Abstract

Digital Radiography (DR) images obtained by OCD-based (optical coupling detector) Micro-CT system usually suffer from low contrast. In this paper, a mathematical model is proposed to describe the image formation process in scintillator. By solving the correlative inverse problem, the quality of DR images is improved, i.e. higher contrast and spatial resolution. By analyzing the radiative transfer process of visible light in scintillator, scattering is recognized as the main factor leading to low contrast. Moreover, involved blurring effect is also concerned and described as point spread function (PSF). Based on these physical processes, the scintillator imaging model is then established. When solving the inverse problem, pre-correction to the intensity of x-rays, dark channel prior based haze removing technique, and an effective blind deblurring approach are employed. Experiments on a variety of DR images show that the proposed approach could improve the contrast of DR images dramatically as well as eliminate the blurring vision effectively. Compared with traditional contrast enhancement methods, such as CLAHE, our method could preserve the relative absorption values well.

© 2015 Optical Society of America

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References

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2015 (1)

D. Chen, H. Li, Q. Wang, P. Zhang, and Y. Zhu, “Computed tomography for high-speed rotation object,” Optics Express 23, 13423–13442 (2015).
[Crossref] [PubMed]

2013 (2)

Y. Zhu, M. Zhao, H. Li, and P. Zhang, “Micro-CT artifacts reduction based on detector random shifting and fast data inpainting,” Medical physics 40, 031114 (2013).
[Crossref] [PubMed]

K. He, J. Sun, and X. Tang, “Guided Image Filtering,” Pattern Analysis and Machine Intelligence, IEEE Transactions on Pattern Analysis and Machine Intelligence, IEEE Transactions on 35, 1397–1409 (2013).
[Crossref] [PubMed]

2012 (2)

S. H. Williams, A. Hilger, N. Kardjilov, I. Manke, M. Strobl, P. A. Douissard, T. Martin, H. Riesemeier, and J. Banhart, “Detection system for microimaging with neutrons,” Journal of Instrumentation 7, P02014 (2012).
[Crossref]

Y. Zhu, M. Zhao, Y. Zhao, H. Li, and P. Zhang, “Noise reduction with low dose CT data based on a modified ROF model,” Opt. Express 20, 17987–18004 (2012).
[Crossref]

2009 (1)

T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM Journal on Imaging Sciences 2, 323–343 (2009).
[Crossref]

2008 (1)

A. Levin, D. Lischinski, and Y. Weiss, “A closed-form solution to natural image matting,” Pattern Analysis and Machine Intelligence, IEEE Transactions on 30, 228–242 (2008).
[Crossref]

2006 (1)

M. Nikl, “Scintillation detectors for X-rays,” Measurement Science and Technology 17, R37 (2006).
[Crossref]

2005 (1)

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Modeling & Simulation 4, 460–489 (2005).
[Crossref]

2003 (3)

E. Samei and M. J. Flynn, “An experimental comparison of detector performance for direct and indirect digital radiography systems,” Medical physics 30, 608–622 (2003).
[Crossref]

M. Jiang and G. Wang, “Development of blind image deconvolution and its applications,” Journal of X-ray Science and Technology 11, 13–19 (2003).
[PubMed]

M. Jiang, G. Wang, M. W. Skinner, J. T. Rubinstein, and M. W. Vannier, “Blind deblurring of spiral CT images,” Medical Imaging, IEEE Transactions on 22, 837–845 (2003).
[Crossref]

2002 (4)

C. W. Van Eijk, “Inorganic scintillators in medical imaging,” Physics in medicine and biology 47, R85–R106 (2002).
[Crossref]

M. Jiang, G. Wang, M. W. Skinner, J. T. Rubinstein, and M. W. Vannier, “Blind deblurring of spiral CT imagesl-comparative studies on edge-to-noise ratios,” Medical physics 29, 821–829 (2002).
[Crossref]

E. Kotter and M. Langer, “Digital radiography with large-area flat-panel detectors,” European radiology 12, 2562–2570 (2002).
[Crossref] [PubMed]

S. G. Narasimhan and S. K. Nayar, “Vision and the atmosphere,” International Journal of Computer Vision 48, 233–254 (2002).
[Crossref]

1998 (1)

A. Koch, C. Raven, P. Spanne, and A. Snigirev, “X-ray imaging with submicrometer resolution employing transparent luminescent screens,” JOSA A 15, 1940–1951 (1998).
[Crossref]

1997 (1)

M. J. Yaffe and J. Rowlands, “X-ray detectors for digital radiography,” Physics in Medicine and Biology 42, 1 (1997).
[Crossref] [PubMed]

1996 (3)

U. Bonse and F. Busch, “X-ray computed microtomography (μ CT) using synchrotron radiation (SR),” Progress in biophysics and molecular biology 65, 133–169 (1996).
[Crossref]

D. Kundur and D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Processing Magazine,  13, 61–63 (1996).
[Crossref]

D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Processing Magazine,  13, 43–64 (1996).
[Crossref]

1994 (2)

U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 μm spatial and 10−4 energy resolution,” Bone and mineral 25, 25–38 (1994).
[Crossref]

A. D. Maidment and M. J. Yaffe, “Analysis of the spatial-frequency-dependent DQE of optically coupled digital mammography detectors,” Medical physics 21, 721–729 (1994).
[Crossref]

1992 (1)

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

1991 (2)

U. Bonse, R. Nusshardt, F. Busch, R. Pahl, J. H. Kinney, Q. C. Johnson, R. A. Saroyan, and M. C. Nichols, “X-ray tomographic microscopy of fibre-reinforced materials,” Journal of materials science 26, 4076–4085 (1991).
[Crossref]

I. Csiszar, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” The annals of statistics 192032–2066 (1991).
[Crossref]

1987 (1)

S. M. Pizer, E. P. Amburn, J. D. Austin, R. Cromartie, A. Geselowitz, T. Greer, B. ter Haar Romeny, J. B. Zimmerman, and K. Zuiderveld, “Adaptive histogram equalization and its variations,” Computer vision, graphics, and image processing 39, 355–368 (1987).
[Crossref]

1975 (1)

J. K. Daugherty, R. C. Hartman, and P. J. Schmidt, “A measurement of cosmic-ray positron and negatron spectra between 50 and 800 MV,” The Astrophysical Journal 198, 493–505 (1975).
[Crossref]

1950 (1)

E. R. Howells, D. C. Phillips, and D. Rogers, “The probability distribution of X-ray intensities. ii. Experimental investigation and the X-ray detection of centres of symmetry,” Acta Crystallographica 3, 210–214 (1950).
[Crossref]

Amburn, E. P.

S. M. Pizer, E. P. Amburn, J. D. Austin, R. Cromartie, A. Geselowitz, T. Greer, B. ter Haar Romeny, J. B. Zimmerman, and K. Zuiderveld, “Adaptive histogram equalization and its variations,” Computer vision, graphics, and image processing 39, 355–368 (1987).
[Crossref]

Annenkov, A.

P. Lecoq, A. Annenkov, A. Gektin, M. Korzhik, and C. Pedrini, Inorganic Scintillators for Detector Systems: Physical Principles and Crystal Engineering (Springer, 2006).

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems (V.H.Winston and Sons, 1977).

Austin, J. D.

S. M. Pizer, E. P. Amburn, J. D. Austin, R. Cromartie, A. Geselowitz, T. Greer, B. ter Haar Romeny, J. B. Zimmerman, and K. Zuiderveld, “Adaptive histogram equalization and its variations,” Computer vision, graphics, and image processing 39, 355–368 (1987).
[Crossref]

Banhart, J.

S. H. Williams, A. Hilger, N. Kardjilov, I. Manke, M. Strobl, P. A. Douissard, T. Martin, H. Riesemeier, and J. Banhart, “Detection system for microimaging with neutrons,” Journal of Instrumentation 7, P02014 (2012).
[Crossref]

Beckmann, F.

U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 μm spatial and 10−4 energy resolution,” Bone and mineral 25, 25–38 (1994).
[Crossref]

Bonse, U.

U. Bonse and F. Busch, “X-ray computed microtomography (μ CT) using synchrotron radiation (SR),” Progress in biophysics and molecular biology 65, 133–169 (1996).
[Crossref]

U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 μm spatial and 10−4 energy resolution,” Bone and mineral 25, 25–38 (1994).
[Crossref]

U. Bonse, R. Nusshardt, F. Busch, R. Pahl, J. H. Kinney, Q. C. Johnson, R. A. Saroyan, and M. C. Nichols, “X-ray tomographic microscopy of fibre-reinforced materials,” Journal of materials science 26, 4076–4085 (1991).
[Crossref]

Burger, M.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Modeling & Simulation 4, 460–489 (2005).
[Crossref]

Busbridge, I. W.

I. W. Busbridge, The Mathematics of Radiative Transfer, 50 (University Press, 1960).

Busch, F.

U. Bonse and F. Busch, “X-ray computed microtomography (μ CT) using synchrotron radiation (SR),” Progress in biophysics and molecular biology 65, 133–169 (1996).
[Crossref]

U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 μm spatial and 10−4 energy resolution,” Bone and mineral 25, 25–38 (1994).
[Crossref]

U. Bonse, R. Nusshardt, F. Busch, R. Pahl, J. H. Kinney, Q. C. Johnson, R. A. Saroyan, and M. C. Nichols, “X-ray tomographic microscopy of fibre-reinforced materials,” Journal of materials science 26, 4076–4085 (1991).
[Crossref]

Campisi, P.

P. Campisi and K. Egiazarian, Blind Image Deconvolution: Theory and Applications (CRC press, 2007).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Courier Corporation, 2013).

Chen, D.

D. Chen, H. Li, Q. Wang, P. Zhang, and Y. Zhu, “Computed tomography for high-speed rotation object,” Optics Express 23, 13423–13442 (2015).
[Crossref] [PubMed]

Cromartie, R.

S. M. Pizer, E. P. Amburn, J. D. Austin, R. Cromartie, A. Geselowitz, T. Greer, B. ter Haar Romeny, J. B. Zimmerman, and K. Zuiderveld, “Adaptive histogram equalization and its variations,” Computer vision, graphics, and image processing 39, 355–368 (1987).
[Crossref]

Csiszar, I.

I. Csiszar, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” The annals of statistics 192032–2066 (1991).
[Crossref]

Daugherty, J. K.

J. K. Daugherty, R. C. Hartman, and P. J. Schmidt, “A measurement of cosmic-ray positron and negatron spectra between 50 and 800 MV,” The Astrophysical Journal 198, 493–505 (1975).
[Crossref]

Delling, G.

U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 μm spatial and 10−4 energy resolution,” Bone and mineral 25, 25–38 (1994).
[Crossref]

Douissard, P. A.

S. H. Williams, A. Hilger, N. Kardjilov, I. Manke, M. Strobl, P. A. Douissard, T. Martin, H. Riesemeier, and J. Banhart, “Detection system for microimaging with neutrons,” Journal of Instrumentation 7, P02014 (2012).
[Crossref]

Draper, N. R.

N. R. Draper and H. Smith, Applied Regression Analysis2nded. (John Wiley, 1981).

Egiazarian, K.

P. Campisi and K. Egiazarian, Blind Image Deconvolution: Theory and Applications (CRC press, 2007).
[Crossref]

Fatemi, E.

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

Fattal, R.

R. Fattal, “Single image dehazing,” in “ACM Transactions on Graphics (TOG),” (ACM, 2008), p. 72.

Flynn, M. J.

E. Samei and M. J. Flynn, “An experimental comparison of detector performance for direct and indirect digital radiography systems,” Medical physics 30, 608–622 (2003).
[Crossref]

Friedman, J.

T. Hastie, R. Tibshirani, J. Friedman, T. Hastie, J. Friedman, and R. Tibshirani, The Elements of Statistical Learning, vol. 2 (Springer, 2009). 1.
[Crossref]

T. Hastie, R. Tibshirani, J. Friedman, T. Hastie, J. Friedman, and R. Tibshirani, The Elements of Statistical Learning, vol. 2 (Springer, 2009). 1.
[Crossref]

Gektin, A.

P. Lecoq, A. Annenkov, A. Gektin, M. Korzhik, and C. Pedrini, Inorganic Scintillators for Detector Systems: Physical Principles and Crystal Engineering (Springer, 2006).

Geselowitz, A.

S. M. Pizer, E. P. Amburn, J. D. Austin, R. Cromartie, A. Geselowitz, T. Greer, B. ter Haar Romeny, J. B. Zimmerman, and K. Zuiderveld, “Adaptive histogram equalization and its variations,” Computer vision, graphics, and image processing 39, 355–368 (1987).
[Crossref]

Goldfarb, D.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Modeling & Simulation 4, 460–489 (2005).
[Crossref]

Goldstein, T.

T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM Journal on Imaging Sciences 2, 323–343 (2009).
[Crossref]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Imaging Processing (Addison-Wesley, Massachusetts1992).

Graeff, W.

U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 μm spatial and 10−4 energy resolution,” Bone and mineral 25, 25–38 (1994).
[Crossref]

Greer, T.

S. M. Pizer, E. P. Amburn, J. D. Austin, R. Cromartie, A. Geselowitz, T. Greer, B. ter Haar Romeny, J. B. Zimmerman, and K. Zuiderveld, “Adaptive histogram equalization and its variations,” Computer vision, graphics, and image processing 39, 355–368 (1987).
[Crossref]

Günnewig, O.

U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 μm spatial and 10−4 energy resolution,” Bone and mineral 25, 25–38 (1994).
[Crossref]

Hahn, M.

U. Bonse, F. Busch, O. Günnewig, F. Beckmann, R. Pahl, G. Delling, M. Hahn, and W. Graeff, “3D computed X-ray tomography of human cancellous bone at 8 μm spatial and 10−4 energy resolution,” Bone and mineral 25, 25–38 (1994).
[Crossref]

Hansen, P. C.

P. C. Hansen, J. G. Nagy, and D. P. O’leary, Deblurring Images: Matrices, Spectra, and Filtering, vol. 3 (SIAM, 2006).
[Crossref]

Hartman, R. C.

J. K. Daugherty, R. C. Hartman, and P. J. Schmidt, “A measurement of cosmic-ray positron and negatron spectra between 50 and 800 MV,” The Astrophysical Journal 198, 493–505 (1975).
[Crossref]

Hastie, T.

T. Hastie, R. Tibshirani, J. Friedman, T. Hastie, J. Friedman, and R. Tibshirani, The Elements of Statistical Learning, vol. 2 (Springer, 2009). 1.
[Crossref]

T. Hastie, R. Tibshirani, J. Friedman, T. Hastie, J. Friedman, and R. Tibshirani, The Elements of Statistical Learning, vol. 2 (Springer, 2009). 1.
[Crossref]

Hatzinakos, D.

D. Kundur and D. Hatzinakos, “Blind image deconvolution revisited,” IEEE Signal Processing Magazine,  13, 61–63 (1996).
[Crossref]

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Figures (15)

Fig. 1
Fig. 1 Imaging systems of Micro-CT: (a) FPD-based system schematic; (b) OCD-based system schematic; (c) Real OCD-based system (model: nanoVoxel-2700; manufacture: Sanying Precision Instrucments Ltd.).
Fig. 2
Fig. 2 The optical magnifying process of OCD-based system.
Fig. 3
Fig. 3 Illustration of the multiple light sources model. A local box of the scintillator is magnified and shown in the yellow wireframe.
Fig. 4
Fig. 4 Illustration of streaming analyses. The difference between in-streaming and out-streaming is shown on the left, which decreases with the increase of the pointolite-element distance. As is shown on the right, the elements of the scintillator can be seperated into two cases: (A) interior element; (B) boundary element.
Fig. 5
Fig. 5 Illustration of the imaging model in a single x-ray path. Depending on the multiple light sources model shown in Fig. 3, the elements are uniformly distributed. Along a single x-ray path, the average emitted distribution function of each element is equal.
Fig. 6
Fig. 6 Illustration of haze imaging model. The light received by camera consists of the reflected light from the scene point and the ambient light. Particularly, the reflected light is attenuated along the light path.
Fig. 7
Fig. 7 The DR image of original specific intensity and its pre-correction fitting surfaces: (a) the image of original specific intensity; fitting surface by using (b) Gaussian function and (c) quintic polynomial.
Fig. 8
Fig. 8 Ratio images between Fig. 7(a) and its fitting surfaces by using (a) Gaussian function and (b) quintic polynomial.
Fig. 9
Fig. 9 Difference images between Fig. 7(a) and its fitting surfaces by using (a) Gaussian function and (b) quintic polynomial.
Fig. 10
Fig. 10 Compared illustrations of pre-correction: (a) original spider DR image; (b) ideal result in standard flat field; (c) pre-correction result by using quintic polynomial. Correlative histograms are shown in (d)–(f).
Fig. 11
Fig. 11 Typical samples: shale (on the left) and bamboo stick (on the right).
Fig. 12
Fig. 12 Enhanced results of Fig. 10(a) by using (a) blind deblurring; (b) the combination of blind deblurring and piecewise-linear intensity level transformation; (c) CLAHE; (d) the proposed method. The zoom-in patches of (a)–(d) (inside the red box) are illustrated in (e)–(h). The correlative histograms of (a)–(d) are illustrated in (i)–(l).
Fig. 13
Fig. 13 Profiles of enhanced results: (a) is spider DR image (marked by green); (b) is beetle DR image (marked by green).
Fig. 14
Fig. 14 Real DR image and its enhanced results: (a) original beetle DR image; enhanced results by using (b) piecewise-linear intensity level transformation; (c) the combination of blind deblurring and piecewise-linear intensity level transformation; (d) CLAHE; (e) the proposed method without blind deblurring process; (f) the proposed method. The correlative histograms of (a)–(f) are illustrated in (g)–(l).
Fig. 15
Fig. 15 The zoom-in patches of Fig. 14(a)–(f) (inside the red box): (a)–(f) is the first group; (g)–(l) is the second group.

Tables (3)

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Table 1 Symbol definition for Eq. (1)

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Table 2 the physical process in a volume element

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Algorithm 1 The Algorithm of Solving Model (24)

Equations (40)

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1 c I λ ( r , v , t ) t = v · I λ ( r , v , t ) b abs ( λ , r ) I λ ( r , v , t ) b sca ( λ , r ) I λ ( r , v , t ) + b sca ( λ , r ) P ( λ , r , u , v ) 4 π I λ ( r , u , t ) d u + S λ ( r , v , t ) ,
Δ V = Δ x Δ y Δ z Δ μ Δ ϕ ,
d r = d x d y d z , d v = sin θ d θ d ϕ = d μ d ϕ .
ξ λ ( r , v , t ) t Δ V = v ξ λ ( r , v , t ) Δ V b abs ( λ , r ) ξ λ ( r , v , t ) Δ V b sca ( λ , r ) ξ λ ( r , v , t ) Δ V + b sca ( λ , r ) P ( λ , r , u , v ) 4 π ξ λ ( r , u , t ) d u Δ V + s λ ( r , v , t ) Δ V ,
ξ λ ( r , v , t ) = I λ ( r , v , t ) λ h c 2 ,
S λ ( r , v , t ) = h c λ s λ ( r , v , t ) ,
ξ λ ( r , v , t ) t Δ V = b sca ( λ , r ) s λ ( r , v , t ) Δ V + b sca ( λ , r ) a λ ¯ ( t ) Δ V + s λ ( r , v , t ) Δ V = ( 1 b sca ( λ , r ) ) s λ ( r , v , t ) Δ V + b sca ( λ , r ) a λ ¯ ( t ) Δ V ,
1 c . I λ ( r , v , t ) t = ( 1 b sca ( λ , r ) ) S λ ( r , v , t ) + b sca ( λ , r ) A λ ¯ ( t ) ,
A λ ¯ ( t ) = h c λ a λ ¯ ( t ) ,
I ( r ) = t t + Δ t i = 0 N 1 ξ λ ( r + i Δ d x , x , t ^ ) t Δ V d t ^ ,
I ( r ) = t t + Δ t i = 0 N 1 [ ( 1 b sca ( λ , r + i Δ d x ) ) s λ ( r + i Δ d x , x , t ^ ) Δ V + b sca ( λ , r + i Δ d x ) a λ ¯ ( t ^ ) Δ V ] d t ^ = t t + Δ t [ s λ ( r , x , t ^ ) Δ V i = 0 N 1 ( 1 b sca ( λ , r + i Δ d x ) ) + a λ ¯ ( t ^ ) Δ V i = 0 N 1 b sca ( λ , r + i Δ d x ) ] d t ^ = t t + Δ t [ s λ ( r , x , t ^ ) N Δ V + ( a λ ¯ ( t ^ ) s λ ( r , x , t ^ ) ) Δ V i = 0 N 1 b sca ( λ , r + i Δ d x ) ] d t ^ = t t + Δ t [ s λ ( r , x , t ^ ) N Δ V + ( a λ ¯ ( t ^ ) s λ ( r , x , t ^ ) ) N Δ V 1 N i = 0 N 1 μ ( r ) N i ] d t ^ ,
μ ( r ) N i = exp ( β ( r ) ( N i ) Δ d ) = exp ( β ( r ) ( d i Δ d ) ) b sca ( λ , r + i Δ d x ) ,
1 N i = 0 N 1 μ ( r ) N i = μ ( r ) ( 1 μ ( r ) N ) N ( 1 μ ( r ) ) = exp ( β ( r ) Δ d ) ( 1 exp ( β ( r ) N Δ d ) ) N ( 1 exp ( β ( r ) Δ d ) ) = exp ( β ( r ) d N ) ( 1 exp ( β ( r ) d ) ) N ( 1 exp ( β ( r ) d N ) ) .
1 N i = 0 N 1 μ ( r ) N i exp ( β ( r ) d N ) ( 1 exp ( β ( r ) d ) ) N [ 1 ( 1 β ( r ) d N ) ] = 1 exp ( β ( r ) d ) β ( r ) d .
1 N i = 0 N 1 μ ( r ) N i 1 [ 1 β ( r ) d + 1 2 ( β ( r ) d ) 2 ] β ( r ) d = 1 1 2 β ( r ) d .
I ( r ) = t t + Δ t [ s λ ( r , x , t ^ ) N Δ V + ( a λ ¯ ( t ^ ) s λ ( r , x , t ^ ) ) N Δ V ( 1 1 2 β ( r ) d ) ] d t ^ = t t + Δ t [ 1 2 β ( r ) d s λ ( r , x , t ^ ) N Δ V + ( 1 1 2 β ( r ) d ) a λ ¯ ( t ^ ) N Δ V ] d t ^ = B ( r ) t t + Δ t s λ ( r , x , t ^ ) N Δ V d t ^ + ( 1 B ( r ) ) t t + Δ t a λ ¯ ( t ^ ) N Δ V d t ^ = B ( r ) J ( r ) + ( 1 B ( r ) ) K ,
B ( r ) = 1 2 β ( r ) d , J ( r ) = t t + Δ t s λ ( r , x , t ^ ) N Δ V d t ^ , K = t t + Δ t a λ ¯ ( t ^ ) N Δ V d t ^ .
I ( r ) * = t t + Δ t s λ ( r , x , t ^ ) N Δ V d t ^ = J ( r ) ,
I ( t ) = t ( t ) J ( r ) + ( 1 t ( r ) ) A ,
I ˙ ( r ) = x Ω ( r ) G σ 0 ( x r ) I ( x ) d x = G σ 0 ( r ) I ( r ) , = G σ 0 ( r ) ( B ( r ) J ( r ) + ( 1 B ( r ) ) K ) ,
I ̆ ( r ) ~ Poission ( I ˙ ( r ) ) = Poission ( G σ 0 ( r ) ( B ( r ) J ( r ) + ( 1 B ( r ) ) K ) ) ,
J dark ( r ) = min m Ω ( r ) J ( m ) ,
J dark ( r ) 0 .
min m Ω ( r ) ( I ˙ ( m ) K ) = min m Ω ( r ) ( G σ 0 ( m ) ( B ¨ ( m ) J ( m ) + ( 1 B ¨ ( m ) ) K ) K ) = min m Ω ( r ) ( B ¨ ( m ) G σ 0 ( m ) J ( m ) K + 1 B ¨ ( m ) ) = B ¨ ( r ) min m Ω ( r ) G σ 0 ( m ) J ( m ) K + 1 B ¨ ( r ) .
min m Ω ( r ) G σ 0 ( m ) J ( m ) K 0 .
B ¨ ( r ) = 1 min m Ω ( r ) I ˙ ( m ) K .
I ˙ dark ( r ) = B ¨ ( r ) ( G σ 0 J ) dark ( r ) + K ( 1 B ¨ ( r ) ) .
I ˙ dark ( r ) K ( 1 B ¨ ( r ) ) .
G σ 0 ( r ) J ( r ) = I ˙ ( r ) K B ¨ ( r ) + K .
B i = a ¯ i I ˙ i + b ¯ i ,
a ¯ i = 1 | ω | k ω i 1 | ω i | I ˙ i B ¨ i μ k B ¨ k ¯ σ k 2 + ε , b ¯ i = 1 | ω | k ω i ( B ¨ k ¯ a k μ k ) ,
G σ 0 ( r ) J ( r ) = I ˙ ( r ) K max ( B ( r ) , B 0 ) + K ,
J ˙ ( r ) = I ˙ ( r ) K max ( B ( r ) , B 0 ) + K ,
G σ 0 J = J ˙ .
arg max σ ENR ( σ ) = E ( σ ) N ( σ ) ,
E ( σ ) = I ( R σ ( J ) , G σ R σ ( J ) ) , N ( σ ) = I ( J , G σ R σ ( J ) ) ,
I ( u , v ) = x u ( x ) log u ( x ) v ( x ) x [ u ( x ) v ( x ) ] ,
min { 1 2 1 G σ J ( G σ J J ˙ ) L 2 2 + μ J L 1 } ,
shrink ( x , a ) = { x a , x ( a , + ) , 0 , x [ a , a ] , x + a , x ( , a ) .
f ( x ) = { a x , 0 x < d , b x c , d x 255 .

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