Abstract

Recent work has shown that tailored overcomplete dictionaries can provide a better image model than standard basis functions for a variety of image processing tasks. Here we propose a modified K-SVD dictionary learning algorithm designed to maintain the advantages of the original approach but with a focus on improved convergence. We then use the learned model to denoise infrared maritime imagery and compare the performance to the original K-SVD algorithm, several overcomplete “fixed” dictionaries, and a standard wavelet denoising algorithm. Results indicate the superiority of overcomplete representations and show that our tailored approach provides similar peak signal-to-noise ratios as the traditional K-SVD at roughly half the computational cost.

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
    [CrossRef]
  2. H. Krim, D. Tucker, and S. Mallat, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225–2238 (1999).
    [CrossRef]
  3. M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10–12 (1996).
    [CrossRef]
  4. J.-L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
    [CrossRef]
  5. M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
    [CrossRef]
  6. R. Rubinstein, M. Zibulevsky, and M. Elad, “double sparsity: learning sparse dictionaries for sparse signal approximation,” IEEE Trans. Signal Process. 58, 1553–1564 (2010).
    [CrossRef]
  7. R. Neff, and A. Zakhor, “Very low bit-rate video coding based on matching pursuits,” IEEE Trans. Circuits Syst. Video Technol. 7, 158–171 (1997).
    [CrossRef]
  8. K. Skretting, and K. Engan, “Recursive least squares dictionary learning algorithm,” IEEE Trans. Signal Process. 58, 2121–2130 (2010).
    [CrossRef]
  9. A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
    [CrossRef]
  10. S. G. Mallat, and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
    [CrossRef]
  11. Y. Pati, R. Rezaifar, and P. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in 1993 Conference Record of the 27th Asilomar Conference on Signals, Systems and Computers (1993), Vol. 1, pp. 40–44.
  12. J. A. Tropp, and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
    [CrossRef]
  13. K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representations,” Neural Comput. 15, 349–396 (2003).
    [CrossRef]
  14. S. Lesage, R. Gribonval, F. Bimbot, and L. Benaroya, “Learning unions of orthonormal bases with thresholded singular value decomposition,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (2005), Vol. 5, pp. 293–296.
  15. R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Technical Report (Technion—Israel Institute of Technology, 2008).
  16. M. Elad, and M. Aharon, “Image denoising via sparse and redundant representations over learned dictionaries,” IEEE Trans. Image Process. 15, 3736–3745 (2006).
    [CrossRef]
  17. J. Yang, J. Wright, T. Huang, and Y. Ma, “Image super-resolution via sparse representation,” IEEE Trans. Image Process. 19, 2861–2873 (2010).
    [CrossRef]
  18. R. Zeyde, M. Elad, and M. Protter, “On single image scale-up using sparse-representations,” Lect. Notes Comput. Sci. 6920/2012, 711–730 (2012).
    [CrossRef]
  19. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, 2010).

2012 (1)

R. Zeyde, M. Elad, and M. Protter, “On single image scale-up using sparse-representations,” Lect. Notes Comput. Sci. 6920/2012, 711–730 (2012).
[CrossRef]

2010 (3)

J. Yang, J. Wright, T. Huang, and Y. Ma, “Image super-resolution via sparse representation,” IEEE Trans. Image Process. 19, 2861–2873 (2010).
[CrossRef]

R. Rubinstein, M. Zibulevsky, and M. Elad, “double sparsity: learning sparse dictionaries for sparse signal approximation,” IEEE Trans. Signal Process. 58, 1553–1564 (2010).
[CrossRef]

K. Skretting, and K. Engan, “Recursive least squares dictionary learning algorithm,” IEEE Trans. Signal Process. 58, 2121–2130 (2010).
[CrossRef]

2009 (1)

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

2007 (1)

J. A. Tropp, and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

2006 (2)

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[CrossRef]

M. Elad, and M. Aharon, “Image denoising via sparse and redundant representations over learned dictionaries,” IEEE Trans. Image Process. 15, 3736–3745 (2006).
[CrossRef]

2003 (1)

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representations,” Neural Comput. 15, 349–396 (2003).
[CrossRef]

2002 (1)

J.-L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

1999 (1)

H. Krim, D. Tucker, and S. Mallat, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225–2238 (1999).
[CrossRef]

1997 (1)

R. Neff, and A. Zakhor, “Very low bit-rate video coding based on matching pursuits,” IEEE Trans. Circuits Syst. Video Technol. 7, 158–171 (1997).
[CrossRef]

1996 (1)

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10–12 (1996).
[CrossRef]

1995 (1)

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

1993 (1)

S. G. Mallat, and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

Aharon, M.

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[CrossRef]

M. Elad, and M. Aharon, “Image denoising via sparse and redundant representations over learned dictionaries,” IEEE Trans. Image Process. 15, 3736–3745 (2006).
[CrossRef]

Benaroya, L.

S. Lesage, R. Gribonval, F. Bimbot, and L. Benaroya, “Learning unions of orthonormal bases with thresholded singular value decomposition,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (2005), Vol. 5, pp. 293–296.

Bimbot, F.

S. Lesage, R. Gribonval, F. Bimbot, and L. Benaroya, “Learning unions of orthonormal bases with thresholded singular value decomposition,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (2005), Vol. 5, pp. 293–296.

Bruckstein, A.

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[CrossRef]

Bruckstein, A. M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Burrus, C. S.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10–12 (1996).
[CrossRef]

Candes, E. J.

J.-L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

Donoho, D. L.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

J.-L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

Elad, M.

R. Zeyde, M. Elad, and M. Protter, “On single image scale-up using sparse-representations,” Lect. Notes Comput. Sci. 6920/2012, 711–730 (2012).
[CrossRef]

R. Rubinstein, M. Zibulevsky, and M. Elad, “double sparsity: learning sparse dictionaries for sparse signal approximation,” IEEE Trans. Signal Process. 58, 1553–1564 (2010).
[CrossRef]

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[CrossRef]

M. Elad, and M. Aharon, “Image denoising via sparse and redundant representations over learned dictionaries,” IEEE Trans. Image Process. 15, 3736–3745 (2006).
[CrossRef]

R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Technical Report (Technion—Israel Institute of Technology, 2008).

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, 2010).

Engan, K.

K. Skretting, and K. Engan, “Recursive least squares dictionary learning algorithm,” IEEE Trans. Signal Process. 58, 2121–2130 (2010).
[CrossRef]

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representations,” Neural Comput. 15, 349–396 (2003).
[CrossRef]

Gilbert, A. C.

J. A. Tropp, and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

Gribonval, R.

S. Lesage, R. Gribonval, F. Bimbot, and L. Benaroya, “Learning unions of orthonormal bases with thresholded singular value decomposition,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (2005), Vol. 5, pp. 293–296.

Guo, H.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10–12 (1996).
[CrossRef]

Huang, T.

J. Yang, J. Wright, T. Huang, and Y. Ma, “Image super-resolution via sparse representation,” IEEE Trans. Image Process. 19, 2861–2873 (2010).
[CrossRef]

Kreutz-Delgado, K.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representations,” Neural Comput. 15, 349–396 (2003).
[CrossRef]

Krim, H.

H. Krim, D. Tucker, and S. Mallat, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225–2238 (1999).
[CrossRef]

Krishnaprasad, P.

Y. Pati, R. Rezaifar, and P. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in 1993 Conference Record of the 27th Asilomar Conference on Signals, Systems and Computers (1993), Vol. 1, pp. 40–44.

Lang, M.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10–12 (1996).
[CrossRef]

Lee, T.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representations,” Neural Comput. 15, 349–396 (2003).
[CrossRef]

Lesage, S.

S. Lesage, R. Gribonval, F. Bimbot, and L. Benaroya, “Learning unions of orthonormal bases with thresholded singular value decomposition,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (2005), Vol. 5, pp. 293–296.

Ma, Y.

J. Yang, J. Wright, T. Huang, and Y. Ma, “Image super-resolution via sparse representation,” IEEE Trans. Image Process. 19, 2861–2873 (2010).
[CrossRef]

Mallat, S.

H. Krim, D. Tucker, and S. Mallat, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225–2238 (1999).
[CrossRef]

Mallat, S. G.

S. G. Mallat, and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

Murray, J. F.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representations,” Neural Comput. 15, 349–396 (2003).
[CrossRef]

Neff, R.

R. Neff, and A. Zakhor, “Very low bit-rate video coding based on matching pursuits,” IEEE Trans. Circuits Syst. Video Technol. 7, 158–171 (1997).
[CrossRef]

Odegard, J. E.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10–12 (1996).
[CrossRef]

Pati, Y.

Y. Pati, R. Rezaifar, and P. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in 1993 Conference Record of the 27th Asilomar Conference on Signals, Systems and Computers (1993), Vol. 1, pp. 40–44.

Protter, M.

R. Zeyde, M. Elad, and M. Protter, “On single image scale-up using sparse-representations,” Lect. Notes Comput. Sci. 6920/2012, 711–730 (2012).
[CrossRef]

Rao, B. D.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representations,” Neural Comput. 15, 349–396 (2003).
[CrossRef]

Rezaifar, R.

Y. Pati, R. Rezaifar, and P. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in 1993 Conference Record of the 27th Asilomar Conference on Signals, Systems and Computers (1993), Vol. 1, pp. 40–44.

Rubinstein, R.

R. Rubinstein, M. Zibulevsky, and M. Elad, “double sparsity: learning sparse dictionaries for sparse signal approximation,” IEEE Trans. Signal Process. 58, 1553–1564 (2010).
[CrossRef]

R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Technical Report (Technion—Israel Institute of Technology, 2008).

Sejnowski, T. J.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representations,” Neural Comput. 15, 349–396 (2003).
[CrossRef]

Skretting, K.

K. Skretting, and K. Engan, “Recursive least squares dictionary learning algorithm,” IEEE Trans. Signal Process. 58, 2121–2130 (2010).
[CrossRef]

Starck, J.-L.

J.-L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

Tropp, J. A.

J. A. Tropp, and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

Tucker, D.

H. Krim, D. Tucker, and S. Mallat, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225–2238 (1999).
[CrossRef]

Wells, R. O.

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10–12 (1996).
[CrossRef]

Wright, J.

J. Yang, J. Wright, T. Huang, and Y. Ma, “Image super-resolution via sparse representation,” IEEE Trans. Image Process. 19, 2861–2873 (2010).
[CrossRef]

Yang, J.

J. Yang, J. Wright, T. Huang, and Y. Ma, “Image super-resolution via sparse representation,” IEEE Trans. Image Process. 19, 2861–2873 (2010).
[CrossRef]

Zakhor, A.

R. Neff, and A. Zakhor, “Very low bit-rate video coding based on matching pursuits,” IEEE Trans. Circuits Syst. Video Technol. 7, 158–171 (1997).
[CrossRef]

Zeyde, R.

R. Zeyde, M. Elad, and M. Protter, “On single image scale-up using sparse-representations,” Lect. Notes Comput. Sci. 6920/2012, 711–730 (2012).
[CrossRef]

Zhang, Z.

S. G. Mallat, and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

Zibulevsky, M.

R. Rubinstein, M. Zibulevsky, and M. Elad, “double sparsity: learning sparse dictionaries for sparse signal approximation,” IEEE Trans. Signal Process. 58, 1553–1564 (2010).
[CrossRef]

R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Technical Report (Technion—Israel Institute of Technology, 2008).

IEEE Signal Process. Lett. (1)

M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Process. Lett. 3, 10–12 (1996).
[CrossRef]

IEEE Trans. Circuits Syst. Video Technol. (1)

R. Neff, and A. Zakhor, “Very low bit-rate video coding based on matching pursuits,” IEEE Trans. Circuits Syst. Video Technol. 7, 158–171 (1997).
[CrossRef]

IEEE Trans. Image Process. (3)

J.-L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670–684 (2002).
[CrossRef]

M. Elad, and M. Aharon, “Image denoising via sparse and redundant representations over learned dictionaries,” IEEE Trans. Image Process. 15, 3736–3745 (2006).
[CrossRef]

J. Yang, J. Wright, T. Huang, and Y. Ma, “Image super-resolution via sparse representation,” IEEE Trans. Image Process. 19, 2861–2873 (2010).
[CrossRef]

IEEE Trans. Inf. Theory (3)

J. A. Tropp, and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

H. Krim, D. Tucker, and S. Mallat, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225–2238 (1999).
[CrossRef]

IEEE Trans. Signal Process. (4)

K. Skretting, and K. Engan, “Recursive least squares dictionary learning algorithm,” IEEE Trans. Signal Process. 58, 2121–2130 (2010).
[CrossRef]

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process. 54, 4311–4322 (2006).
[CrossRef]

R. Rubinstein, M. Zibulevsky, and M. Elad, “double sparsity: learning sparse dictionaries for sparse signal approximation,” IEEE Trans. Signal Process. 58, 1553–1564 (2010).
[CrossRef]

S. G. Mallat, and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

Lect. Notes Comput. Sci. (1)

R. Zeyde, M. Elad, and M. Protter, “On single image scale-up using sparse-representations,” Lect. Notes Comput. Sci. 6920/2012, 711–730 (2012).
[CrossRef]

Neural Comput. (1)

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representations,” Neural Comput. 15, 349–396 (2003).
[CrossRef]

SIAM Rev. (1)

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Other (4)

S. Lesage, R. Gribonval, F. Bimbot, and L. Benaroya, “Learning unions of orthonormal bases with thresholded singular value decomposition,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (2005), Vol. 5, pp. 293–296.

R. Rubinstein, M. Zibulevsky, and M. Elad, “Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit,” CS Technical Report (Technion—Israel Institute of Technology, 2008).

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, 2010).

Y. Pati, R. Rezaifar, and P. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in 1993 Conference Record of the 27th Asilomar Conference on Signals, Systems and Computers (1993), Vol. 1, pp. 40–44.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1.
Fig. 1.

Basic overview of dictionary-based image processing. The image under study is divided into possibly overlapping tiles of size N×N (tiles shown here are nonoverlapping). Each tile is transformed into an N-point vector and modeled using an N×M dictionary (M>N). The model coefficients xi are obtained by solving Eq. (4).

Fig. 2.
Fig. 2.

Several fixed overcomplete dictionaries used in representing N=64 point image patches. The number of atoms used is M=441. Shown are the (a) cosine, (b) Haar, and (c) Gabor dictionaries.

Fig. 3.
Fig. 3.

Influence of DUCs on algorithm convergence. (a) Progression in RMSE as a function of iteration in training a dictionary using 20,000 training patches with a sparsity constraint of K=6 atoms. (b) Expected percentage gain in RMSE (averaged over 32 independent trials) obtained after four training iterations and using a cardinality of K=6 as a function of the number of training patches used. (c) Expected percentage gain in RMSE after four training iterations as a function of the sparsity constraint K using 50,000 training patches. (d) Expected percentage gain in RMSE as a function of the number of iterations for K=6. In all cases, the dictionary was trained to model 8×8 image patches and consisted of 384 atoms.

Fig. 4.
Fig. 4.

SWIR camera used to collect all imagery used in this study.

Fig. 5.
Fig. 5.

Several different SWIR images used in training the dictionaries shown in Fig. 6.

Fig. 6.
Fig. 6.

Dictionaries generated using the (a) original and (b) modified (fast) K-SVD algorithms.

Fig. 7.
Fig. 7.

(a) Original and (b) noise-corrupted imagery.

Fig. 8.
Fig. 8.

Close up of denoised imagery. (a) Overcomplete cosine, (b) overcomplete Haar, (c) overcomplete Gabor, (e) K-SVD, (f) modified K-SVD, and (g) standard wavelet.

Fig. 9.
Fig. 9.

(a) Difference in dictionary training time between standard and modified K-SVD algorithm and (b) corresponding denoising performance on several different SWIR images using 16 iterations of the standard algorithm and eight iterations of the modified algorithm with four DUCs. In this example, 80,000 image patches were used to train a M=320 atom dictionary using a sparsity constraint of K=7.

Fig. 10.
Fig. 10.

Difference in PSNR values between learned and fixed (cosine) dictionaries for one image as a function of the number of iterations used in training the dictionary. Performance is consistently improved using the learned approach and the modified K-SVD algorithm produces better results for a fixed number of iterations.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

y=z+n=Ax+n,
AAij={1/N:i=1N1j=02/Ncos(πij/N):i=0N1j=1N1
AAij={1/N   :i=1N1j=02/Ncos(πij/M):i=0N1j=1M1
x^=minx{yAx22subjecttox0<K}.
A^,X^=minA,X{YAX22subject toX0<K}.
PSNR=10log10(max(y)21#pixelsall pixels(z^z).2).

Metrics