Skip to main content
SpringerLink
Log in

Denoising color images by reduced quaternion matrix singular value decomposition

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

We propose a color-image-denoising algorithm that is based on the reduced quaternion matrix (RQM) of singular value decomposition (SVD). The new algorithm represents a color image as an RQM and handles such an image in a holistic manner. This algorithm can combine similar blocks from a noisy image by using a similar criterion. The proposed framework computes the optimal unitary matrix pair by using RQMSVD, and the coefficients of RQMSVD are obtained by projecting each block onto unitary matrices. The final filtered image is then obtained by manipulating the coefficients with hard threshold. The performance of the proposed algorithm is experimentally verified by using a variety of images and noise levels. Results demonstrate that the proposed algorithm is at par with or exceeds current state-of-the-art algorithms in both visual and quantitative performance.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  • Awate, S. P., & Whitaker, R. T. (2006). Unsupervised information-theoretic adaptive image filtering for image restoration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(3), 364–376.

    Article  Google Scholar 

  • Baudes, A., Coll, B., & Morel, J. M. (2005). A review of image denoising methods, with a new one. Multiscale Model Simulation, 4(2), 490–530.

    Article  Google Scholar 

  • Buades, A., Coll, B., & Morel, J. M. (2006). Neighboring filters and PDEs. Numerische Mathematik, 105(1), 1–34.

    Article  MATH  MathSciNet  Google Scholar 

  • Chatterjee, P., & Milanfar, P. (2009). Clustering-based denoising with locally learned dictionaries. IEEE Transactions on Image Processing, 18(7), 1438–1451.

    Article  MathSciNet  Google Scholar 

  • Chatterjee, P., & Milanfar, P. (2012). Patch-based near-optimal image denoising. IEEE Transactions on Image Processing, 21(4), 1635–1649.

    Article  MathSciNet  Google Scholar 

  • Coifman, D., & Meer, P. (2002). Mean shift: A robust approach toward feature space analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(5), 603–619.

    Article  Google Scholar 

  • Dabov, K., Foi, A., Katkovnik, V., & Egiazarian, K. O. (2007). Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Transactions on Image Processing, 16(8), 2080–3745.

    Article  MathSciNet  Google Scholar 

  • Donoho, D., & Johnstone, I. (1993). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(6), 425–455.

    MathSciNet  Google Scholar 

  • Elad, M., & Aharon, M. (2006). Image denoising via sparse and redundant representations over learned dictionaries. IEEE Transactions on Image Processing, 15(12), 3736–3745.

    Article  MathSciNet  Google Scholar 

  • Haji, M., Bui, T. D., & Suen, C. Y. (2012). Removal of noise patterns in handwritten images using expectation maximization and fuzzy inferences systems. Pattern Recognition, 45(12), 4237–4249.

    Article  Google Scholar 

  • Hamilton, W. R. (1866). Elements of quaternions. Green and London: Longmans.

    Google Scholar 

  • Hsia, C. H., Jing, G. M., & Chiang, J. S. (2012). A fast discrete wavelet transform algorithm for visual processing application. Signal Processing, 92(1), 89–106.

    Article  Google Scholar 

  • Kervrann, C., & Boulanger, J. (2006). Optimal spatial adaptation for patch based image denoising. IEEE Transactions on Image Processing, 15(10), 2866–2878.

    Article  Google Scholar 

  • Kervrann, C., & Boulanger, J. (2008). Local adaptivity to variable smoothness for exemplar-based image denoising and representation. International Journal of Computation Vision, 79(1), 45–69.

    Article  Google Scholar 

  • Liu, C., Szeliski, R., Kang, S. B., Zitnick, C. L., & Freeman, W. T. (2008). Automatic estimation and removal of noise from a single image. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(2), 299–314.

    Article  Google Scholar 

  • Muresan, D. D., & Parks, T. W. (2003). Adaptive principle components and image denoising. In: IEEE Conference on Image Processing (pp. 101–104). Bacelona: Spain, Sept 2003.

  • Perona, P., & Malik, J. (1990). Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7), 629–639.

    Article  Google Scholar 

  • Portilla, J., Strela, V., Waiwright, M., & Simoncelli, E. (2003). Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Transactions on Image Processing, 12(11), 1338–1351.

    Article  MATH  MathSciNet  Google Scholar 

  • Rajpoot, N., & Butt, T. (2012). A multiresolution framework for local similarity based image denoising. Pattern Recognition, 45(8), 2983–2951.

    Article  Google Scholar 

  • Rajwade, A., & Rangarajan, A. (2013). Image denoising using the higher order singular value decomposition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(4), 849–862.

    Article  Google Scholar 

  • Schtte, H. D., & Wenzel, J. (1990). Hypercomplex numbers in digital signal processing. IEEE International Symposium on Circuits and Systems, 8(2), 1557–1560.

    Article  Google Scholar 

  • Sheng, L. Q., & Zhen, C. S. (2009). The translation invariant Contourlet-like transform for image denoising. Acta Automatica Sinica, 35(5), 505–508.

    Article  Google Scholar 

  • Takeda, H., Farsiu, S., & Milanfar, P. (2007). Kernel regression for image processing and reconstruction. IEEE Transactions on Image Processing, 16(2), 349–366.

    Article  MathSciNet  Google Scholar 

  • Tomasi, C., & Manduchi, M. (1998). Bilateral Filtering for Gray and Color Images. In: IEEE Conference on Computer Vision (pp. 839–846). Bombay, India, Jan 1998.

  • Tschumperle, D., & Deriche, R. (2005). Vector-valued image regularization with PDEs: A common framework for different applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(4), 506–517.

    Article  Google Scholar 

  • Vo, A., Oraintara, S., & Vonn, N. N. (2011). Distribution of relative phase for statistical image modeling in complex wavelet domain. Signal Processing, 91(1), 114–125.

    Article  MATH  Google Scholar 

  • Wang, Z., & Bovik, A. (2004). Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13(4), 600–612.

    Article  Google Scholar 

  • Ye, J. (2005). Generalized low rank approximation of matrices. Machine Learning, 61(1), 167–191.

    Article  MATH  Google Scholar 

  • Zhang, L., Dong, W., Zhang, D., & Shi, G. (2010). Two-stage image denoising by principle component analysis with local pixel grouping. Pattern Recognition, 43(4), 1531–1549.

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (61202319, 61272077, 61203243, 61201439, 61162002,); Natural Science Foundation of Jiangxi (20114BAB201034, 20122BAB211025); Department of Education of Jiangxi (GJJ13481).

Author information

Authors and Affiliations

  1. Department of Information Engineering, University of Nanchang Hangkong, Nanchang, Jiangxi, 330063, People’s Republic of China

    Shan Gai, Guowei Yang, Minghua Wan & Lei Wang

Authors
  1. Shan Gai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guowei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Minghua Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shan Gai.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gai, S., Yang, G., Wan, M. et al. Denoising color images by reduced quaternion matrix singular value decomposition. Multidim Syst Sign Process 26, 307–320 (2015). https://doi.org/10.1007/s11045-013-0268-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-013-0268-x

Keywords

Search

Navigation