.. _references: ========== References ========== This page provides academic references, resources, and citations related to topological data analysis and its application to medical imaging. .. contents:: Contents :local: :depth: 2 Persistent Homology Foundations ================================ Core Theory ----------- .. [Edel2010] Edelsbrunner, H., & Harer, J. (2010). *Computational Topology: An Introduction*. American Mathematical Society. **The foundational textbook for computational topology and persistent homology.** .. [Zomo2005] Zomorodian, A., & Carlsson, G. (2005). Computing persistent homology. *Discrete & Computational Geometry*, 33(2), 249-274. **Original paper on computing persistent homology efficiently.** .. [Carls2009] Carlsson, G. (2009). Topology and data. *Bulletin of the American Mathematical Society*, 46(2), 255-308. **Influential survey paper on topological data analysis.** .. [Ghris2008] 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 --------------------- .. [Bauer2021] 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.** .. [Maria2014] 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.** .. [Morozov2008] 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 -------------------------------- .. [Ali2023] 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 ------------------ .. [Adams2017] 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.** .. [Perea2018] 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 ---------------------- .. [Bubenik2015] Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. *Journal of Machine Learning Research*, 16(1), 77-102. **Persistence landscapes for statistical analysis.** .. [Chazal2014] 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 -------------------- .. [Chung2022] 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.** .. [Reininghaus2015] 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.** .. [Kusano2016] 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 ---------------------------- .. [Adcock2016] 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.** .. [Clough2020] 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.** .. [Qaiser2019] 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 ------------- .. [Chung2018] 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.** .. [Lee2019] 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.** .. [Songdechakraiwut2021] 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 -------------------------- .. [Li2021] 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.** .. [Qaiser2018] 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.** .. [Lawson2019] 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 ---------------- .. [Bendich2016] 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.** .. [Garside2021] 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 --------------- .. [Byrne2020] 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.** .. [Qaiser2022] 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 ------------------------------------ .. [Fasy2014] 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.** .. [Robinson2017] 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.** .. [Chazal2017] 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-Steiner2007] 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.** .. [Kerber2017] 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 -------------------------- .. [Hofer2017] 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.** .. [Moor2020] Moor, M., Horn, M., Rieck, B., & Borgwardt, K. (2020). Topological autoencoders. *International Conference on Machine Learning*, 7045-7054. **Autoencoders with topological constraints.** .. [Clough2022] 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 ---------------- .. [Carriere2020] 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.** .. [Chevyrev2018] 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 -------------- .. [GUDHI] The GUDHI Project. *GUDHI User and Reference Manual*. GUDHI Editorial Board, 2015. https://gudhi.inria.fr/ **Main computational library used by MedTDA.** .. [Ripser] Bauer, U. *Ripser: Efficient computation of Vietoris-Rips persistence barcodes*. https://github.com/Ripser/ripser **Fast persistence computation library.** .. [Dionysus] Morozov, D. *Dionysus 2*. https://mrzv.org/software/dionysus2/ **Python library for persistent homology.** .. [Giotto-TDA] 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 ------------- .. [Persim] Saul, N., & Tralie, C. *Persim: Persistence-based similarity and visualization*. https://github.com/scikit-tda/persim **Visualization tools for persistence diagrams.** .. [Kepler-Mapper] 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 -------------------------- .. [NiBabel] 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.** .. [SimpleITK] 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.** .. [PyDicom] Mason, D. *pydicom: An open source DICOM library*. *Medical Physics*, 38(6), 3493. **Python library for DICOM files.** Textbooks and Tutorials ======================== Topological Data Analysis -------------------------- .. [Oudot2015] Oudot, S. Y. (2015). *Persistence Theory: From Quiver Representations to Data Analysis*. American Mathematical Society. **Mathematical foundations of persistence theory.** .. [Otter2017] 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.** .. [Rabadan2019] 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 ---------------------- .. [Bankman2008] Bankman, I. (2008). *Handbook of Medical Image Processing and Analysis*. Academic Press. **Comprehensive reference for medical image processing.** .. [Suetens2017] 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/ - **PyTDA** https://github.com/LimenResearch/pytda - **TDA-API** https://github.com/FatemehTarashi/awesome-tda Citing MedTDA ============= If you use MedTDA in your research, please cite: .. code-block:: bibtex @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 Related Projects ================ - **Giotto-TDA**: General TDA library for machine learning - **Scikit-TDA**: Collection of TDA tools for Python - **TTK**: Topological analysis and visualization - **Persim**: Persistence diagram visualization - **DREiMac**: Dimensionality reduction with Rips complexes - **Teaspoon**: TDA for time series - **MedPy**: Medical image processing in Python - **ITK-SNAP**: Medical image segmentation and visualization Contributing References ======================== If you find a relevant paper missing from this list, please: 1. Open an issue or pull request on GitHub 2. Include full citation in BibTeX format 3. Brief description of relevance 4. Category where it should be listed See :doc:`contributing` for how to contribute. See Also ======== - :doc:`theory/index` - Theoretical background - :doc:`user_guide` - User guide and tutorials - :doc:`faq` - Frequently asked questions - :doc:`contributing` - How to contribute