the Biomedical Engineering Department at Carnegie Mellon University in 2015.

Gustavo K. Rohde is an associate professor of Biomedical Engineering, and Electrical and Computer Engineering at the University of Virginia, Charlottesville, VA, USA. He has authored over 60 peer reviewed publications and is currently serving as an associate editor for IEEE Transactions on Image Processing, BMC Bioinformatics, and IEEE Journal of Biomedical and Health Informatics, in addition to being on the editorial board of Cytometry A. His research and teaching interests include predictive modeling in medicine and biology, cytometry, signal and image processing, computer vision, machine learning, and mobile and remote sensing.