Abstract

We present advanced techniques for the restoration of images obtained by soft x-ray laser microscopy. We show two methods. One method is based on adaptive thresholding, while the other uses local Wiener filtering in the wavelet domain to achieve high noise gains. These wavelet based denoising techniques are improved using spatial noise modeling. The accurate noise model is built up from two consecutive images of the object and respective background images. To our knowledge, the results of both proposed approaches over-perform competitive methods. The analysis is robust to enable image acquisition with significantly lower exposure times, which is critical in samples that are sensitive to radiation damage as is the case of biological samples imaged by SXR microscopy.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
  4. O. von Hofsten, M. Bertilson, J. Reinspach, A. Holmberg, H. M. Hertz, and U. Vogt, “Sub-25-nm laboratory x-ray microscopy using a compound Fresnel zone plate,” Opt. Lett. 34(17), 2631–2633 (2009).
    [Crossref] [PubMed]
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    [Crossref]
  21. J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image Denoising using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Trans. Image Process. 12(11), 1338–1351 (2003).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  23. A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
    [Crossref] [PubMed]

2013 (1)

2012 (3)

2011 (1)

2010 (1)

M. Vrankić, D. Seršić, and V. Sučić, “Adaptive 2-D wavelet transform based on the lifting scheme with preserved vanishing moments,” IEEE Trans. Image Process. 19(8), 1987–2004 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (2)

2007 (2)

F. Luisier, T. Blu, and M. Unser, “A New SURE Approach to Image Denoising: Interscale Orthonormal Wavelet Thresholding,” IEEE Trans. Image Process. 16(3), 593–606 (2007).
[Crossref] [PubMed]

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (1)

2004 (1)

H. Choi and R. Baraniuk, “Multiple Wavelet Basis Image Denoising Using Besov Ball Projections,” IEEE Signal Process. Lett. 11(9), 717–720 (2004).
[Crossref]

2003 (3)

R. L. Claypoole, G. M. Davis, W. Sweldens, and R. G. Baraniuk, “Nonlinear wavelet transforms for image coding via lifting,” IEEE Trans. Image Process. 12(12), 1449–1459 (2003).
[Crossref] [PubMed]

H. Stollberg, J. Boutet de Monvel, A. Holmberg, and H. M. Hertz, “Wavelet-based image restoration for compact X-ray microscopy,” J. Microsc. 211(2), 154–160 (2003).
[Crossref] [PubMed]

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image Denoising using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Trans. Image Process. 12(11), 1338–1351 (2003).
[Crossref] [PubMed]

1995 (1)

D. L. Donoho, “De-noising by soft thresholding,” IEEE Trans. Inf. Theory 41(3), 613–627 (1995).
[Crossref]

1994 (1)

D. L. Donoho and I. M. Johnstone, “Ideal adaptation via wavelet shrinkage,” Biometrika 81(3), 425–455 (1994).
[Crossref]

Anderson, E.

Anderson, E. H.

S. Carbajo, I. D. Howlett, F. Brizuela, K. S. Buchanan, M. C. Marconi, W. Chao, E. H. Anderson, I. Artioukov, A. Vinogradov, J. J. Rocca, and C. S. Menoni, “Sequential single-shot imaging of nanoscale dynamic interactions with a table-top soft x-ray laser,” Opt. Lett. 37(14), 2994–2996 (2012).
[Crossref] [PubMed]

C. A. Brewer, F. Brizuela, P. Wachulak, D. H. Martz, W. Chao, E. H. Anderson, D. T. Attwood, A. V. Vinogradov, I. A. Artyukov, A. G. Ponomareko, V. V. Kondratenko, M. C. Marconi, J. J. Rocca, and C. S. Menoni, “Single-shot extreme ultraviolet laser imaging of nanostructures with wavelength resolution,” Opt. Lett. 33(5), 518–520 (2008).
[Crossref] [PubMed]

G. Vaschenko, C. Brewer, F. Brizuela, Y. Wang, M. A. Larotonda, B. M. Luther, M. C. Marconi, J. J. Rocca, C. S. Menoni, E. H. Anderson, W. Chao, B. D. Harteneck, J. A. Liddle, Y. Liu, and D. T. Attwood, “Sub-38 nm resolution tabletop microscopy with 13 nm wavelength laser light,” Opt. Lett. 31(9), 1214–1216 (2006).
[Crossref] [PubMed]

E. H. Anderson, “Specialized electron beam nanolithography for EUV and X-ray diffractive optics,” IEEE J. Quantum Electron. 42(1), 27–35 (2006).
[Crossref]

G. Vaschenko, F. Brizuela, C. Brewer, M. Grisham, H. Mancini, C. S. Menoni, M. C. Marconi, J. J. Rocca, W. Chao, J. A. Liddle, E. H. Anderson, D. T. Attwood, A. V. Vinogradov, I. A. Artioukov, Y. P. Pershyn, and V. V. Kondratenko, “Nanoimaging with a compact extreme-ultraviolet laser,” Opt. Lett. 30(16), 2095–2097 (2005).
[Crossref] [PubMed]

Artioukov, I.

Artioukov, I. A.

Artyukov, I. A.

Attwood, D. T.

Baraniuk, R.

H. Choi and R. Baraniuk, “Multiple Wavelet Basis Image Denoising Using Besov Ball Projections,” IEEE Signal Process. Lett. 11(9), 717–720 (2004).
[Crossref]

Baraniuk, R. G.

R. L. Claypoole, G. M. Davis, W. Sweldens, and R. G. Baraniuk, “Nonlinear wavelet transforms for image coding via lifting,” IEEE Trans. Image Process. 12(12), 1449–1459 (2003).
[Crossref] [PubMed]

S. P. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved Wavelet Denoising via Empirical Wiener Filtering,” in Proceedings of SPIE, Mathematical Imaging, San Diego, (1997).
[Crossref]

Bertilson, M.

Blu, T.

F. Luisier, T. Blu, and M. Unser, “A New SURE Approach to Image Denoising: Interscale Orthonormal Wavelet Thresholding,” IEEE Trans. Image Process. 16(3), 593–606 (2007).
[Crossref] [PubMed]

Boutet de Monvel, J.

H. Stollberg, J. Boutet de Monvel, A. Holmberg, and H. M. Hertz, “Wavelet-based image restoration for compact X-ray microscopy,” J. Microsc. 211(2), 154–160 (2003).
[Crossref] [PubMed]

Brewer, C.

Brewer, C. A.

Brizuela, F.

Buchanan, K. S.

Carbajo, S.

Chao, W.

S. Carbajo, I. D. Howlett, F. Brizuela, K. S. Buchanan, M. C. Marconi, W. Chao, E. H. Anderson, I. Artioukov, A. Vinogradov, J. J. Rocca, and C. S. Menoni, “Sequential single-shot imaging of nanoscale dynamic interactions with a table-top soft x-ray laser,” Opt. Lett. 37(14), 2994–2996 (2012).
[Crossref] [PubMed]

W. Chao, P. Fischer, T. Tyliszczak, S. Rekawa, E. Anderson, and P. Naulleau, “Real space soft x-ray imaging at 10 nm spatial resolution,” Opt. Express 20(9), 9777–9783 (2012).
[Crossref] [PubMed]

C. A. Brewer, F. Brizuela, P. Wachulak, D. H. Martz, W. Chao, E. H. Anderson, D. T. Attwood, A. V. Vinogradov, I. A. Artyukov, A. G. Ponomareko, V. V. Kondratenko, M. C. Marconi, J. J. Rocca, and C. S. Menoni, “Single-shot extreme ultraviolet laser imaging of nanostructures with wavelength resolution,” Opt. Lett. 33(5), 518–520 (2008).
[Crossref] [PubMed]

G. Vaschenko, C. Brewer, F. Brizuela, Y. Wang, M. A. Larotonda, B. M. Luther, M. C. Marconi, J. J. Rocca, C. S. Menoni, E. H. Anderson, W. Chao, B. D. Harteneck, J. A. Liddle, Y. Liu, and D. T. Attwood, “Sub-38 nm resolution tabletop microscopy with 13 nm wavelength laser light,” Opt. Lett. 31(9), 1214–1216 (2006).
[Crossref] [PubMed]

G. Vaschenko, F. Brizuela, C. Brewer, M. Grisham, H. Mancini, C. S. Menoni, M. C. Marconi, J. J. Rocca, W. Chao, J. A. Liddle, E. H. Anderson, D. T. Attwood, A. V. Vinogradov, I. A. Artioukov, Y. P. Pershyn, and V. V. Kondratenko, “Nanoimaging with a compact extreme-ultraviolet laser,” Opt. Lett. 30(16), 2095–2097 (2005).
[Crossref] [PubMed]

Choi, H.

H. Choi and R. Baraniuk, “Multiple Wavelet Basis Image Denoising Using Besov Ball Projections,” IEEE Signal Process. Lett. 11(9), 717–720 (2004).
[Crossref]

Claypoole, R. L.

R. L. Claypoole, G. M. Davis, W. Sweldens, and R. G. Baraniuk, “Nonlinear wavelet transforms for image coding via lifting,” IEEE Trans. Image Process. 12(12), 1449–1459 (2003).
[Crossref] [PubMed]

Davis, G. M.

R. L. Claypoole, G. M. Davis, W. Sweldens, and R. G. Baraniuk, “Nonlinear wavelet transforms for image coding via lifting,” IEEE Trans. Image Process. 12(12), 1449–1459 (2003).
[Crossref] [PubMed]

Donoho, D. L.

D. L. Donoho, “De-noising by soft thresholding,” IEEE Trans. Inf. Theory 41(3), 613–627 (1995).
[Crossref]

D. L. Donoho and I. M. Johnstone, “Ideal adaptation via wavelet shrinkage,” Biometrika 81(3), 425–455 (1994).
[Crossref]

Egiazarian, K.

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[Crossref] [PubMed]

Etkin, L. D.

D. Y. Parkinson, G. McDermott, L. D. Etkin, M. A. Le Gros, and C. A. Larabell, “Quantitative 3-D imaging of eukaryotic cells using soft X-ray tomography,” J. Struct. Biol. 162(3), 380–386 (2008).
[Crossref] [PubMed]

Fischer, P.

Foi, A.

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[Crossref] [PubMed]

Ghael, S. P.

S. P. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved Wavelet Denoising via Empirical Wiener Filtering,” in Proceedings of SPIE, Mathematical Imaging, San Diego, (1997).
[Crossref]

Grisham, M.

Gu, M.

Guo, X. J.

Harteneck, B. D.

Hertz, H. M.

O. von Hofsten, M. Bertilson, J. Reinspach, A. Holmberg, H. M. Hertz, and U. Vogt, “Sub-25-nm laboratory x-ray microscopy using a compound Fresnel zone plate,” Opt. Lett. 34(17), 2631–2633 (2009).
[Crossref] [PubMed]

H. Stollberg, J. Boutet de Monvel, A. Holmberg, and H. M. Hertz, “Wavelet-based image restoration for compact X-ray microscopy,” J. Microsc. 211(2), 154–160 (2003).
[Crossref] [PubMed]

Holmberg, A.

O. von Hofsten, M. Bertilson, J. Reinspach, A. Holmberg, H. M. Hertz, and U. Vogt, “Sub-25-nm laboratory x-ray microscopy using a compound Fresnel zone plate,” Opt. Lett. 34(17), 2631–2633 (2009).
[Crossref] [PubMed]

H. Stollberg, J. Boutet de Monvel, A. Holmberg, and H. M. Hertz, “Wavelet-based image restoration for compact X-ray microscopy,” J. Microsc. 211(2), 154–160 (2003).
[Crossref] [PubMed]

Howlett, I. D.

Huang, S. M.

Johnstone, I. M.

D. L. Donoho and I. M. Johnstone, “Ideal adaptation via wavelet shrinkage,” Biometrika 81(3), 425–455 (1994).
[Crossref]

Katkovnik, V.

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[Crossref] [PubMed]

Kondratenko, V. V.

Larabell, C. A.

D. Y. Parkinson, G. McDermott, L. D. Etkin, M. A. Le Gros, and C. A. Larabell, “Quantitative 3-D imaging of eukaryotic cells using soft X-ray tomography,” J. Struct. Biol. 162(3), 380–386 (2008).
[Crossref] [PubMed]

Larotonda, M. A.

Le Gros, M. A.

D. Y. Parkinson, G. McDermott, L. D. Etkin, M. A. Le Gros, and C. A. Larabell, “Quantitative 3-D imaging of eukaryotic cells using soft X-ray tomography,” J. Struct. Biol. 162(3), 380–386 (2008).
[Crossref] [PubMed]

Lee, K. H.

Liddle, J. A.

Liu, B.

Liu, X. L.

Liu, Y.

Luisier, F.

F. Luisier, T. Blu, and M. Unser, “A New SURE Approach to Image Denoising: Interscale Orthonormal Wavelet Thresholding,” IEEE Trans. Image Process. 16(3), 593–606 (2007).
[Crossref] [PubMed]

Luther, B. M.

Mancini, H.

Marconi, M. C.

Martz, D. H.

McDermott, G.

D. Y. Parkinson, G. McDermott, L. D. Etkin, M. A. Le Gros, and C. A. Larabell, “Quantitative 3-D imaging of eukaryotic cells using soft X-ray tomography,” J. Struct. Biol. 162(3), 380–386 (2008).
[Crossref] [PubMed]

Menoni, C. S.

Nam, C. H.

Naulleau, P.

Ni, C.

Park, S. B.

Parkinson, D. Y.

D. Y. Parkinson, G. McDermott, L. D. Etkin, M. A. Le Gros, and C. A. Larabell, “Quantitative 3-D imaging of eukaryotic cells using soft X-ray tomography,” J. Struct. Biol. 162(3), 380–386 (2008).
[Crossref] [PubMed]

Pershyn, Y. P.

Ponomareko, A. G.

Portilla, J.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image Denoising using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Trans. Image Process. 12(11), 1338–1351 (2003).
[Crossref] [PubMed]

Reinspach, J.

Rekawa, S.

Rocca, J. J.

Sayed, A. M.

S. P. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved Wavelet Denoising via Empirical Wiener Filtering,” in Proceedings of SPIE, Mathematical Imaging, San Diego, (1997).
[Crossref]

Seršic, D.

M. Tomić and D. Seršić, “Adaptive edge-preserving denoising by point-wise wavelet basis selection,” IET Signal Processing 6(1), 1–7 (2012).
[Crossref]

M. Vrankić, D. Seršić, and V. Sučić, “Adaptive 2-D wavelet transform based on the lifting scheme with preserved vanishing moments,” IEEE Trans. Image Process. 19(8), 1987–2004 (2010).
[Crossref] [PubMed]

D. Seršić, “A realization of wavelet filter bank with adaptive filter parameters,” In Proc. EUSPICOTampere, Finland, pp. 1733–1736, (2000).

Simoncelli, E. P.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image Denoising using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Trans. Image Process. 12(11), 1338–1351 (2003).
[Crossref] [PubMed]

Singhal, H.

Stollberg, H.

H. Stollberg, J. Boutet de Monvel, A. Holmberg, and H. M. Hertz, “Wavelet-based image restoration for compact X-ray microscopy,” J. Microsc. 211(2), 154–160 (2003).
[Crossref] [PubMed]

Strela, V.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image Denoising using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Trans. Image Process. 12(11), 1338–1351 (2003).
[Crossref] [PubMed]

Sucic, V.

M. Vrankić, D. Seršić, and V. Sučić, “Adaptive 2-D wavelet transform based on the lifting scheme with preserved vanishing moments,” IEEE Trans. Image Process. 19(8), 1987–2004 (2010).
[Crossref] [PubMed]

Sweldens, W.

R. L. Claypoole, G. M. Davis, W. Sweldens, and R. G. Baraniuk, “Nonlinear wavelet transforms for image coding via lifting,” IEEE Trans. Image Process. 12(12), 1449–1459 (2003).
[Crossref] [PubMed]

Tomic, M.

M. Tomić and D. Seršić, “Adaptive edge-preserving denoising by point-wise wavelet basis selection,” IET Signal Processing 6(1), 1–7 (2012).
[Crossref]

Tyliszczak, T.

Unser, M.

F. Luisier, T. Blu, and M. Unser, “A New SURE Approach to Image Denoising: Interscale Orthonormal Wavelet Thresholding,” IEEE Trans. Image Process. 16(3), 593–606 (2007).
[Crossref] [PubMed]

Vaschenko, G.

Vinogradov, A.

Vinogradov, A. V.

Vogt, U.

von Hofsten, O.

Vrankic, M.

M. Vrankić, D. Seršić, and V. Sučić, “Adaptive 2-D wavelet transform based on the lifting scheme with preserved vanishing moments,” IEEE Trans. Image Process. 19(8), 1987–2004 (2010).
[Crossref] [PubMed]

Wachulak, P.

Wainwright, M. J.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image Denoising using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Trans. Image Process. 12(11), 1338–1351 (2003).
[Crossref] [PubMed]

Wang, Y.

Biometrika (1)

D. L. Donoho and I. M. Johnstone, “Ideal adaptation via wavelet shrinkage,” Biometrika 81(3), 425–455 (1994).
[Crossref]

IEEE J. Quantum Electron. (1)

E. H. Anderson, “Specialized electron beam nanolithography for EUV and X-ray diffractive optics,” IEEE J. Quantum Electron. 42(1), 27–35 (2006).
[Crossref]

IEEE Signal Process. Lett. (1)

H. Choi and R. Baraniuk, “Multiple Wavelet Basis Image Denoising Using Besov Ball Projections,” IEEE Signal Process. Lett. 11(9), 717–720 (2004).
[Crossref]

IEEE Trans. Image Process. (5)

R. L. Claypoole, G. M. Davis, W. Sweldens, and R. G. Baraniuk, “Nonlinear wavelet transforms for image coding via lifting,” IEEE Trans. Image Process. 12(12), 1449–1459 (2003).
[Crossref] [PubMed]

M. Vrankić, D. Seršić, and V. Sučić, “Adaptive 2-D wavelet transform based on the lifting scheme with preserved vanishing moments,” IEEE Trans. Image Process. 19(8), 1987–2004 (2010).
[Crossref] [PubMed]

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image Denoising using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Trans. Image Process. 12(11), 1338–1351 (2003).
[Crossref] [PubMed]

F. Luisier, T. Blu, and M. Unser, “A New SURE Approach to Image Denoising: Interscale Orthonormal Wavelet Thresholding,” IEEE Trans. Image Process. 16(3), 593–606 (2007).
[Crossref] [PubMed]

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[Crossref] [PubMed]

IEEE Trans. Inf. Theory (1)

D. L. Donoho, “De-noising by soft thresholding,” IEEE Trans. Inf. Theory 41(3), 613–627 (1995).
[Crossref]

IET Signal Processing (1)

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[Crossref]

J. Microsc. (1)

H. Stollberg, J. Boutet de Monvel, A. Holmberg, and H. M. Hertz, “Wavelet-based image restoration for compact X-ray microscopy,” J. Microsc. 211(2), 154–160 (2003).
[Crossref] [PubMed]

J. Struct. Biol. (1)

D. Y. Parkinson, G. McDermott, L. D. Etkin, M. A. Le Gros, and C. A. Larabell, “Quantitative 3-D imaging of eukaryotic cells using soft X-ray tomography,” J. Struct. Biol. 162(3), 380–386 (2008).
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Opt. Express (2)

Opt. Lett. (6)

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G. Vaschenko, C. Brewer, F. Brizuela, Y. Wang, M. A. Larotonda, B. M. Luther, M. C. Marconi, J. J. Rocca, C. S. Menoni, E. H. Anderson, W. Chao, B. D. Harteneck, J. A. Liddle, Y. Liu, and D. T. Attwood, “Sub-38 nm resolution tabletop microscopy with 13 nm wavelength laser light,” Opt. Lett. 31(9), 1214–1216 (2006).
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[Crossref]

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[Crossref]

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Figures (10)

Fig. 1
Fig. 1

Two images of the same object, yinp1 and yinp2. Salt & peppers background noise is clearly visible, exhibiting the same spatial distribution in both images. The object represents a wheel test pattern. The width of the features in the inner ring is 100 nm.

Fig. 2
Fig. 2

From left to right: the background corrected images (y1 and y2), and its average y. The deterministic noise pattern is canceled out by subtraction, and the residual noise is reduced by preprocessing and averaging.

Fig. 3
Fig. 3

Left: sample estimate of the total noise level yd. Center: resulting total noise level model σ ^ w (m,n) built from two images difference. Right: total noise level estimated from a single image σ ^ w1 (m,n) .

Fig. 4
Fig. 4

Left: wavelet based background noise model σ wb0 . Right: polynomial background noise model σ wb .

Fig. 5
Fig. 5

Left: PDF estimate of the background noise level. Right: PDF estimate of the resulting pure input noise level.

Fig. 6
Fig. 6

Left: wavelet denoised image using fixed threshold. Right: wavelet denoised image using adaptive thresholds.

Fig. 7
Fig. 7

Proposed wavelet denoising method using adaptive thresholds. WT denotes wavelet transform, WT−1 denotes its inverse, and indices correspond to choice of wavelets.

Fig. 8
Fig. 8

Proposed variant of empirical Wiener denoising in the wavelet domain.

Fig. 9
Fig. 9

Denoised image using the proposed empirical Wiener denoising in the wavelet domain.

Fig. 10
Fig. 10

Synthetic test image analysis. From left to right: noisy image, exact noise levels (Gaussian + Poisson), estimated noise levels, restored image using Wiener filtering in the wavelet domain.

Tables (2)

Tables Icon

Table 1 Residual noise levels and noise gains (in brackets) for different denoising methods and two residual noise estimators on the real image example*

Tables Icon

Table 2 Signal to noise ratios (SNR), residual noise levels and noise gains (in brackets) for different denoising methods and two residual noise estimators on the synthetic image example*

Equations (14)

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

y(m,n)=x(m,n)+w(m,n),
y= y 1 + y 2 2 ,
z(m,n)= j=1,...J [ h j (m,n) H m,n,j + v j (m,n) V m,n,j + d j (m,n) D m,n,j ] + a j (m,n) A m,n,j ,
T=K σ N ,
t j (m,n)=| ψ j (m,n) |T, ψ ^ j (m,n)=sign[ ψ j (m,n) ]( t j (m,n)+| t j (m,n) | )/2,
σ ^ wb = p 5 (M×M)+ p 4 (M×N)+ p 3 (N×N)+ p 2 M+ p 1 N+ p 0 1
Φ T Φ P * = Φ T σ wb0 ,
σ ^ w1 (m,n)= medFilt 15×15 ( | d 1 (m,n) | )/0.6745.
T(m,n)=K σ w (m,n),
t j (m,n)=| ψ j (m,n) |T(m,n), ψ ^ j (m,n)=sign[ ψ j (m,n) ]( t j (m,n)+| t j (m,n) | )/2,
ψ ^ j (m,n)= | ψ pj (m,n) | 2 /( | ψ pj (m,n) | 2 + σ N 2 ) ψ j (m,n),
ψ ^ j (m,n)= | ψ pj (m,n) | 2 /( | ψ pj (m,n) | 2 + σ w 2 (m,n) ) ψ j (m,n),
d ^ m =mean(| y ^ 1 y ^ 2 |)/1.131,
d m =mean(| y 1 y 2 |)/1.131, g e =20 log 10 ( d m / d ^ m ).

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