Skip to main content
SpringerLink
Log in
Book cover

International Conference on Multimedia and Network Information System

MISSI 2018: Multimedia and Network Information Systems pp 322–331Cite as

Sparse Solution to Inverse Problem of Nonlinear Dimensionality Reduction

  • Conference paper
  • First Online:

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 833))

Abstract

In this paper we propose a sparse solution to the inverse problem of nonlinear dimensionality reduction (NLDR), which holds potential high-performance applications in data representation, compression, generation, and visualization. Firstly, the sparse solution model of the inverse problem of NLDR is established, which consists of four components: classical NLDR, sparse dictionary learning, NLDR embedding, and sparse NLDR reconstruction. Secondly, the special sparse solution to the inverse problem of isometric feature mapping (ISOMAP), a classical NDLR algorithm, is presented. ISOMAP embedding and sparse ISOMAP reconstruction algorithms are raised, and the alternating directions method of multipliers (ADMM) is adopted to resolve the minimization problem of the special sparse solution. Finally, it is revealed by the experimental results that, in the situation of very low dimensional representation, the proposed method is superior to the state of the art methods, such as discrete cosine transformation (DCT) and sparse representation (SR), in the reconstruction performance of image and video data.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Sullivan, G.J., Ohm, J.-R., Han, W.-J., Wiegand, T.: Overview of the high efficiency video coding (HEVC) standard. IEEE Trans. Circ. Syst. Video Technol. 22(12), 1649–1668 (2012)

    Google Scholar 

  2. Sam, T.: Roweis: nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)

    Article  Google Scholar 

  3. Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)

    Article  Google Scholar 

  4. Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. Adv. Neural. Inf. Process. Syst. 14(6), 585–591 (2001)

    Google Scholar 

  5. Hinton, G.E., Salakhutdinov, R.R.: Reducing the dimensionality of data with neural networks. Science 313(5786), 504–507 (2006)

    Article  MathSciNet  Google Scholar 

  6. Zhang, C.: Reconstruction and analysis of multi-pose face images based on nonlinear dimensionality reduction. Pattern Recognit. 37(2), 325–336 (2004)

    Article  Google Scholar 

  7. Monniga, N.D., Fornberga, B., Meyerb, F.G.: Inverting nonlinear dimensionality reduction with scale-free radial basis function interpolation. Appl. Comput. Harmonic Anal. 37(1), 162–170 (2014)

    Article  MathSciNet  Google Scholar 

  8. Liu, J., Hu, Y., Yang, J., Chen, Y., Shu, H., Luo, L., Feng, Q., Gui, Z., Coatrieux, G.: 3D feature constrained reconstruction for low dose CT imaging. IEEE Trans. Circ. Syst. Video Technol. 28(5), 1232–1247 (2018)

    Google Scholar 

  9. Bai, T., Yan, H., Jia, X., Jiang, S., Wang, G., Mou, X.: Z-index parameterization (ZIP) for volumetric CT image reconstruction via 3D dictionary learning. IEEE Trans. Med. Imaging 36(12), 2466–2478 (2017)

    Article  Google Scholar 

  10. Wen, F., Pei, L., Yang, Y., Wenxian, Yu., Liu, P.: Efficient and robust recovery of sparse signal and image using generalized nonconvex regularization. IEEE Trans. Comput. Imaging 3(4), 566–579 (2017)

    Article  MathSciNet  Google Scholar 

  11. Cui, X., Mili, L., Hengyong, Yu.: Sparse-prior-based projection distance optimization method for joint CT-MRI reconstruction. IEEE Access 5, 20099–20110 (2017)

    Article  Google Scholar 

  12. Bettens, S., Schretter, C., Deligiannis, N., Schelkens, P.: Bounds and conditions for compressive digital holography using wavelet sparsifying bases. IEEE Trans. Comput. Imaging 3(4), 592–604 (2017)

    Article  MathSciNet  Google Scholar 

  13. Ozkan, E., Vishnevsky, V., Goksel, O.: Inverse problem of ultrasound beamforming with sparsity constraints and regularization. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65(3), 356–365 (2018)

    Article  Google Scholar 

  14. Kim, K., Ye, J.C., Worstell, W., Ouyang, J., Rakvongthai, Y., Fakhri, G.E., Li, Q.: Sparse-view spectral CT reconstruction using spectral patch-based low-rank penalty. IEEE Trans. Med. Imaging 34(3), 748–760 (2015)

    Google Scholar 

  15. Cai, J.-F., Jia, X., Gao, H., Jiang, S.B., Shen, Z., Zhao, H.: Cine cone beam CT reconstruction using low-rank matrix factorization: algorithm and a proof-of-principle study. IEEE Trans. Med. Imaging 33(8), 1581–1591 (2014)

    Article  Google Scholar 

  16. Jin, K.H., Lee, D., Ye, J.C.: A general framework for compressed sensing and parallel MRI using annihilating filter based low-rank Hankel matrix. IEEE Trans. Comput. Imaging 2(4), 480–495 (2016)

    Google Scholar 

  17. Lin, X., Wang, C., Chen, W., Liu, X.: Denoising multi-channel images in parallel MRI by low rank matrix decomposition. IEEE Trans. Appl. Supercond. 24(5), 480–494 (2014)

    Google Scholar 

  18. Lingala, S.G., Hu, Y., DiBella, E., Jacob, M.: Accelerated dynamic MRI exploiting sparsity and low-rank structure: k-t SLR. IEEE Trans. Med. Imaging 30(5), 1042–1054 (2011)

    Google Scholar 

  19. Christodoulou, A.G., Zhang, H., Zhao, B., Hitchens, T.K., Ho, C., Liang, Z.-P.: High-resolution cardiovascular MRI by integrating parallel imaging with low-rank and sparse modeling. IEEE Trans. Biomed. Eng. 60(11), 3083–3092 (2013)

    Google Scholar 

  20. Chen, H., Zhang, Y., Kalra, M.K., Lin, F., Chen, Y., Liao, P., Zhou, J., Wang, G.: Low-dose CT with a residual Encoder-Decoder Convolutional Neural Network (RED-CNN). IEEE Trans. Med. Imaging 36(12), 2524–2535 (2017)

    Article  Google Scholar 

  21. Schlemper, J., Caballero, J., Hajnal, J.V., Price, A., Rueckert, D.: A deep cascade of convolutional neural networks for dynamic MR image reconstruction. IEEE Trans. Med. Imaging 37(2), 491–503 (2017)

    Article  Google Scholar 

  22. Dufan, W., Kim, K., El Fakhri, G., Li, Q.: Iterative low-dose CT reconstruction with priors trained by artificial neural network. IEEE Trans. Med. Imaging 36(12), 2479–2486 (2017)

    Article  Google Scholar 

  23. Afonso, M.V., Bioucas-Dias, J.M., Figueiredo, M.A.T.: Fast image recovery using variable splitting and constrained optimization. IEEE Trans. Image Process. 19(9), 2345–2356 (2010)

    Google Scholar 

  24. Aharon, M., Elad, M., Bruckstein, A.: K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54(11), 4311–4322 (2006)

    Article  Google Scholar 

  25. Jain, A.K.: Fundamentals of Digital Image Processing, 1st edn. Prentice Hall, Englewood Cliffs (1989)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yangzhou University, Yangzhou, 225002, Jiangsu, China

    Honggui Li

  2. Institut Supérieur d’Électronique de Paris, 75006, Paris, France

    Maria Trocan

Authors
  1. Honggui Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Trocan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Honggui Li .

Editor information

Editors and Affiliations

  1. Faculty of Computer Science and Management, Wrocław University of Science and Technology, Wrocław, Poland

    Kazimierz Choroś

  2. Faculty of Computer Science and Management, Wrocław University of Science and Technology, Wrocław, Poland

    Marek Kopel

  3. Faculty of Computer Science and Management, Wrocław University of Science and Technology, Wrocław, Poland

    Elżbieta Kukla

  4. Faculty of Computer Science and Management, Wrocław University of Science and Technology, Wrocław, Poland

    Andrzej Siemiński

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Li, H., Trocan, M. (2019). Sparse Solution to Inverse Problem of Nonlinear Dimensionality Reduction. In: Choroś, K., Kopel, M., Kukla, E., Siemiński, A. (eds) Multimedia and Network Information Systems. MISSI 2018. Advances in Intelligent Systems and Computing, vol 833. Springer, Cham. https://doi.org/10.1007/978-3-319-98678-4_33

Download citation

Publish with us

Policies and ethics

Search

Navigation