Abstract

Many denoising approaches extend image processing to a hyperspectral cube structure, but do not take into account a sensor model nor the format of the recording. We propose a denoising framework for hyperspectral images that uses sensor data to convert an acquisition to a representation facilitating the noise-estimation, namely the photon-corrected image. This photon corrected image format accounts for the most common noise contributions and is spatially proportional to spectral radiance values. The subsequent denoising is based on an extended variational denoising model, which is suited for a Poisson distributed noise. A spatially and spectrally adaptive total variation regularisation term accounts the structural proposition of a hyperspectral image cube. We evaluate the approach on a synthetic dataset that guarantees a noise-free ground truth, and the best results are achieved when the dark current is taken into account.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
  3. H. Othman and S.-E. Qian, “Noise reduction of hyperspectral imagery using hybrid spatial-spectral derivative-domain wavelet shrinkage,” IEEE Trans. Geosci. Remote Sens. 44, 397–408 (2006).
    [Crossref]
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    [Crossref]
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    [Crossref]
  6. Q. Yuan, L. Zhang, and H. Shen, “Hyperspectral image denoising employing a spectral spatial adaptive total variation model,” IEEE Trans. Geosci. Remote Sens. 50, 3660–3677 (2012).
    [Crossref]
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  8. X. Gong, B. Lai, and Z. Xiang, “A L0 sparse analysis prior for blind poissonian image deconvolution,” Opt. Express 22, 370–375 (2014).
    [Crossref]
  9. F. Deger, A. Mansouri, M. Pedersen, J. Y. Hardeberg, and Y. Voisin, “A variational approach for denoising hyperspectral images corrupted by Poisson distributed noise,” in Image Signal Process (Springer, 2014), pp. 106–114.
    [Crossref]
  10. H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote Sens. 52, 4729–4743 (2014).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]

2014 (2)

X. Gong, B. Lai, and Z. Xiang, “A L0 sparse analysis prior for blind poissonian image deconvolution,” Opt. Express 22, 370–375 (2014).
[Crossref]

H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote Sens. 52, 4729–4743 (2014).
[Crossref]

2012 (3)

Q. Yuan, L. Zhang, and H. Shen, “Hyperspectral image denoising employing a spectral spatial adaptive total variation model,” IEEE Trans. Geosci. Remote Sens. 50, 3660–3677 (2012).
[Crossref]

X. Liu, S. Bourennane, and C. Fossati, “Denoising of hyperspectral images using the PARAFAC model and statistical performance analysis,” IEEE Trans. Geosci. Remote Sens. 50, 3717–3724 (2012).
[Crossref]

T. Skauli, “An upper-bound metric for characterizing spectral and spatial coregistration errors in spectral imaging,” Opt. Express 20, 918–933 (2012).
[Crossref] [PubMed]

2011 (2)

T. Skauli, “Sensor noise informed representation of hyperspectral data, with benefits for image storage and processing,” Opt. Express 19, 13031–13046 (2011).
[Crossref] [PubMed]

H. Li and L. Zhang, “A hybrid automatic endmember extraction algorithm based on a local window,” IEEE Trans. Geosci. Remote Sens. 49, 4223–4238 (2011).
[Crossref]

2009 (2)

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25, 1–24 (2009).
[Crossref]

S. Osher and T. Goldstein, “The Split Bregman method for L1 regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[Crossref]

2008 (1)

D. Letexier and S. Bourennane, “Noise removal from hyperspectral images by multidimensional filtering,” IEEE Trans. Geosci. Remote Sens. 46, 2061–2069 (2008).
[Crossref]

2007 (2)

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

J. Martín-Herrero, “Anisotropic diffusion in the hypercube,” IEEE Trans. Geosci. Remote Sens. 45, 1386–1398 (2007).
[Crossref]

2006 (1)

H. Othman and S.-E. Qian, “Noise reduction of hyperspectral imagery using hybrid spatial-spectral derivative-domain wavelet shrinkage,” IEEE Trans. Geosci. Remote Sens. 44, 397–408 (2006).
[Crossref]

2004 (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

1997 (1)

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Asaki, T. J.

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

Bertero, M.

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25, 1–24 (2009).
[Crossref]

Boccacci, P.

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25, 1–24 (2009).
[Crossref]

Boreman, G. D.

E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996).

Bourennane, S.

X. Liu, S. Bourennane, and C. Fossati, “Denoising of hyperspectral images using the PARAFAC model and statistical performance analysis,” IEEE Trans. Geosci. Remote Sens. 50, 3717–3724 (2012).
[Crossref]

D. Letexier and S. Bourennane, “Noise removal from hyperspectral images by multidimensional filtering,” IEEE Trans. Geosci. Remote Sens. 46, 2061–2069 (2008).
[Crossref]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Chartrand, R.

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

Deger, F.

F. Deger, A. Mansouri, M. Pedersen, J. Y. Hardeberg, and Y. Voisin, “A variational approach for denoising hyperspectral images corrupted by Poisson distributed noise,” in Image Signal Process (Springer, 2014), pp. 106–114.
[Crossref]

Dereniak, E. L.

E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996).

Fairchild, M. D.

M. D. Fairchild and G. M. Johnson, “Metacow: a public-domain, high-extended-dynamic-range, spectral test target for imaging system analysis and simulation,” in “Color Imaging Conf.”, (IS&T, 2004), pp. 239–245.

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Fossati, C.

X. Liu, S. Bourennane, and C. Fossati, “Denoising of hyperspectral images using the PARAFAC model and statistical performance analysis,” IEEE Trans. Geosci. Remote Sens. 50, 3717–3724 (2012).
[Crossref]

García-Beltran, A.

George, S.

R. Shrestha, R. Pillay, S. George, and J. Y. Hardeberg, “Quality evaluation in spectral imaging–quality factors and metrics,” JAIC-Journal of the International Colour Association12 (2014).

Getreuer, P.

P. Getreuer, “Rudin-Osher-Fatemi total variation denoising using Split Bregman,” Image Process. Line (2012).

Goldstein, T.

S. Osher and T. Goldstein, “The Split Bregman method for L1 regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[Crossref]

Gong, X.

X. Gong, B. Lai, and Z. Xiang, “A L0 sparse analysis prior for blind poissonian image deconvolution,” Opt. Express 22, 370–375 (2014).
[Crossref]

Hardeberg, J. Y.

F. Deger, A. Mansouri, M. Pedersen, J. Y. Hardeberg, and Y. Voisin, “A variational approach for denoising hyperspectral images corrupted by Poisson distributed noise,” in Image Signal Process (Springer, 2014), pp. 106–114.
[Crossref]

R. Shrestha, R. Pillay, S. George, and J. Y. Hardeberg, “Quality evaluation in spectral imaging–quality factors and metrics,” JAIC-Journal of the International Colour Association12 (2014).

He, W.

H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote Sens. 52, 4729–4743 (2014).
[Crossref]

Hernández-Andrés, J.

Johnson, G. M.

M. D. Fairchild and G. M. Johnson, “Metacow: a public-domain, high-extended-dynamic-range, spectral test target for imaging system analysis and simulation,” in “Color Imaging Conf.”, (IS&T, 2004), pp. 239–245.

Lai, B.

X. Gong, B. Lai, and Z. Xiang, “A L0 sparse analysis prior for blind poissonian image deconvolution,” Opt. Express 22, 370–375 (2014).
[Crossref]

Le, T.

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

Letexier, D.

D. Letexier and S. Bourennane, “Noise removal from hyperspectral images by multidimensional filtering,” IEEE Trans. Geosci. Remote Sens. 46, 2061–2069 (2008).
[Crossref]

Li, H.

H. Li and L. Zhang, “A hybrid automatic endmember extraction algorithm based on a local window,” IEEE Trans. Geosci. Remote Sens. 49, 4223–4238 (2011).
[Crossref]

Liu, X.

X. Liu, S. Bourennane, and C. Fossati, “Denoising of hyperspectral images using the PARAFAC model and statistical performance analysis,” IEEE Trans. Geosci. Remote Sens. 50, 3717–3724 (2012).
[Crossref]

Mansouri, A.

F. Deger, A. Mansouri, M. Pedersen, J. Y. Hardeberg, and Y. Voisin, “A variational approach for denoising hyperspectral images corrupted by Poisson distributed noise,” in Image Signal Process (Springer, 2014), pp. 106–114.
[Crossref]

Martín-Herrero, J.

J. Martín-Herrero, “Anisotropic diffusion in the hypercube,” IEEE Trans. Geosci. Remote Sens. 45, 1386–1398 (2007).
[Crossref]

Osher, S.

S. Osher and T. Goldstein, “The Split Bregman method for L1 regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[Crossref]

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Othman, H.

H. Othman and S.-E. Qian, “Noise reduction of hyperspectral imagery using hybrid spatial-spectral derivative-domain wavelet shrinkage,” IEEE Trans. Geosci. Remote Sens. 44, 397–408 (2006).
[Crossref]

Pedersen, M.

F. Deger, A. Mansouri, M. Pedersen, J. Y. Hardeberg, and Y. Voisin, “A variational approach for denoising hyperspectral images corrupted by Poisson distributed noise,” in Image Signal Process (Springer, 2014), pp. 106–114.
[Crossref]

Pillay, R.

R. Shrestha, R. Pillay, S. George, and J. Y. Hardeberg, “Quality evaluation in spectral imaging–quality factors and metrics,” JAIC-Journal of the International Colour Association12 (2014).

Qian, S.-E.

H. Othman and S.-E. Qian, “Noise reduction of hyperspectral imagery using hybrid spatial-spectral derivative-domain wavelet shrinkage,” IEEE Trans. Geosci. Remote Sens. 44, 397–408 (2006).
[Crossref]

Romero, J.

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Shen, H.

H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote Sens. 52, 4729–4743 (2014).
[Crossref]

Q. Yuan, L. Zhang, and H. Shen, “Hyperspectral image denoising employing a spectral spatial adaptive total variation model,” IEEE Trans. Geosci. Remote Sens. 50, 3660–3677 (2012).
[Crossref]

Shrestha, R.

R. Shrestha, R. Pillay, S. George, and J. Y. Hardeberg, “Quality evaluation in spectral imaging–quality factors and metrics,” JAIC-Journal of the International Colour Association12 (2014).

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Skauli, T.

Voisin, Y.

F. Deger, A. Mansouri, M. Pedersen, J. Y. Hardeberg, and Y. Voisin, “A variational approach for denoising hyperspectral images corrupted by Poisson distributed noise,” in Image Signal Process (Springer, 2014), pp. 106–114.
[Crossref]

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Xiang, Z.

X. Gong, B. Lai, and Z. Xiang, “A L0 sparse analysis prior for blind poissonian image deconvolution,” Opt. Express 22, 370–375 (2014).
[Crossref]

Yang, J.

J. Yang and Y. Zhao, “Poisson-Gaussian mixed noise removing for hyperspectral image via spatial-spectral structure similarity,” in “32nd Chinese Control Conf.” (Xi’an, 2013), pp. 3715–3720.

Yuan, Q.

H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote Sens. 52, 4729–4743 (2014).
[Crossref]

Q. Yuan, L. Zhang, and H. Shen, “Hyperspectral image denoising employing a spectral spatial adaptive total variation model,” IEEE Trans. Geosci. Remote Sens. 50, 3660–3677 (2012).
[Crossref]

Zanella, R.

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25, 1–24 (2009).
[Crossref]

Zanni, L.

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25, 1–24 (2009).
[Crossref]

Zhang, H.

H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote Sens. 52, 4729–4743 (2014).
[Crossref]

Zhang, L.

H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote Sens. 52, 4729–4743 (2014).
[Crossref]

Q. Yuan, L. Zhang, and H. Shen, “Hyperspectral image denoising employing a spectral spatial adaptive total variation model,” IEEE Trans. Geosci. Remote Sens. 50, 3660–3677 (2012).
[Crossref]

H. Li and L. Zhang, “A hybrid automatic endmember extraction algorithm based on a local window,” IEEE Trans. Geosci. Remote Sens. 49, 4223–4238 (2011).
[Crossref]

Zhao, Y.

J. Yang and Y. Zhao, “Poisson-Gaussian mixed noise removing for hyperspectral image via spatial-spectral structure similarity,” in “32nd Chinese Control Conf.” (Xi’an, 2013), pp. 3715–3720.

IEEE Trans. Geosci. Remote Sens. (7)

H. Li and L. Zhang, “A hybrid automatic endmember extraction algorithm based on a local window,” IEEE Trans. Geosci. Remote Sens. 49, 4223–4238 (2011).
[Crossref]

X. Liu, S. Bourennane, and C. Fossati, “Denoising of hyperspectral images using the PARAFAC model and statistical performance analysis,” IEEE Trans. Geosci. Remote Sens. 50, 3717–3724 (2012).
[Crossref]

H. Othman and S.-E. Qian, “Noise reduction of hyperspectral imagery using hybrid spatial-spectral derivative-domain wavelet shrinkage,” IEEE Trans. Geosci. Remote Sens. 44, 397–408 (2006).
[Crossref]

J. Martín-Herrero, “Anisotropic diffusion in the hypercube,” IEEE Trans. Geosci. Remote Sens. 45, 1386–1398 (2007).
[Crossref]

D. Letexier and S. Bourennane, “Noise removal from hyperspectral images by multidimensional filtering,” IEEE Trans. Geosci. Remote Sens. 46, 2061–2069 (2008).
[Crossref]

Q. Yuan, L. Zhang, and H. Shen, “Hyperspectral image denoising employing a spectral spatial adaptive total variation model,” IEEE Trans. Geosci. Remote Sens. 50, 3660–3677 (2012).
[Crossref]

H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote Sens. 52, 4729–4743 (2014).
[Crossref]

IEEE Trans. Image Process. (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Inverse Probl. (1)

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25, 1–24 (2009).
[Crossref]

J. Math. Imaging Vis. (1)

T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imaging Vis. 27, 257–263 (2007).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (3)

Phys. D (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[Crossref]

SIAM J. Imaging Sci. (1)

S. Osher and T. Goldstein, “The Split Bregman method for L1 regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[Crossref]

Other (8)

E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996).

J. Yang and Y. Zhao, “Poisson-Gaussian mixed noise removing for hyperspectral image via spatial-spectral structure similarity,” in “32nd Chinese Control Conf.” (Xi’an, 2013), pp. 3715–3720.

(HySpex / Norsk Elektro Optikk AS), “Imaging spectrometer (user manual),” Tech. Rep. (2013).

F. Deger, A. Mansouri, M. Pedersen, J. Y. Hardeberg, and Y. Voisin, “A variational approach for denoising hyperspectral images corrupted by Poisson distributed noise,” in Image Signal Process (Springer, 2014), pp. 106–114.
[Crossref]

M. D. Fairchild and G. M. Johnson, “Metacow: a public-domain, high-extended-dynamic-range, spectral test target for imaging system analysis and simulation,” in “Color Imaging Conf.”, (IS&T, 2004), pp. 239–245.

J. Padfield, “Library of illumination spectral power distributions,” http://research.ng-london.org.uk/scientific/spd/ . Accessed: 2014-12-10.

R. Shrestha, R. Pillay, S. George, and J. Y. Hardeberg, “Quality evaluation in spectral imaging–quality factors and metrics,” JAIC-Journal of the International Colour Association12 (2014).

P. Getreuer, “Rudin-Osher-Fatemi total variation denoising using Split Bregman,” Image Process. Line (2012).

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Figures (4)

Fig. 1
Fig. 1 Different stages of the proposed denoising framework for HSI. Knowledge of the sensor characteristics allows the conversion to a photon corrected image (presented in Section 3). This is a better representation to find an appropriate noise model and to estimate the corresponding parameters.
Fig. 2
Fig. 2 Spectral power distributions (SPDs) of the applied illuminants and radiance values of the synthetic dataset. a) CIE D65 and CIE A are standardized SPDs, while GE 4100K is measured by [21]. b) Spectral radiance values L of the synthetic data set [20]. The image width is 600 px and every cow is 100 px × 100 px. In total there are 24 cows in different colors. For this visualization, the bands 40, 30, 9 are assigned to the red, green and blue channel.
Fig. 3
Fig. 3 Parameters of a hyperspectral pushbroom scanner. a) Quantum efficiency includes effects of the optics and the photodetector. It is clearly visible that the spectral variation is much larger than the spatial non-linearities. b) The dark current Id[i, j]t is a constant offset of photoelectrons, and averaged over 200 measurements. It shows few outliers with higher values. The variance is Var(Id[i, j]t) = 6.5.
Fig. 4
Fig. 4 Band-wise evaluation of the PSNR results for different illuminants. Lc is the denoised result of the proposed photon corrected image format. a) Dataset generated with a CIE D65 illuminant. b) Dataset generated with a CIE A illuminant. c) Dataset generated with a GE 4100K illuminant.

Tables (3)

Tables Icon

Table 1 PSNR for different illuminants, intensities gl, applied to different HSI image formats. The grey columns correspond to Fig. 2(a) and Fig. 4, and bold values show the best result for each column. In a low-photon environment (gl = 0.5) the difference between Lc and Lcs is larger, and it is more important to take the dark current into account. In brighter environments (gl > 1) the denoising does not improve the results.

Tables Icon

Table 2 SSIM for different illuminants, intensities, applied to different HSI image formats. The results coincide with Table 1.

Tables Icon

Table 3 GFC [24] for the three illuminants. A GFC of 1 signifies a perfect reconstruction of the spectral feature. For each illuminant we denote the mean value and the minimum that describes the worst spectral feature reconstruction.

Equations (18)

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

N ph [ i , j ] = L [ i , j ] t A Ω Δ λ λ [ j ] h c ,
N [ i , j ] = η [ i , j ] N ph [ i , j ] + I d [ i , j ] t + δ N ,
D raw [ i , j ] = round ( g f N [ i , j ] ) ,
L n [ i , j ] = ( D raw g f I d [ i , j ] t ) h c η [ i , j ] t A Ω Δ λ λ .
η [ j ] = 1 M i = 1 M η [ i , j ] ,
f rc [ i , j ] = L n [ i , j ] k [ j ] 1 , k [ j ] = h c s dw η [ j ] t A Ω Δ λ [ j ] λ ,
f c [ i , j ] = f rc [ i , j ] s dw + I ¯ d t ,
f cs [ i , j ] = f rc [ i , j ] / s dw .
σ n [ i , j ] = f c [ i , j ] f cs [ i , j ] ,
u ^ = arg min u u TV ( H ) + β H ( u ( x ) f c ( x ) log u ( x ) ) d x ,
u SSATV = i M 1 M 2 W i G i ,
G i = j B ( x u ) i , j 2 + ( y u ) i , j 2 ,
W i = ( 1 + G i ) 1 1 M 1 M 2 k M 1 M 2 ( 1 + G k ) 1 .
u ^ = arg min u i M 1 M 2 W i G i + β i M 1 M 2 j B ( u i , j f i , j log u i , j ) .
arg min u u TV ( H ) subject to H ( u ( x ) f c ( x ) log u ( x ) ) d x = mean ( σ [ i , j ] 2 )
PSNR ( L , L ^ ) = 10 log 10 ( max ( L ) MSE ) , MSE = 1 M 1 M 2 B i M 1 M 2 B ( L i L ^ i ) 2 , RMSE = MSE .
SSIM ( L , L ^ ) = ( 2 μ L μ L ^ + c 1 ) ( 2 σ L , L ^ + c 2 ) ( μ L 2 + μ L ^ 2 + c 1 ) ( σ L 2 + σ L ^ 2 + c 2 ) ,
GFC ( p , p ^ ) = | j p j p ^ j | | j ( p j ) 2 | 1 / 2 | j ( p ^ j ) 2 | 1 / 2 ,

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