Skip to main content


Sparse Approximation of Images Inspired from the Functional Architecture of the Primary Visual Areas


Several drawbacks of critically sampled wavelets can be solved by overcomplete multiresolution transforms and sparse approximation algorithms. Facing the difficulty to optimize such nonorthogonal and nonlinear transforms, we implement a sparse approximation scheme inspired from the functional architecture of the primary visual cortex. The scheme models simple and complex cell receptive fields through log-Gabor wavelets. The model also incorporates inhibition and facilitation interactions between neighboring cells. Functionally these interactions allow to extract edges and ridges, providing an edge-based approximation of the visual information. The edge coefficients are shown sufficient for closely reconstructing the images, while contour representations by means of chains of edges reduce the information redundancy for approaching image compression. Additionally, the ability to segregate the edges from the noise is employed for image restoration.


  1. 1.

    Simoncelli EP, Freeman WT, Adelson EH, Heeger DJ: Shiftable multiscale transforms. IEEE Transactions on Information Theory 1992,38(2):587-607. 10.1109/18.119725

  2. 2.

    Do MN, Vetterli M: The contourlet transform: an efficient directional multiresolution image representation. IEEE Transactions on Image Processing 2005,14(12):2091–2106.

  3. 3.

    Kingsbury N: Complex wavelets for shift invariant analysis and filtering of signals. Applied and Computational Harmonic Analysis 2001,10(3):234–253. 10.1006/acha.2000.0343

  4. 4.

    Donoho DL, Flesia AG: Can recent innovations in harmonic analysis 'explain' key findings in natural image statistics? Network: Computation in Neural Systems 2001,12(3):371–393.

  5. 5.

    Mallat SG, Zhang Z: Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing 1993,41(12):3397–3415. 10.1109/78.258082

  6. 6.

    Chen SS, Donoho DL, Saunders MA: Atomic decomposition by basis pursuit. SIAM Journal of Scientific Computing 1998,20(1):33–61. 10.1137/S1064827596304010

  7. 7.

    Perrinet L, Samuelides M, Thorpe S: Coding static natural images using spiking event times: do neurons cooperate? IEEE Transactions on Neural Networks 2004,15(5):1164–1175. 10.1109/TNN.2004.833303

  8. 8.

    Fischer S, Cristóbal G, Redondo R: Sparse overcomplete Gabor wavelet representation based on local competitions. IEEE Transactions on Image Processing 2006,15(2):265–272.

  9. 9.

    Pece AEC: The problem of sparse image coding. Journal of Mathematical Imaging and Vision 2002,17(2):89–108. 10.1023/A:1020677318841

  10. 10.

    Perrinet L: Feature detection using spikes: the greedy approach. Journal of Physiology Paris 2004,98(4–6):530–539.

  11. 11.

    Daugman JG: Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. Journal of the Optical Society of America. A, Optics and Image Science 1985,2(7):1160–1169. 10.1364/JOSAA.2.001160

  12. 12.

    De Valois RL, Albrecht DG, Thorell LG: Spatial frequency selectivity of cells in macaque visual cortex. Vision Research 1982,22(5):545–559. 10.1016/0042-6989(82)90113-4

  13. 13.

    Olshausen BA, Field DJ: Sparse coding with an overcomplete basis set: a strategy employed by V1? Vision Research 1997,37(23):3311–3325. 10.1016/S0042-6989(97)00169-7

  14. 14.

    Kapadia MK, Westheimer G, Gilbert CD: Spatial distribution of contextual interactions in primary visual cortex and in visual perception. Journal of Neurophysiology 2000,84(4):2048–2062.

  15. 15.

    Mandon S, Kreiter AK: Rapid contour integration in macaque monkeys. Vision Research 2005,45(3):291–300. 10.1016/j.visres.2004.08.010

  16. 16.

    Hess RF, Hayes A, Field DJ: Contour integration and cortical processing. Journal of Physiology Paris 2003,97(2–3):105–119. 10.1016/j.jphysparis.2003.09.013

  17. 17.

    Grossberg S, Mingolla E, Williamson J: Synthetic aperture radar processing by a multiple scale neural system for boundary and surface representation. Neural Networks 1995,8(7–8):1005–1028. 10.1016/0893-6080(95)00079-8

  18. 18.

    Hansen T, Sepp W, Neumann H: Recurrent long-range interactions in early vision. In Emergent Neural Computational Architectures Based on Neuroscience, LNAI. Volume 2036. Edited by: Wermter S, Austin J, Willshaw D. Springer, Heidelberg, Germany; 2001:127–138. 10.1007/3-540-44597-8_9

  19. 19.

    Heitger F, Rosenthaler L, Von der Heydt R, Peterhans E, Kubler O: Simulation of neutral contour mechanisms: from simple to end-stopped cells. Vision Research 1992,32(5):963–981. 10.1016/0042-6989(92)90039-L

  20. 20.

    Yen S-C, Finkel LH: Extraction of perceptually salient contours by striate cortical networks. Vision Research 1998,38(5):719–741. 10.1016/S0042-6989(97)00197-1

  21. 21.

    VanRullen R, Delorme A, Thorpe SJ: Feed-forward contour integration in primary visual cortex based on asynchronous spike propagation. Neurocomputing 2001,38–40(1–4):1003–1009.

  22. 22.

    Coifman RR, Donoho D: Translation-invariant de-noising. In Wavelets and Statistics, Lecture Notes in Statistics. Volume 103. Edited by: Antoniadis A, Oppenheim G. Springer, New York, NY, USA; 1995:125–150. 10.1007/978-1-4612-2544-7_9

  23. 23.

    Fischer S, Sroubek F, Perrinet L, Redondo R, Cristóbal G: Self-invertible 2D log-Gabor wavelets. International Journal of Computer Vision, to appear

  24. 24.

    Fischer S, Redondo R, Perrinet L, Cristóbal G: Sparse Gabor wavelets by local operations. In Bioengineered and Bioinspired Systems II, May 2005, Sevilla, Spain, Proceedings of SPIE Edited by: Carmona RA. 5839: 75–86.

  25. 25.

    Portilla J, Strela V, Wainwright MJ, Simoncelli EP: Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Transactions on Image Processing 2003,12(11):1338–1351. 10.1109/TIP.2003.818640

  26. 26.

    Mallat S, Zhong S: Characterization of signals from multiscale edges. IEEE Transactions on Pattern Analysis and Machine Intelligence 1992,14(7):710–732. 10.1109/34.142909

  27. 27.

    Elder JH: Are edges incomplete? International Journal of Computer Vision 1999,34(2–3):97–122.

  28. 28.

    Peotta L, Granai L, Vandergheynst P: Image compression using an edge adapted redundant dictionary and wavelets. Signal Processing 2006,86(3):444–456. special issue on Sparse Approximations in Signal and Image Processing 10.1016/j.sigpro.2005.05.023

  29. 29.

    Starck J-L, Elad M, Donoho DL: Image decomposition via the combination of sparse representations and a variational approach. IEEE Transactions on Image Processing 2005,14(10):1570–1582.

  30. 30.

    Wakin M, Romberg J, Choi H, Baraniuk R:Image compression using an efficient edge cartoon texture model. Proceedings of Data Compression Conference (DCC '02), April 2002, Snowbird, Utah, USA 43–52.

  31. 31.

    Canny J: Computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence 1986,8(6):679–698.

  32. 32.

    Morrone MC, Burr DC: Feature detection in human vision: a phase-dependent energy model. Proceedings of the Royal Society of London. Series B. Biological Sciences 1988,235(1280):221–245. 10.1098/rspb.1988.0073

  33. 33.

    Rust BW, Rushmeier HE: A new representation of the contrast sensitivity function for human vision. In Proceedings of the International Conference on Imaging Science, Systems, and Technology (CISST '97), June 1997, Las Vegas, Nev, USA Edited by: Arabnia HR. 1–15.

  34. 34.

    Atick JJ: Could information theory provide an ecological theory of sensory processing? Network: Computation in Neural Systems 1992,3(2):213–251. 10.1088/0954-898X/3/2/009

  35. 35.

    Chang SG, Yu B, Vetterli M: Adaptive wavelet thresholding for image denoising and compression. IEEE Transactions on Image Processing 2000,9(9):1532–1546. 10.1109/83.862633

  36. 36.

    Bosking WH, Zhang Y, Schofield B, Fitzpatrick D: Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. Journal of Neuroscience 1997,17(6):2112–2127.

  37. 37.

    Krüger N: Collinearity and parallelism are statistically significant second-order relations of complex cell responses. Neural Processing Letters 1998,8(2):117–129. 10.1023/A:1009688428205

  38. 38.

    Geisler WS, Perry JS, Super BJ, Gallogly DP: Edge co-occurrence in natural images predicts contour grouping performance. Vision Research 2001,41(6):711–724. 10.1016/S0042-6989(00)00277-7

  39. 39.

    Kovesi P: Phase congruency detects corners and edges. Proceedings of the 7th International Conference on Digital Image Computing: Techniques and Applications (DICTA '03), December 2003, Sydney, NSW, Australia 309–318.

  40. 40.

    Hubel D: Eye, Brain, and Vision, Scientific American Library Series. W. H. Freeman, New York, NY, USA; 1988.

  41. 41.

    Dobbins A, Zucker SW, Cynader MS: Endstopping and curvature. Vision Research 1989,29(10):1371–1387. 10.1016/0042-6989(89)90193-4

  42. 42.

    Pasupathy A, Connor CE: Population coding of shape in area V4. Nature Neuroscience 2002,5(12):1332–1338. 10.1038/nn972

  43. 43.

    Redondo R, Cristóbal G: Lossless chain coder for gray edge images. Proceedings of IEEE International Conference on Image Processing (ICIP '03), September 2003, Barcelona, Spain 2: 201–204.

  44. 44.

    Freeman H: On the encoding of arbitrary geometric configurations. IRE Transactions on Electronic Computers 1961, 10: 260–268.

  45. 45.

    Howard GP: The design and analysis of efficient lossless data compression systems. In Tech. Rep. CS-93-28. Department of Computer Science, Brown University, Providence, RI, USA; 1993.

  46. 46.

    Portilla J, Simoncelli EP: A parametric texture model based on joint statistics of complex wavelet coefficients. International Journal of Computer Vision 2000,40(1):49–70. 10.1023/A:1026553619983

  47. 47.

    Fischer S: New contributions in overcomplete image representations inspired from the functional architecture of the primary visual cortex, Ph.D. thesis. Technical University Madrid High Technical School of Telecommunication Engineering, Department of Electronic Engineering, Spain; 2007.

Download references

Author information

Correspondence to Sylvain Fischer.

Rights and permissions

Reprints and Permissions

About this article


  • Approximation Algorithm
  • Visual Cortex
  • Receptive Field
  • Approximation Scheme
  • Image Compression