Hybrid Compressive Sampling via a New Total Variation TVL1

  • Xianbiao Shu
  • Narendra Ahuja
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6316)


Compressive sampling (CS) is aimed at acquiring a signal or image from data which is deemed insufficient by Nyquist/Shannon sampling theorem. Its main idea is to recover a signal from limited measurements by exploring the prior knowledge that the signal is sparse or compressible in some domain. In this paper, we propose a CS approach using a new total-variation measure TVL1, or equivalently TV\(_{\ell_1}\), which enforces the sparsity and the directional continuity in the gradient domain. Our TV\(_{\ell_1}\) based CS is characterized by the following attributes. First, by minimizing the ℓ1-norm of partial gradients, it can achieve greater accuracy than the widely-used TV\(_{\ell_1\ell_2}\) based CS. Second, it, named hybrid CS, combines low-resolution sampling (LRS) and random sampling (RS), which is motivated by our induction that these two sampling methods are complementary. Finally, our theoretical and experimental results demonstrate that our hybrid CS using TV\(_{\ell_1}\) yields sharper and more accurate images.


Gradient Magnitude Compressive Sampling Hybrid Sampling Diagonal Edge Partial Gradient 
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. 1.
    Candes, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59(8), 1208–1223 (2006)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Candes, E., Tao, T.: Near-optimal signal recovery from random projections and universal encoding strategies? IEEE Transactions on Information Theory 52(12), 5406–5245 (2006)Google Scholar
  3. 3.
    Candes, E., Romberg, J.: Sparsity and incoherence in compressive sampling. Inverse Prob. 23(3), 969–986 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Candes, E., Tao, T.: Decoding by linear programming. IEEE Transactions on Information Theory 51, 4203–4215 (2005)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, S., Donoho, D.: Atomic decomposition by basic pursuit. SIAM J. Sci. Comp. 20, 33–61 (1998)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Coifman, R., Geshwind, F., Meyer, Y.: Noiselets. Appl. Comp. Harmonic Analysis 10, 27–44 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Donoho, D.: Compressed sensing. IEEE Trans. on Information Theory (2006)Google Scholar
  8. 8.
    Duarte, M.F., Davenport, M.A., Takhar, D., et al.: Single-pixel imaging via compressive sampling. IEEE Signal Processing Magazine 25(2), 83–91 (2008)CrossRefGoogle Scholar
  9. 9.
    Gan, L., Do, T., Tran, T.: Fast compressive imaging using scrambled block hadamard ensemble. EUSIPCOGoogle Scholar
  10. 10.
    He, L., Chang, T.C., Osher, S., Fang, T., Speier, P.: Mr image reconstruction by using the iterative refinement method and nonlinear inverse scale space methods. UCLA CAM Report, pp. 06–35 (2006)Google Scholar
  11. 11.
  12. 12.
    Lustig, M., Donoho, D., Santos, J., Pauly, J.: Compressed sensing mri. IEEE Sig. Proc. Magazine (2007)Google Scholar
  13. 13.
    Ma, S., Yin, W., Zhang, Y., Chakraborty, A.: An efficient algorithm for compressed mr imaging using total variation and wavelets. CVPR (2008)Google Scholar
  14. 14.
    Maleh, R., Gilbert, A.C., Strauss, M.J.: Sparse gradient image reconstruction done faster. ICIP 2, 77–80 (2007)Google Scholar
  15. 15.
    Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM Journal on Computing 24, 227–234 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Romberg, J.: Variational methods for compressive sampling. Proc. SPIE 6498, 64980J–2–5 (2007)Google Scholar
  17. 17.
    Romberg, J.: Imaging via compressive sampling. Comm. Pure Appl. Math, 14–20 (2008)Google Scholar
  18. 18.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D Nonlinear Phenomena 60(1), 259–268 (1992)zbMATHCrossRefGoogle Scholar
  19. 19.
    Saunders, M.A.: Pdco: Primal-dual interior-point method for convex objectives. Systems Optimization Laboratory, Stanford University (2002)Google Scholar
  20. 20.
    Skodras, A., Christopoulos, C., Ebrahimi, T.: The jpeg2000 still image compression standard. IEEE Signal Processing Mag. 18, 36–58 (2001)CrossRefGoogle Scholar
  21. 21.
  22. 22.
    Takhar, D., Laska, J., Wakin, M., Duarte, M., et al.: A new compressive imaging camera architecture using optical-domain compression. Proc. of Computational Imaging IV at SPIE Electronic Imaging 6065, 43–52 (2006)Google Scholar
  23. 23.
    Tropp, J.A., Gilbert, A.C.: Signal recovery from partial information via orthogonal matching pursuit. IEEE Transactions on Information Theory 53, 4655–4666 (2007)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Yang, J., Zhang, Y., Yin, W.: A fast tv-l1-l2 algorithm for image reconstruction from partial fourier data. To be submitted to IEEE Trans. on Special Topics (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Xianbiao Shu
    • 1
  • Narendra Ahuja
    • 1
  1. 1.University of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations