Image Representation

  • René Vidal
  • Yi Ma
  • S. Shankar Sastry
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 40)


In this and the following chapters, we demonstrate why multiple subspaces can be a very useful class of models for image processing and how the subspace clustering techniques may facilitate many important image processing tasks, such as image representation, compression, image segmentation, and video segmentation.


Mean Square Error Discrete Cosine Transform Sparse Representation Image Representation Image Patch 
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.


  1. Bruckstein, A., Donoho, D., & Elad, M. (2009). From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Review, 51(1), 34–81.MathSciNetCrossRefMATHGoogle Scholar
  2. Burt, P. J., & Adelson, E. H. (1983). The Laplacian pyramid as a compact image code. IEEE Transactions on Communications, 31(4), 532–540.CrossRefGoogle Scholar
  3. Candès, E. (2006). Compressive sampling. In Proceedings of the International Congress of Mathematics.Google Scholar
  4. Candès, E., & Donoho, D. (2002). New tight frames of curvelets and optimal representations of objects with smooth singularities. Technical Report. Stanford University.MATHGoogle Scholar
  5. Candès, E., & Wakin, M. (2008). An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), 21–30.CrossRefGoogle Scholar
  6. Chen, J.-Q., Pappas, T. N., Mojsilovic, A., & Rogowitz, B. E. (2003). Image segmentation by spatially adaptive color and texture features. In IEEE International Conference on Image Processing.Google Scholar
  7. Chen, S., Donoho, D., & Saunders, M. (1998). Atomic decomposition by basis pursuit. SIAM Journal of Scientific Computing, 20(1), 33–61.MathSciNetCrossRefMATHGoogle Scholar
  8. Coifman, R., & Wickerhauser, M. (1992). Entropy-based algorithms for best bases selection. IEEE Transactions on Information Theory, 38(2), 713–718.CrossRefMATHGoogle Scholar
  9. Delsarte, P., Macq, B., & Slock, D. (1992). Signal-adapted multiresolution transform for image coding. IEEE Transactions on Information Theory, 38, 897–903.CrossRefGoogle Scholar
  10. DeVore, R. (1998). Nonlinear approximation. Acta Numerica, 7, 51–150.MathSciNetCrossRefMATHGoogle Scholar
  11. DeVore, R., Jawerth, B., & Lucier, B. (1992). Image compression through wavelet transform coding. IEEE Transactions on Information Theory, 38(2), 719–746.MathSciNetCrossRefMATHGoogle Scholar
  12. Do, M. N., & Vetterli, M. (2002). Contourlets: A directional multiresolution image representation. In IEEE International Conference on Image Processing.Google Scholar
  13. Donoho, D. (1995). Cart and best-ortho-basis: A connection. Manuscript.MATHGoogle Scholar
  14. Donoho, D. (1998). Sparse components analysis and optimal atomic decomposition. Technical Report, Department of Statistics, Stanford University.Google Scholar
  15. Donoho, D. L. (1999). Wedgelets: Nearly-minimax estimation of edges. Annals of Statistics, 27, 859–897.MathSciNetCrossRefMATHGoogle Scholar
  16. Donoho, D. L., & Elad, M. (2003). Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization. Proceedings of National Academy of Sciences, 100(5), 2197–2202.MathSciNetCrossRefMATHGoogle Scholar
  17. Donoho, D. L., Vetterli, M., DeVore, R., & Daubechies, I. (1998). Data compression and harmonic analysis. IEEE Transactions on Information Theory, 44(6), 2435–2476.MathSciNetCrossRefMATHGoogle Scholar
  18. Effros, M., & Chou, P. (1995). Weighted universal transform coding: Universal image compression with the Karhunen-Loéve transform. In IEEE International Conference on Image Processing (Vol. 2, pp. 61–64).Google Scholar
  19. Elad, M., & Bruckstein, A. (2001). On sparse signal representations. In IEEE International Conference on Image Processing.Google Scholar
  20. Elad, M., & Bruckstein, A. (2002). A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Transactions on Information Theory, 48(9), 2558–2567.MathSciNetCrossRefMATHGoogle Scholar
  21. Elad, M., Figueiredo, M. A. T., & Ma, Y. (2010). On the role of sparse and redundant representations in image processing. Proceedings of the IEEE, 98(6), 972–982.CrossRefGoogle Scholar
  22. Feuer, A., Nemirovski, A. (2003). On sparse representation in pairs of bases. IEEE Transactions on Information Theory, 49(6), 1579–1581.MathSciNetCrossRefMATHGoogle Scholar
  23. Fisher, Y. (1995). Fractal Image Compression: Theory and Application. Springer-Verlag Telos.CrossRefGoogle Scholar
  24. Gersho, A., & Gray, R. M. (1992). Vector Quantization and Signal Compression. Boston: Kluwer Academic.CrossRefMATHGoogle Scholar
  25. Hong, W., Wright, J., Huang, K., & Ma, Y. (2006). Multi-scale hybrid linear models for lossy image representation. IEEE Transactions on Image Processing, 15(12), 3655–3671.MathSciNetCrossRefGoogle Scholar
  26. Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417–441.CrossRefMATHGoogle Scholar
  27. Jolliffe, I. (2002). Principal Component Analysis (2nd ed.). New York: Springer.MATHGoogle Scholar
  28. LePennec, E., & Mallat, S. (2005). Sparse geometric image representation with bandelets. IEEE Transactions on Image Processing, 14(4), 423–438.MathSciNetCrossRefGoogle Scholar
  29. Mallat, S. (1999). A Wavelet Tour of Signal Processing (2nd ed.). London: Academic.MATHGoogle Scholar
  30. Meyer, F. (2000). Fast adaptive wavelet packet image compression. IEEE Transactions on Image Processing, 9(5), 792–800.CrossRefGoogle Scholar
  31. Meyer, F. (2002). Image compression with adaptive local cosines. IEEE Transactions on Image Processing, 11(6), 616–629.CrossRefGoogle Scholar
  32. Muresan, D., & Parks, T. (2003). Adaptive principal components and image denoising. In IEEE International Conference on Image Processing.Google Scholar
  33. Olshausen, B., & D.J.Field (1996). Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381(6583), 607–609.CrossRefGoogle Scholar
  34. Pavlovic, V., Moulin, P., & Ramchandran, K. (1998). An integrated framework for adaptive subband image coding. IEEE Transactions on Signal Processing, 47(4), 1024–1038.CrossRefGoogle Scholar
  35. Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. The London, Edinburgh and Dublin Philosphical Magazine and Journal of Science, 2, 559–572.CrossRefMATHGoogle Scholar
  36. Rabiee, H., Kashyap, R., & Safavian, S. (1996). Adaptive multiresolution image coding with matching and basis pursuits. In IEEE International Conference on Image Processing.Google Scholar
  37. Ramchandran, K., & Vetterli, M. (1993). Best wavelet packets bases in a rate-distortion sense. IEEE Transactions on Image Processing, 2, 160–175.CrossRefGoogle Scholar
  38. Ramchandran, K., Vetterli, M., & Herley, C. (1996). Wavelets, subband coding, and best basis. Proceedings of the IEEE, 84(4), 541–560.CrossRefGoogle Scholar
  39. Shapiro, J. M. (1993). Embedded image coding using zerotrees of wavelet coefficients. IEEE Transactions on Signal Processing, 41(12), 3445–3463.CrossRefMATHGoogle Scholar
  40. Sikora, T., & Makai, B. (1995). Shape-adaptive DCT for generic coding of video. IEEE Transactions on Circuits and Systems For Video Technology, 5, 59–62.CrossRefGoogle Scholar
  41. Spielman, D., Wang, H., & Wright, J. (2012). Exact recovery of sparsity-used dictionaries. Conference on Learning Theory (COLT).Google Scholar
  42. Starck, J.-L., Elad, M., & Donoho, D. (2003). Image decomposition: Separation of texture from piecewise smooth content. In Proceedings of the SPIE (Vol. 5207, pp. 571–582).Google Scholar
  43. Sun, J., Qu, Q., & Wright, J. (2015). Complete dictionary recovery over the sphere. Preprint. CrossRefGoogle Scholar
  44. Vetterli, M., & Kovacevic, J. (1995). Wavelets and subband coding. Upper Saddle River: Prentice-Hall.MATHGoogle Scholar
  45. Wallace, G. K. (1991). The JPEG still picture compression standard. Communications of the ACM. Special issue on digital multimedia systems, 34(4), 30–44.Google Scholar
  46. Yang, J., Wright, J., Huang, T., & Ma, Y. (2010). Image super-resolution via sparse representation. IEEE Transactions on Image Processing, 19(11), 2861–2873.MathSciNetCrossRefGoogle Scholar
  47. Yu, G., Sapiro, G., & Mallat, S. (2010). Image modeling and enhancement via structured sparse model selection. In International Conference on Image Processing.Google Scholar
  48. Yu, G., Sapiro, G., & Mallat, S. (2012). Solving inverse problems with piecewise linear estimators: From gaussian mixture models to structured sparsity. IEEE Transactions on Image Processing, 21(5), 2481–2499.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York 2016

Authors and Affiliations

  • René Vidal
    • 1
  • Yi Ma
    • 2
  • S. Shankar Sastry
    • 3
  1. 1.Center for Imaging Science Department of Biomedical EngineeringJohns Hopkins UniversityBaltimoreUSA
  2. 2.School of Information Science and Technology ShanghaiTech UniversityShanghaiChina
  3. 3.Department of Electrical Engineering and Computer ScienceUniversity of California BerkeleyBerkeleyUSA

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