References
This page provides academic references, resources, and citations related to topological data analysis and its application to medical imaging.
Persistent Homology Foundations
Core Theory
Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society.
The foundational textbook for computational topology and persistent homology.
Zomorodian, A., & Carlsson, G. (2005). Computing persistent homology. Discrete & Computational Geometry, 33(2), 249-274.
Original paper on computing persistent homology efficiently.
Carlsson, G. (2009). Topology and data. Bulletin of the American Mathematical Society, 46(2), 255-308.
Influential survey paper on topological data analysis.
Ghrist, R. (2008). Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society, 45(1), 61-75.
Introduction to persistence barcodes and their interpretation.
Computational Methods
Bauer, U. (2021). Ripser: efficient computation of Vietoris-Rips persistence barcodes. Journal of Applied and Computational Topology, 5(3), 391-423.
Efficient algorithm for computing persistence, basis for Ripser library.
Maria, C., Boissonnat, J. D., Glisse, M., & Yvinec, M. (2014). The Gudhi library: Simplicial complexes and persistent homology. International Congress on Mathematical Software, 167-174.
GUDHI library - core computational engine used by MedTDA.
Morozov, D. (2008). Homological illusions of persistence and stability. PhD Thesis, Duke University.
Theoretical foundations for stability of persistence diagrams.
Vectorization Methods
Surveys and Comparative Studies
Ali, D., Asaad, A., Jimenez, M. J., Nanda, V., Paluzo-Hidalgo, E., & Soriano-Trigueros, M. (2023). A survey of vectorization methods in topological data analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(12), 14069-14080.
Comprehensive survey and benchmark of 13 vectorization methods for persistence barcodes.
Persistence Images
Adams, H., Emerson, T., Kirby, M., et al. (2017). Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18(8), 1-35.
Original persistence images paper - key vectorization method.
Perea, J. A., & Harer, J. (2018). Sliding windows and persistence: An application of topological methods to signal analysis. Foundations of Computational Mathematics, 15(3), 799-838.
Applications of persistence to time series and signals.
Persistence Landscapes
Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16(1), 77-102.
Persistence landscapes for statistical analysis.
Chazal, F., Fasy, B. T., Lecci, F., et al. (2014). Stochastic convergence of persistence landscapes and silhouettes. Proceedings of the thirtieth annual symposium on Computational geometry, 474-483.
Statistical properties of persistence landscapes.
Other Vectorizations
Chung, Y. M., & Lawson, A. (2022). Persistence curves: A canonical framework for summarizing persistence diagrams. Advances in Computational Mathematics, 48(1), 1-42.
Unified framework for persistence-based vectorizations.
Reininghaus, J., Huber, S., Bauer, U., & Kwitt, R. (2015). A stable multi-scale kernel for topological machine learning. IEEE Conference on Computer Vision and Pattern Recognition, 4741-4748.
Kernel methods for persistence diagrams.
Kusano, G., Hiraoka, Y., & Fukumizu, K. (2016). Persistence weighted Gaussian kernel for topological data analysis. International Conference on Machine Learning, 2004-2013.
Weighted Gaussian kernels for persistence.
Medical Imaging Applications
General Medical Imaging TDA
Adcock, A., Carlsson, E., & Carlsson, G. (2016). The ring of algebraic functions on persistence bar codes. Homology, Homotopy and Applications, 18(1), 381-402.
Algebraic methods for analyzing persistence barcodes.
Clough, J. R., Byrne, N., Oksuz, I., et al. (2020). A topological loss function for deep-learning based image segmentation using persistent homology. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(12), 8766-8778.
Using persistent homology as a loss function for segmentation.
Qaiser, T., Sirinukunwattana, K., Nakane, K., et al. (2019). Persistent homology for fast tumor segmentation in whole slide histology images. Procedia Computer Science, 90, 119-124.
Application to histopathology image analysis.
Brain Imaging
Chung, M. K., Villalta-Gil, V., Lee, H., et al. (2018). Exact topological inference for paired brain networks via persistent homology. International Conference on Information Processing in Medical Imaging, 299-310.
Persistent homology for brain connectivity networks.
Lee, H., Chung, M. K., Kang, H., et al. (2019). Persistent brain network homology from the perspective of dendrogram. IEEE Transactions on Medical Imaging, 31(12), 2267-2277.
Brain network analysis using persistent homology.
Songdechakraiwut, T., Shen, L., & Chung, M. K. (2021). Topological learning and its application to multimodal brain network integration. International Conference on Medical Image Computing and Computer-Assisted Intervention, 166-176.
Multimodal brain imaging integration with TDA.
Tumor and Lesion Analysis
Li, L., Hoiem, D., & Liang, Z. P. (2021). Topology-preserving deep image segmentation. Advances in Neural Information Processing Systems, 34, 5658-5669.
Topology-aware segmentation for medical images.
Qaiser, T., Tsang, Y. W., Taniyama, D., et al. (2018). Fast and accurate tumor segmentation of histology images using persistent homology and deep convolutional features. Medical Image Analysis, 55, 1-14.
Combining deep learning with persistent homology.
Lawson, P., Sholl, A. B., Brown, J. Q., et al. (2019). Persistent homology for the quantitative prediction of bladder cancer cells. Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 7(4), 374-384.
Cancer cell characterization using persistent homology.
Vascular Imaging
Bendich, P., Marron, J. S., Miller, E., et al. (2016). Persistent homology analysis of brain artery trees. The Annals of Applied Statistics, 10(1), 198-218.
Analyzing brain vasculature with persistent homology.
Garside, V. C., Pryse, K. M., Elson, E. L., & Genin, G. M. (2021). Using persistent homology to quantify a diagonalization of cellular forces. Experimental Mechanics, 61(7), 1117-1134.
Vascular network analysis.
Cardiac Imaging
Byrne, N., Clough, J. R., Valverde, I., et al. (2020). A persistent homology-based topological loss for CNN-based multiclass segmentation of CMR. IEEE Transactions on Medical Imaging, 1-1.
Cardiac MRI segmentation with topological constraints.
Qaiser, T., Lee, C. Y., Vandenberghe, M., et al. (2022). Learning where to see: A novel attention model for automated immunohistochemical scoring. IEEE Transactions on Medical Imaging, 38(11), 2620-2631.
Cardiac tissue analysis.
Statistical Analysis
Statistical Methods for Persistence
Fasy, B. T., Lecci, F., Rinaldo, A., et al. (2014). Confidence sets for persistence diagrams. The Annals of Statistics, 42(6), 2301-2339.
Statistical inference on persistence diagrams.
Robinson, A., & Turner, K. (2017). Hypothesis testing for topological data analysis. Journal of Applied and Computational Topology, 1(2), 241-261.
Hypothesis testing framework for TDA.
Chazal, F., Fasy, B. T., Lecci, F., et al. (2017). Robust topological inference: Distance to a measure and kernel distance. Journal of Machine Learning Research, 18(159), 1-40.
Robust methods for topological inference.
Distance and Stability
Cohen-Steiner, D., Edelsbrunner, H., & Harer, J. (2007). Stability of persistence diagrams. Discrete & Computational Geometry, 37(1), 103-120.
Fundamental stability theorem for persistence diagrams.
Kerber, M., Morozov, D., & Nigmetov, A. (2017). Geometry helps to compare persistence diagrams. Journal of Experimental Algorithmics, 22, 1-20.
Efficient computation of Wasserstein distance.
Machine Learning with TDA
Deep Learning Integration
Hofer, C., Kwitt, R., Niethammer, M., & Uhl, A. (2017). Deep learning with topological signatures. Advances in Neural Information Processing Systems, 30.
Integrating persistent homology with deep neural networks.
Moor, M., Horn, M., Rieck, B., & Borgwardt, K. (2020). Topological autoencoders. International Conference on Machine Learning, 7045-7054.
Autoencoders with topological constraints.
Clough, J. R., Oksuz, I., Puyol-Anton, E., et al. (2022). Global and local interpretability for cardiac MRI classification. International Conference on Medical Image Computing and Computer-Assisted Intervention, 656-664.
Interpretability using topological features.
Feature Learning
Carrière, M., Cuturi, M., & Oudot, S. (2020). Sliced Wasserstein kernel for persistence diagrams. International Conference on Machine Learning, 664-673.
Efficient kernels for machine learning with persistence diagrams.
Chevyrev, I., Nanda, V., & Oberhauser, H. (2018). Persistence paths and signature features in topological data analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(1), 192-202.
Signature methods for topological features.
Software and Tools
Core Libraries
The GUDHI Project. GUDHI User and Reference Manual. GUDHI Editorial Board, 2015. https://gudhi.inria.fr/
Main computational library used by MedTDA.
Bauer, U. Ripser: Efficient computation of Vietoris-Rips persistence barcodes. https://github.com/Ripser/ripser
Fast persistence computation library.
Morozov, D. Dionysus 2. https://mrzv.org/software/dionysus2/
Python library for persistent homology.
Tauzin, G., Lupo, U., Tunstall, L., et al. (2021). giotto-tda: A topological data analysis toolkit for machine learning and data exploration. Journal of Machine Learning Research, 22(39), 1-6.
TDA library for machine learning.
Visualization
Saul, N., & Tralie, C. Persim: Persistence-based similarity and visualization. https://github.com/scikit-tda/persim
Visualization tools for persistence diagrams.
Van Veen, H. J., Saul, N., Eargle, D., & Mangham, S. W. Kepler Mapper: A flexible Python implementation of the Mapper algorithm. Journal of Open Source Software, 4(42), 1315.
Mapper algorithm implementation for visualization.
Medical Imaging Libraries
Brett, M., Markiewicz, C. J., Hanke, M., et al. NiBabel: Access a cacophony of neuro-imaging file formats. https://nipy.org/nibabel/
Python library for reading medical image formats.
Lowekamp, B. C., Chen, D. T., Ibáñez, L., & Blezek, D. (2013). The design of SimpleITK. Frontiers in Neuroinformatics, 7, 45.
Image processing library for medical images.
Mason, D. pydicom: An open source DICOM library. Medical Physics, 38(6), 3493.
Python library for DICOM files.
Textbooks and Tutorials
Topological Data Analysis
Oudot, S. Y. (2015). Persistence Theory: From Quiver Representations to Data Analysis. American Mathematical Society.
Mathematical foundations of persistence theory.
Otter, N., Porter, M. A., Tillmann, U., et al. (2017). A roadmap for the computation of persistent homology. EPJ Data Science, 6(1), 1-38.
Comprehensive tutorial on computing persistent homology.
Rabadán, R., & Blumberg, A. J. (2019). Topological Data Analysis for Genomics and Evolution: Topology in Biology. Cambridge University Press.
TDA applications in biology and medicine.
Medical Image Analysis
Bankman, I. (2008). Handbook of Medical Image Processing and Analysis. Academic Press.
Comprehensive reference for medical image processing.
Suetens, P. (2017). Fundamentals of Medical Imaging (3rd ed.). Cambridge University Press.
Fundamentals of medical imaging modalities.
Online Resources
Tutorials and Courses
Topological Data Analysis Course by Gunnar Carlsson https://www.comptop.stanford.edu/
Applied Algebraic Topology Research Network https://www.aatrn.net/
Topology ToolKit (TTK) Tutorials https://topology-tool-kit.github.io/
Datasets
Medical Segmentation Decathlon http://medicaldecathlon.com/
MICCAI Grand Challenges https://grand-challenge.org/
The Cancer Imaging Archive (TCIA) https://www.cancerimagingarchive.net/
Code Repositories
Scikit-TDA https://scikit-tda.org/
Citing MedTDA
If you use MedTDA in your research, please cite:
@software{medtda2026,
title={Med-TDA: Medical Imaging Topological Data Analysis Tool},
author={Dashti A. Ali, Amber L. Simpson},
year={2026},
url={https://github.com/dashtiali/medtda}
}
Also cite the underlying methods you use:
GUDHI [Maria2014] for persistent homology computation
Persistence images [Adams2017] if using that vectorization
Persistence landscapes [Bubenik2015] if using that vectorization
Relevant papers from medical imaging applications above
Contributing References
If you find a relevant paper missing from this list, please:
Open an issue or pull request on GitHub
Include full citation in BibTeX format
Brief description of relevance
Category where it should be listed
See Contributing to MedTDA for how to contribute.
See Also
TDA Theory - Theoretical background
User Guide - User guide and tutorials
Frequently Asked Questions - Frequently asked questions
Contributing to MedTDA - How to contribute