Skip to main content

Radon Cumulative Distribution Transform Subspace Modeling for Image Classification


We present a new supervised image classification method applicable to a broad class of image deformation models. The method makes use of the previously described Radon Cumulative Distribution Transform (R-CDT) for image data, whose mathematical properties are exploited to express the image data in a form that is more suitable for machine learning. While certain operations such as translation, scaling, and higher-order transformations are challenging to model in native image space, we show the R-CDT can capture some of these variations and thus render the associated image classification problems easier to solve. The method—utilizing a nearest-subspace algorithm in the R-CDT space—is simple to implement, non-iterative, has no hyper-parameters to tune, is computationally efficient, label efficient, and provides competitive accuracies to state-of-the-art neural networks for many types of classification problems. In addition to the test accuracy performances, we show improvements (with respect to neural network-based methods) in terms of computational efficiency (it can be implemented without the use of GPUs), number of training samples needed for training, as well as out-of-distribution generalization. The Python code for reproducing our results is available at Shifat-E-Rabbi et al. (Python code implementing the Radon cumulative distribution transform subspace model for image classification.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. 1.

    We are using a slightly different definition of the CDT than in [31]. The properties of the CDT outlined here hold in both definitions.

  2. 2.

    Rigorously speaking, if \(\widehat{\mathbb {V}}^{(p)}\) is a closed subspace, then \(d^2( \widehat{s},\widehat{\mathbb {V}}^{(p)})>0\) if and only if \(\widehat{s}\notin \widehat{\mathbb {V}}^{(p)} \). In practice, \(\widehat{\mathbb {V}}^{(p)}\) will be a finite dimensional space and hence, the closedness condition is satisfied.

  3. 3.

    The same grid is chosen for all images. mn are positive integers.


  1. 1.

    Shifat-E-Rabbi, M., Yin, X., Rubaiyat, A.H.M., Li, S., Kolouri, S., Aldroubi, A., Nichols, J.M., Rohde, G.K: Python code implementing the Radon cumulative distribution transform subspace model for image classification.

  2. 2.

    Sertel, O., Kong, J., Shimada, H., Catalyurek, U.V., Saltz, J.H., Gurcan, M.N.: Computer-aided prognosis of neuroblastoma on whole-slide images: classification of stromal development. Pattern Recognit. 42(6), 1093–1103 (2009)

    Article  Google Scholar 

  3. 3.

    Basu, S., Kolouri, S., Rohde, G.K.: Detecting and visualizing cell phenotype differences from microscopy images using transport-based morphometry. Proc. Natl. Acad. Sci. 111(9), 3448–3453 (2014)

    Article  Google Scholar 

  4. 4.

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

    Article  Google Scholar 

  5. 5.

    Schulz, J.B., Borkert, J., Wolf, S., Schmitz-Hübsch, T., Rakowicz, M., Mariotti, C., Schoels, L., Timmann, D., Warrenburg, B., Dürr, A., Pandolfo, M., Kang, J., Mandly, A.G., Nagele, T., Grisoli, M., Boguslawska, R., Bauer, P., Klockgether, T., Hauser, T.: Visualization, quantification and correlation of brain atrophy with clinical symptoms in spinocerebellar ataxia types 1, 3 and 6. Neuroimage 49(1), 158–168 (2010)

    Article  Google Scholar 

  6. 6.

    Hadid, A., Heikkila, J.Y., Silvén, O., Pietikainen, M.: Face and eye detection for person authentication in mobile phones. In: 2007 First ACM/IEEE International Conference on Distributed Smart Cameras, pp. 101–108 (2007)

  7. 7.

    Shifat-E-Rabbi, M., Yin, X., Fitzgerald, C.E., Rohde, G.K.: Cell image classification: a comparative overview. Cytometry A 97A(4), 347–362 (2020)

    Article  Google Scholar 

  8. 8.

    Rawat, W., Wang, Z.: Deep convolutional neural networks for image classification: a comprehensive review. Neural Comput. 29(9), 2352–2449 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Lu, D., Weng, Q.: A survey of image classification methods and techniques for improving classification performance. Int. J. Remote Sens. 28(5), 823–870 (2007)

    Article  Google Scholar 

  10. 10.

    Prewitt, J.M.S., Mendelsohn, M.L.: The analysis of cell images. Ann. N. Y. Acad. Sci. 128(3), 1035–1053 (1966)

    Article  Google Scholar 

  11. 11.

    Orlov, N., Shamir, L., Macura, T., Johnston, J., Eckley, D.M., Goldberg, I.G.: WND-CHARM: multi-purpose image classification using compound image transforms. Pattern Recognit. Lett. 29(11), 1684–1693 (2008)

    Article  Google Scholar 

  12. 12.

    Ponomarev, G.V., Arlazarov, V.L., Gelfand, M.S., Kazanov, M.D.: Ana hep-2 cells image classification using number, size, shape and localization of targeted cell regions. Pattern Recognit. 47(7), 2360–2366 (2014)

    Article  Google Scholar 

  13. 13.

    Bandos, T.V., Bruzzone, L., Camps-Valls, G.: Classification of hyperspectral images with regularized linear discriminant analysis. IEEE Trans. Geosci. Remote Sens. 47(3), 862–873 (2009)

    Article  Google Scholar 

  14. 14.

    Muldoon, T.J., Thekkek, N., Roblyer, D.M., Maru, D., Harpaz, N., Potack, J., Anandasabapathy, S., Richards-Kortum, R.R.: Evaluation of quantitative image analysis criteria for the high-resolution microendoscopic detection of neoplasia in Barrett’s esophagus. J. Biomed. Opt. 15(2), 026027 (2010)

    Article  Google Scholar 

  15. 15.

    Zhang, J., Marszałek, M., Lazebnik, S., Schmid, C.: Local features and kernels for classification of texture and object categories: a comprehensive study. Int. J. Comput. Vis. 73(2), 213–238 (2007)

    Article  Google Scholar 

  16. 16.

    Perronnin, F., Sánchez, J., Mensink, T.: Improving the fisher kernel for large-scale image classification. In: European Conference on Computer Vision, pp. 143–156 (2010)

  17. 17.

    Bosch, A., Zisserman, A., Munoz, X.: Image classification using random forests and ferns. In: 2007 IEEE 11th International Conference on Computer Vision, pp. 1–8. IEEE (2007)

  18. 18.

    Du, P., Samat, A., Waske, B., Liu, S., Li, Z.: Random forest and rotation forest for fully polarized SAR image classification using polarimetric and spatial features. ISPRS J. Photogramm. Remote Sens. 105, 38–53 (2015)

    Article  Google Scholar 

  19. 19.

    LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436–444 (2015)

    Article  Google Scholar 

  20. 20.

    Shin, H.-C., Roth, H.R., Gao, M., Lu, L., Xu, Z., Nogues, I., Yao, J., Mollura, D., Summers, R.M.: Deep convolutional neural networks for computer-aided detection: CNN architectures, dataset characteristics and transfer learning. IEEE Trans. Med. Imaging 35(5), 1285–1298 (2016)

    Article  Google Scholar 

  21. 21.

    Szegedy, C., Vanhoucke, V., Ioffe, S., Shlens, J., Wojna, Z.: Rethinking the inception architecture for computer vision. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2818–2826 (2016)

  22. 22.

    Szegedy, C., Liu, W., Jia, Y., Sermanet, P., Reed, S., Anguelov, D., Erhan, D., Vanhoucke, V., Rabinovich, A.: Going deeper with convolutions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–9 (2015)

  23. 23.

    Wolberg, G.: Image morphing: a survey. Vis. Comput. 14(8), 360–372 (1998)

    Article  Google Scholar 

  24. 24.

    Kolouri, S., Park, S.R., Rohde, G.K.: The radon cumulative distribution transform and its application to image classification. IEEE Trans. Image Process. 25(2), 920–934 (2016)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Kolouri, S., Park, S.R., Thorpe, M., Slepcev, D., Rohde, G.K.: Optimal mass transport: signal processing and machine-learning applications. IEEE Signal Process. Mag. 34(4), 43–59 (2017)

    Article  Google Scholar 

  26. 26.

    Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin (2008)

    MATH  Google Scholar 

  27. 27.

    Wang, W., Slepčev, D., Basu, S., Ozolek, J.A., Rohde, G.K.: A linear optimal transportation framework for quantifying and visualizing variations in sets of images. Int. J. Comput. Vis. 101(2), 254–269 (2013)

    MathSciNet  Article  Google Scholar 

  28. 28.

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

  29. 29.

    Park, S.R., Cattell, L., Nichols, J.M., Watnik, A., Doster, T., Rohde, G.K.: De-multiplexing vortex modes in optical communications using transport-based pattern recognition. Opt. Express 26(4), 4004–4022 (2018)

    Article  Google Scholar 

  30. 30.

    Fitzgerald, C.E., Cattell, L., Rohde, G.K.: Training classifiers with limited data using the Radon cumulative distribution transform. Med. Imaging Image Process. 10574, 105742 (2018)

    Google Scholar 

  31. 31.

    Park, S.R., Kolouri, S., Kundu, S., Rohde, G.K.: The cumulative distribution transform and linear pattern classification. Appl. Comput. Harmon. Anal. 45(3), 616–641 (2018)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Bracewell, R.N.: The Fourier Transform and Its Applications, vol. 31999. McGraw-Hill, New York (1986)

    MATH  Google Scholar 

  33. 33.

    Yang, I.: A convex optimization approach to distributionally robust Markov decision processes with Wasserstein distance. IEEE Control Syst. Lett. 1(1), 164–9 (2017)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Quinto, E.T.: An introduction to x-ray tomography and radon transforms. In: Proceedings of Symposia in Applied Mathematics, vol. 63, p. 1 (2006)

  35. 35.

    Natterer, F.: The Mathematics of Computerized Tomography. SIAM, Philadelphia (2001)

    Book  Google Scholar 

  36. 36.

    LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324 (1998)

    Article  Google Scholar 

  37. 37.

    Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition (2014). arXiv:1409.1556

  38. 38.

    He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 770–778 (2016)

  39. 39.

    Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization (2014). arXiv:1412.6980

  40. 40.

    Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J.: Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–30 (2011)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Dalal, N., Triggs, B.: Histograms of oriented gradients for human detection. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2005)

  42. 42.

    Lowe, D.G.: Object recognition from local scale-invariant features. In: Proceedings of the Seventh IEEE International Conference on Computer Vision, vol. 2, pp. 1150–1157 (1999)

  43. 43.

    Lee, G.R., Gommers, R., Waselewski, F., Wohlfahrt, K., O’Leary, A.: PyWavelets: a Python package for wavelet analysis. J. Open Source Softw. 4(36), 1237 (2019)

    Article  Google Scholar 

  44. 44.

    Kaggle: Sign Language MNIST. Accessed 10 Mar 2020

  45. 45.

    Vondrick, C., Khosla, A., Malisiewicz, T., Torralba, A.: Hoggles: Visualizing object detection features. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 1–8 (2013)

  46. 46.

    Marcus, D.S., Wang, T.H., Parker, J., Csernansky, J.G., Morris, J.C., Buckner, R.L.: Open access series of imaging studies (OASIS): cross-sectional MRI data in young, middle aged, nondemented, and demented older adults. J. Cogn. Neurosci. 19(9), 1498–1507 (2007)

    Article  Google Scholar 

  47. 47.

    Krizhevsky, A., Hinton, G.: Learning multiple layers of features from tiny images (2009)

  48. 48.

    Gardner, M.W., Dorling, S.R.: Artificial neural networks (the multilayer perceptron)—a review of applications in the atmospheric sciences. Atmos. Environ. 32(14–15), 2627–2636 (1998)

    Article  Google Scholar 

  49. 49.

    Pampel, F.C.: Logistic Regression: A Primer. SAGE Publications Incorporated, Thousand Oaks (2020)

    MATH  Google Scholar 

  50. 50.

    Rubaiyat, A.H., Hallam, K.M., Nichols, J.M., Hutchinson, M.N., Li, S., Rohde, G.K.: Parametric signal estimation using the cumulative distribution transform. IEEE Trans. Signal Process. 68, 3312–24 (2020)

    MathSciNet  Article  Google Scholar 

  51. 51.

    Nichols, J.M., Emerson, T.H., Cattell, L., Park, S., Kanaev, A., Bucholtz, F., Watnik, A., Doster, T., Rohde, G.K.: Transport-based model for turbulence-corrupted imagery. Appl. Opt. 57(16), 4524–36 (2018)

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Mohammad Shifat-E-Rabbi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported in part by NIH Grants GM130825, GM090033.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 226 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shifat-E-Rabbi, M., Yin, X., Rubaiyat, A.H.M. et al. Radon Cumulative Distribution Transform Subspace Modeling for Image Classification. J Math Imaging Vis 63, 1185–1203 (2021).

Download citation


  • R-CDT
  • Nearest subspace
  • Image classification
  • Generative model