Skip to main content
SpringerLink
Log in

Sparse Modeling of Textures

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

This paper presents a generative model for textures that uses a local sparse description of the image content. This model enforces the sparsity of the expansion of local texture patches on adapted atomic elements. The analysis of a given texture within this framework performs the sparse coding of all the patches of the texture into the dictionary of atoms. Conversely, the synthesis of a new texture is performed by solving an optimization problem that seeks for a texture whose patches are sparse in the dictionary. This paper explores several strategies to choose this dictionary. A set of hand crafted dictionaries composed of edges, oscillations, lines or crossings elements allows to synthesize synthetic images with geometric features. Another option is to define the dictionary as the set of all the patches of an input exemplar. This leads to computer graphics methods for synthesis and shares some similarities with non-local means filtering. The last method we explore learns the dictionary by an optimization process that maximizes the sparsity of a set of exemplar patches. Applications of all these methods to texture synthesis, inpainting and classification shows the efficiency of the proposed texture model.

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

Similar content being viewed by others

References

  1. Aharon, M., Elad, M.: Sparse and redundant modeling of image content using an image-signature-dictionary. J. Imag. Sci. 1(3), 228–247 (2008)

    Article  Google Scholar 

  2. Aharon, M., Elad, M., Bruckstein, A.M.: The k-svd: An algorithm for designing of overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54(11), 4311–4322 (2006)

    Article  Google Scholar 

  3. Ashikhmin, M.: Synthesizing natural textures. In: SI3D’01: Proceedings of the 2001 Symposium on Interactive 3D Graphics, pp. 217–226. Assoc. Comput. Mach., New York (2001)

    Chapter  Google Scholar 

  4. Ballester, C., Caselles, V., Verdera, J.: Disocclusion by joint interpolation of vector fields and gray levels. Multiscale Model. Simul. 2(1), 80–123 (2003)

    Article  MathSciNet  Google Scholar 

  5. Bar-Joseph, Z., El-Yaniv, R., Lischinski, D., Werman, M.: Texture mixing and texture movie synthesis using statistical learning. IEEE Trans. Vis. Comput. Graph. 7(2), 120–135 (2001)

    Article  Google Scholar 

  6. Bell, A.J., Sejnowski, T.J.: The independent components of natural scenes are edge filters. Vis. Res. 37, 3327–3338 (1997)

    Article  Google Scholar 

  7. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proc. of Siggraph 2000, pp. 417–424 (2000)

  8. Bovik, A.C.: Analysis of multichannel narrow-band filters for image texture segmentation. IEEE Trans. Signal Process. 39(9), 2025 (1991)

    Article  Google Scholar 

  9. Brox, T., Cremers, D.: Iterated nonlocal means for texture restoration. In: Proc. International Conference on Scale Space and Variational Methods in Computer Vision, Ischia, Italy. LNCS. Springer, Berlin (2007)

    Google Scholar 

  10. Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998)

    Article  MathSciNet  Google Scholar 

  12. Clausi, D.A., Jernigan, M.E.: Designing Gabor filters for optimal texture separability. Pattern Recogn. 33(11), 1835–1849 (2000)

    Article  Google Scholar 

  13. Cohen, L.D.: A new approach of vector quantization for image data compression and texture detection. In: International Conference on Pattern Recognition, pp. 1250–1254 (1988)

  14. Cook, R.L., DeRose, T.: Wavelet noise. ACM Trans. Graph. 24(3), 803–811 (2005)

    Article  Google Scholar 

  15. Criminisi, A., Pérez, P., Toyama, K.: Region filling and object removal by exemplar-based image inpainting. IEEE Trans. Image Process. 13(9), 1200–1212 (2004)

    Article  Google Scholar 

  16. Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57, 1413–1541 (2004)

    Article  MATH  Google Scholar 

  17. De Bonet, J.S.: Multiresolution sampling procedure for analysis and synthesis of texture images. In: Proc. of SIGGRAPH’97, pp. 361–368. Assoc. Comput. Mach./Addison Wesley, New York/Reading (1997)

    Google Scholar 

  18. Donoho, D.: Wedgelets: Nearly-minimax estimation of edges. Ann. Stat. 27, 353–382 (1999)

    Article  MathSciNet  Google Scholar 

  19. Efros, A.A., Leung, T.K.: Texture synthesis by non-parametric sampling. In: ICCV’99: Proceedings of the International Conference on Computer Vision, vol. 2, p. 1033. IEEE Computer Society, Los Alamitos (1999)

    Chapter  Google Scholar 

  20. Efros, A.A., Freeman, W.T.: Image quilting for texture synthesis and transfer. In: Proc. of SIGGRAPH 2001, pp. 341–346 (2001)

  21. Engan, K., Aase, S.O., Hakon Husoy, J.: Method of optimal directions for frame design. In: Proc. ICASSP’99, Washington, DC, pp. 2443–2446. IEEE Computer Society, Los Alamitos (1999)

    Google Scholar 

  22. Fadili, M.J., Starck, J.-L., Murtagh, F.: Inpainting and zooming using sparse representations. Comput. J. (2006, revised)

  23. Heeger, D.J., Bergen, J.R.: Pyramid-Based texture analysis/synthesis. In: Cook, R. (ed.) SIGGRAPH 95 Conference Proceedings. Annual Conference Series, pp. 229–238. Assoc. Comput. Mach./Addison Wesley, New York/Reading (1995)

    Chapter  Google Scholar 

  24. Jain, A.K., Farrokhnia, F.: Unsupervised texture segmentation using Gabor filters. Pattern Recogn. 24(12), 1167–1186 (1991)

    Article  Google Scholar 

  25. Julesz, B.: Visual pattern discrimination. IRE Trans. Inf. Theory 8(2), 84–92 (1962)

    Article  Google Scholar 

  26. Kreutz-Delgado, K., Murray, J.F., Rao, B.D., Engan, K., Lee, T.-W., Sejnowski, T.J.: Dictionary learning algorithms for sparse representation. Neural Comput. 15(2), 349–396 (2003)

    Article  MATH  Google Scholar 

  27. Kwatra, V., Essa, I., Bobick, A., Kwatra, N.: Texture optimization for example-based synthesis. ACM Trans. Graph. 24(3), 795–802 (2005). Proc. of SIGGRAPH 2005

    Article  Google Scholar 

  28. Kwatra, V., Schödl, A., Essa, I., Turk, G., Bobick, A.: Graphcut textures: Image and video synthesis using graph cuts. ACM Trans. Graph. 22(3), 277–286 (2003). SIGGRAPH 2003

    Article  Google Scholar 

  29. Lee, D.D., Seung, H.S.: Algorithms for non-negative matrix factorization. In: Advances in Neural Information Processing Systems 13. MIT Press, Cambridge (2001)

    Google Scholar 

  30. Lefebvre, S., Hoppe, H.: Parallel controllable texture synthesis. ACM Trans. Graph. 24(3), 777–786 (2005)

    Article  Google Scholar 

  31. Lefebvre, S., Hoppe, H.: Appearance-space texture synthesis. ACM Trans. Graph. 25(3), 541–548 (2006)

    Article  Google Scholar 

  32. Lewicki, M.S., Sejnowski, T.J.: Learning overcomplete representations. Neural Comput. 12(2), 337–365 (2000)

    Article  Google Scholar 

  33. Lu, S.Y., Hernandez, J.E., Clark, G.A.: Texture segmentation by clustering of Gabor feature vectors. In: Proc. IJCNN’91, International Joint Conference on Neural Networks, vol. I, pp. 683–688. IEEE Press, New York (1991)

    Google Scholar 

  34. Mairal, J., Elad, M., Sapiro, G.: Sparse representation for color image restoration. IEEE Trans. Image Process. 17(1), 53–69 (2008)

    Article  MathSciNet  Google Scholar 

  35. Mairal, J., Sapiro, G., Elad, M.: Learning multiscale sparse representations for image and video restoration. Preprint (2007)

  36. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (1998)

    MATH  Google Scholar 

  37. Manduchi, R., Portilla, J.: Independent component analysis of textures. In: ICCV, pp. 1054–1060 (1999)

  38. Matusik, W., Zwicker, M., Durand, F.: Texture design using a simplicial complex of morphable textures. ACM Trans. Graph. 24(3), 787–794 (2005)

    Article  Google Scholar 

  39. Olshausen, B.A., Field, D.J.: Emergence of simple-cell receptive-field properties by learning a sparse code for natural images. Nature 381(6583), 607–609 (1996)

    Article  Google Scholar 

  40. Paragios, N., Deriche, R.: Geodesic active regions and level set methods for supervised texture segmentation. Int. J. Comput. Vis. 46(3), 223–247 (2002)

    Article  MATH  Google Scholar 

  41. Perlin, K.: An image synthesizer. In: Proc. of SIGGRAPH’85, pp. 287–296. Assoc. Comput. Mach., New York (1985)

    Google Scholar 

  42. Peyré, G.: Non-negative sparse modeling of textures. In: Proc. of SSVM’07, pp. 628–639 (2007)

  43. Peyré, G.: Manifold models for signals and images. Comput. Vis. Image Underst. (2008, to appear)

  44. Portilla, J., Simoncelli, E.P.: A parametric texture model based on joint statistics of complex wavelet coefficients. Int. J. Comput. Vis. 40(1), 49–70 (2000)

    Article  MATH  Google Scholar 

  45. Sallee, P., Olshausen, B.A.: Learning sparse multiscale image representations. In: Becker, S., Thrun, S., Obermayer, K. (eds.) NIPS, pp. 1327–1334. MIT Press, Cambridge (2002)

    Google Scholar 

  46. Simoncelli, E.P., Olshausen, B.A.: Natural image statistics and neural representation. Ann. Rev. Neurosci. 24, 1193–1215 (2001)

    Article  Google Scholar 

  47. Skretting, K., Husoy, J.H.: Texture classification using sparse frame based representations. EURASIP J. Appl. Signal Process. 2006(1), 102–102 (2006)

    Google Scholar 

  48. Starck, J.-L., Elad, M., Donoho, D.L.: Redundant multiscale transforms and their application for morphological component analysis. Adv. Imaging Electron Phys. 132, 287–348 (2004)

    Google Scholar 

  49. Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)

    Article  MathSciNet  Google Scholar 

  50. Tropp, J.A.: Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans. Inf. Theory 52(3), 1030–1051 (2006)

    Article  MathSciNet  Google Scholar 

  51. Tschumperlé, D., Deriche, R.: Vector-valued image regularization with PDEs: A common framework for different applications. IEEE Trans. Pattern Anal. Mach. Intell. 27(4), 506–517 (2005)

    Article  Google Scholar 

  52. Tseng, P.: Convergence of a block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109(3), 475–494 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  53. Tuceryan, M.: Moment-based texture segmentation. Pattern Recogn. Lett. 15(7), 659–668 (1994)

    Article  Google Scholar 

  54. Wei, L.-Y., Levoy, M.: Fast texture synthesis using tree-structured vector quantization. In: SIGGRAPH’00: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 479–488. Assoc. Comput. Mach./Addison-Wesley, New York/Reading (2000)

    Chapter  Google Scholar 

  55. Wei, L.-Y., Han, J., Zhou, K., Bao, H., Guo, B., Shum, H.-Y.: Inverse texture synthesis. ACM Trans. Graph. 27(3), 1–9 (2008). http://doi.acm.org/10.1145/1360612.1360651

    Google Scholar 

  56. Zeng, X.-Y., Chen, Y.-W., van Alphen, D., Nakao, Z.: Selection of ICA features for texture classification. In: ISNN (2), pp. 262–267 (2005)

  57. Zhu, S.C., Liu, X.W., Wu, Y.N.: Exploring texture ensembles by efficient Markov chain Monte Carlo-toward a ‘trichromacy’ theory of texture. IEEE Trans. Pattern Anal. Mach. Intell. 22(6), 554–569 (2000)

    Article  Google Scholar 

  58. Zhu, S.C., Wu, Y., Mumford, D.: Filters, random fields and maximum entropy (FRAME): Towards a unified theory for texture modeling. Int. J. Comput. Vis. 27(2), 107–126 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS and Ceremade, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    Gabriel Peyré

Authors
  1. Gabriel Peyré
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriel Peyré.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peyré, G. Sparse Modeling of Textures. J Math Imaging Vis 34, 17–31 (2009). https://doi.org/10.1007/s10851-008-0120-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-008-0120-3

Keywords

Search

Navigation