Skip to main content

FCNN: Fourier Convolutional Neural Networks

Part of the Lecture Notes in Computer Science book series (LNAI,volume 10534)


The Fourier domain is used in computer vision and machine learning as image analysis tasks in the Fourier domain are analogous to spatial domain methods but are achieved using different operations. Convolutional Neural Networks (CNNs) use machine learning to achieve state-of-the-art results with respect to many computer vision tasks. One of the main limiting aspects of CNNs is the computational cost of updating a large number of convolution parameters. Further, in the spatial domain, larger images take exponentially longer than smaller image to train on CNNs due to the operations involved in convolution methods. Consequently, CNNs are often not a viable solution for large image computer vision tasks. In this paper a Fourier Convolution Neural Network (FCNN) is proposed whereby training is conducted entirely within the Fourier domain. The advantage offered is that there is a significant speed up in training time without loss of effectiveness. Using the proposed approach larger images can therefore be processed within viable computation time. The FCNN is fully described and evaluated. The evaluation was conducted using the benchmark Cifar10 and MNIST datasets, and a bespoke fundus retina image dataset. The results demonstrate that convolution in the Fourier domain gives a significant speed up without adversely affecting accuracy. For simplicity the proposed FCNN concept is presented in the context of a basic CNN architecture, however, the FCNN concept has the potential to improve the speed of any neural network system involving convolution.


  • Fourier Convolution
  • Convolutional Neural Network (CNNs)
  • Fourier Domain
  • Large Image
  • Spatial Kernel

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-71249-9_47
  • Chapter length: 13 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-71249-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.


  1. Krizhevsky, A., Sutskever, I., Hinton, G.E.: ImageNet classification with deep convolutional neural networks. In: Pereira, F., Burges, C.J.C., Bottou, L., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, vol. 25, pp. 1097–1105. Curran Associates Inc. (2012)

    Google Scholar 

  2. LeCun, Y., Boser, B., Denker, J.S., Howard, R.E., Habbard, W., Jackel, L.D., Henderson, D.: Advances in neural information processing systems, vol. 2, pp. 396–404. Citeseer (1990)

    Google Scholar 

  3. Kaggle: Kaggle datasets.

  4. He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. CoRR, abs/1512.03385 (2015)

    Google Scholar 

  5. Szegedy, C., Liu, W., Jia, Y., Sermanet, P., Reed, S., Anguelov, D., Erhan, D., Vanhoucke, V., Rabinovich, A.: Going deeper with convolutions. In: Computer Vision and Pattern Recognition (CVPR) (2015)

    Google Scholar 

  6. Sermanet, P., Eigen, D., Zhang, X., Mathieu, M., Fergus, R., LeCun, Y.: Overfeat: integrated recognition, localization and detection using convolutional networks. CoRR, abs/1312.6229 (2013)

    Google Scholar 

  7. Vasilache, N., Johnson, J., Mathieu, M., Chintala, S., Piantino, S., LeCun, Y.: Fast convolutional nets when fbfft: a GPU performance evaluation (2015)

    Google Scholar 

  8. Chan, T.F., Wong, C.K.: Total variation blind deconvolution. IEEE Trans. Image Process. 7(3), 370–375 (1998)

    CrossRef  Google Scholar 

  9. Persch, N., Elhayek, A., Welk, M., Bruhn, A., Grewenig, S., Böse, K., Kraegeloh, A., Weickert, J.: Enhancing 3-D cell structures in confocal and STED microscopy: a joint model for interpolation, deblurring and anisotropic smoothing. Measur. Sci. Technol. 24(12), 125703 (2013)

    CrossRef  Google Scholar 

  10. Williams, B.M., Chen, K., Harding, S.P.: A new constrained total variational deblurring model and its fast algorithm. Numer. Algorithms 69(2), 415–441 (2015)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex fourier series. Math. comput. 19(90), 297–301 (1965)

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Campisi, P., Egiazarian, K.: Blind Image Deconvolution. CRC Press, Boca Raton (2007)

    CrossRef  Google Scholar 

  13. Kumar, R., Gothwal, H., Kedawat, S.: Cardiac arrhythmias detection in an ECG beat signal using fast fourier transform and artificial neural network. J. Biomed. Sci. Eng. 4, 289–296 (2011)

    CrossRef  Google Scholar 

  14. LeCun, Y., Mathieu, M., Henaff, M.: Fast training of convolutional networks through FFTs (2014)

    Google Scholar 

  15. Adams, R.P., Rippel, O., Snoek, J.: Spectral representations for convolutional neural networks (2015)

    Google Scholar 

  16. Chollet, F.: Keras. (2015)

  17. Theano Development Team. Theano: a python framework for fast computation of mathematical expressions. arXiv e-prints abs/1605.02688, May 2016

  18. LeCun, Y., Cortes, C.: MNIST handwritten digit database (2010)

    Google Scholar 

  19. Krizhevsky, A.: Learning multiple layers of features from tiny images.

  20. Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial nets. In: Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N.D., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, vol. 27, pp. 2672–2680. Curran Associates Inc. (2014)

    Google Scholar 

  21. Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Proceedings of the International Conference on Artificial Intelligence and Statistics (AISTATS 2010). Society for Artificial Intelligence and Statistics (2010)

    Google Scholar 

Download references


The authors would like to acknowledge everyone in the Centre for Research in Image Analysis (CRiA) imaging team at the Institute of Ageing and Chronic Disease at the University of Liverpool and the Fight for Sight charity who have supported this work through funding.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Harry Pratt .

Editor information

Editors and Affiliations

1 Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 113 KB)

Rights and permissions

Reprints and Permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Pratt, H., Williams, B., Coenen, F., Zheng, Y. (2017). FCNN: Fourier Convolutional Neural Networks. In: Ceci, M., Hollmén, J., Todorovski, L., Vens, C., Džeroski, S. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2017. Lecture Notes in Computer Science(), vol 10534. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71248-2

  • Online ISBN: 978-3-319-71249-9

  • eBook Packages: Computer ScienceComputer Science (R0)