Abstract

Phase Contrast Microscopy (PCM) is an important tool for the long term study of living cells. Unlike fluorescence methods which suffer from photobleaching of fluorophore or dye molecules, PCM image contrast is generated by the natural variations in optical index of refraction. Unfortunately, the same physical principles which allow for these studies give rise to complex artifacts in the raw PCM imagery. Of particular interest in this paper are neuron images where these image imperfections manifest in very different ways for the two structures of specific interest: cell bodies (somas) and dendrites. To address these challenges, we introduce a novel parametric image model using the level set framework and an associated variational approach which simultaneously restores and segments this class of images. Using this technique as the basis for an automated image analysis pipeline, results for both the synthetic and real images validate and demonstrate the advantages of our approach.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (1)

J. Pang, J. B. Driban, T. E. McAlindon, J. G. Tamez-Pena, J. Fripp, and E. L. Miller, “On the use of directional edge information and coupled shape priors for segmentation of magnetic resonance images of the knee,” IEEE J. of Biomed. and Health Inform. 19, 1153–1167 (2015).

2014 (2)

D. Nam, J. Mantell, D. Bull, P. Verkade, and A. Achim, “A novel framework for segmentation of secretory granules in electron micrographs,” MedIA 18, 411–424(2014).

S. B. Mehta and R. Oldenbourg, “Image simulation for bilogical microscopy: microlith,” Biomed. Opt. Express 5, 1822–1838 (2014).
[Crossref] [PubMed]

2013 (1)

H. Su, Z. Yin, S. Huh, and T. Kanade, “Cell segmentation in phase contrast microscopy images via semi-supervised clustering over optics-related feataures,” MedIA 17, 746–765 (2013).

2012 (3)

Z. Yin, T. Kanade, and M. Chen, “Understanding the phase contrast optics to restore artifact-free microscopy images for segmentation,” MedIA 16, 1047–1062 (2012).

M. E. Ambuhl, C. Brepsant, J. Meister, A. B. Verkhovsky, and I. F. Sbalzarini, “High-resolution cell outline segmentation and tracking from phase-contrast microscopy images,” J. Microscopy 245, 161–170 (2012).
[Crossref]

J. S. Vestergaard, A. L. Dahl, P. Holm, and R. Larsen, “Pipeline for tracking neural progenitor cells,” MICCAI Workshop WCV,  7766155–164 (2012).

2011 (2)

L. Dehmelt, G. Poplawski, E. Hwang, and S. Halpain, “NeuriteQuant: An open source toolkit for high content screens of neuronal morphogenesis,” BMC Neurosci. 12, 100 (2011).
[Crossref] [PubMed]

S-Y Ho, C-Y Chao, H-L Huang, T-W Chiu, P. Charoenkwan, and E. Hwang, “NeurphologyJ: An automatic neuronal morphology quantification method and its application in pharmacological discovery,” BMC Bioinformatics 12, 1471–2105 (2011).
[Crossref]

2010 (2)

E. Meijering, “Neuron tracing in perspective,” Cytometry Part A 77, 693–704 (2010).
[Crossref]

Z. Yin, K. Li, T. Kanade, and M. Chen, “Understanding the Optics to Aid Microscopy Image Segmentation,” MICCAI,  6361209–217 (2010).
[PubMed]

2008 (6)

I. Ersoy, F. Bunyak, K. Palaniappan, M. Sun, and G. Forgacs, “Cell spreading analysis with directed edge profile-guided level set active contours,” MICCAI,  5241376–383 (2008).
[PubMed]

S. Lankton and A. Tannenbaum, “Localizing region-based active contours,” IEEE Trans. Image Process. 17, 2029–2039 (2008).
[Crossref] [PubMed]

A. Gooya, H. Liao, K. Matsumiya, K. Masamune, Y. Masutani, and T. Dohi, “A variational method for geometric regularization of vascular segmentation in medical images,” IEEE Trans. Image Process. 17, 1295–1312 (2008).
[Crossref] [PubMed]

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total vairation image reconstruction,” SIAM J. Imaging Sciences 1, 248–272 (2008).
[Crossref]

C. Li, C.Y Kao, J.C. Gore, and Z. Ding, “Minimization of region-scalable fitting energy for image segmentation,” IEEE Trans. Image Process. 17, 1940–1949 (2008).
[Crossref] [PubMed]

Y. Shi and W. C. Karl, “A real-time algorithm for the approximation of level-set-based curve evolution,” IEEE Trans. Image Process. 17, 645–656 (2008).
[Crossref] [PubMed]

2006 (1)

O. Al-Kofahi, R. J. Radke, B. Roysam, and G. Banker, “Automated semantic analysis of changes in image sequences of neurons in culture,” IEEE Trans. on Biomed. Eng. 53, 1109–1123 (2006).
[Crossref]

2005 (2)

N. Ozkucur, K. P. Quinn, J. Pang, I. Georgakoudi, E. Miller, M. Levin, and D. L. Kaplan, “Membrane potential depolarization causes alterations in neuron arrangement and connectivity in cocultures,” Brain and Behav. 5, e00295 (2005).

K. Zimmer and J. Oliver-Marin, “Compled Parametric Active Contours,” IEEE Trans. on PAMI 27, 1838–1842 (2005).
[Crossref]

2004 (2)

S. Serra-Capizzano, “A note on Antireflective boundary Conditions and fast deblurring models,” SIAM J. Sci. Comput. 25, 1307–1325 (2004).
[Crossref]

E. Meijering, M. Jacob, J.-C.F. Sarria, P. Steiner, H. Hirling, and M. Unser, “Design and validation of a tool for neurite tracing and analysis in fluorescence microscopy images,” Cytometry Part A 58A, 167–176 (2004).
[Crossref]

2002 (3)

L. Vese and T. Chan, “A multiphase level set framework for image segmentation using the Mumford and Shah model,” IJCV 50, 271–293 (2002).
[Crossref]

A. Yezzi, A. Tsai, and A. Willsky, “A fully global approach to image segmenation via coupled curve evolution equation,” J. Vis. Comm. Image Rep. 13, 195–216 (2002).
[Crossref]

A. Vasilevskiy, “Flux maximizing geometric flows,” IEEE Trans. PAMI 24,1565–1578 (2002).
[Crossref]

2001 (2)

T. F. Chan and L. A. Vese, “Active contours without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[Crossref]

L. M. Lorigo, O. D. Faugeras, W. E. L. Grimson, R. Kervien, R. Kikinis, A. Nabavi, and C. F. Westin, “CURVES: Curve evolution for vessel segmentation,” MedIA 5, 195–206 (2001).

2000 (1)

K. Krissian, “Model based detection of tubular strucutres in 3D images,” Comput. Vis. Image Und. 80, 130–171 (2000).
[Crossref]

1999 (2)

A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, “Rapid automated tracing and feature extraction from retinal fundus images using direct exploratory algorithms,” IEEE Trans. Inf. Tech. Biomed. 3, 125–138 (1999).
[Crossref]

M. K. Ng, R. H. Chan, and W. C. Tang, “A fast algorithm for deblurring models with Neuman boundary conditions,” SIAM J. Sci. Comput. 21, 851–8666 (1999).
[Crossref]

1998 (1)

R. F. Frangi, W. J. Niessen, K. L. Vincken, and M. A. Viergever, “Multiscale vessel enhancement filtring,” MICCAI,  1496130–137 (1998).

1997 (1)

V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” IJCV 22, 61–79 (1997).
[Crossref]

1995 (1)

K. Wu, D. Gauthier, and M. Levine, “Live cell image segmentation,” IEEE Trans. on Biomed. Eng. 42, 1–12 (1995).
[Crossref]

1991 (1)

W. Freeman and E. Adelson, “The design and use of steerable filters,” IEEE Trans. PAMI 13, 891–906 (1991).
[Crossref]

1989 (2)

J. E. Vaughn, “Fine structure of synaptogenesis in the vertebrate central nervous system,” Synapse 3, 255–285 (1989).
[Crossref] [PubMed]

G. L. Gerstein, P. Bedenbaugh, and M. H. Aertsen, “Neuronal assemblies,” IEEE Trans. Biomed. Eng. 36, 4–14 (1989).
[Crossref] [PubMed]

1986 (1)

J. Kittler and J. Illingworth, “Minimum error thresholding,” Pattern Recogn. 19, 41–47 (1986).
[Crossref]

1979 (1)

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Sys. Man. Cyber. 9, 62–66 (1979).
[Crossref]

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[Crossref] [PubMed]

Achim, A.

D. Nam, J. Mantell, D. Bull, P. Verkade, and A. Achim, “A novel framework for segmentation of secretory granules in electron micrographs,” MedIA 18, 411–424(2014).

Adelson, E.

W. Freeman and E. Adelson, “The design and use of steerable filters,” IEEE Trans. PAMI 13, 891–906 (1991).
[Crossref]

Aertsen, M. H.

G. L. Gerstein, P. Bedenbaugh, and M. H. Aertsen, “Neuronal assemblies,” IEEE Trans. Biomed. Eng. 36, 4–14 (1989).
[Crossref] [PubMed]

Al-Kofahi, O.

O. Al-Kofahi, R. J. Radke, B. Roysam, and G. Banker, “Automated semantic analysis of changes in image sequences of neurons in culture,” IEEE Trans. on Biomed. Eng. 53, 1109–1123 (2006).
[Crossref]

Ambuhl, M. E.

M. E. Ambuhl, C. Brepsant, J. Meister, A. B. Verkhovsky, and I. F. Sbalzarini, “High-resolution cell outline segmentation and tracking from phase-contrast microscopy images,” J. Microscopy 245, 161–170 (2012).
[Crossref]

Banker, G.

O. Al-Kofahi, R. J. Radke, B. Roysam, and G. Banker, “Automated semantic analysis of changes in image sequences of neurons in culture,” IEEE Trans. on Biomed. Eng. 53, 1109–1123 (2006).
[Crossref]

Bedenbaugh, P.

G. L. Gerstein, P. Bedenbaugh, and M. H. Aertsen, “Neuronal assemblies,” IEEE Trans. Biomed. Eng. 36, 4–14 (1989).
[Crossref] [PubMed]

Boles, W.

C. J. Bradhurst, W. Boles, and Y. Xiao, “Segmentation of bone marrow stromal cells in phase contrast microscopy images,” IEEE Inter. Conf. Image Vis. Computing, 1–6 (2008).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interfence and Diffraction of Light(Cambridge University Press, 1999).
[Crossref]

Bradhurst, C. J.

C. J. Bradhurst, W. Boles, and Y. Xiao, “Segmentation of bone marrow stromal cells in phase contrast microscopy images,” IEEE Inter. Conf. Image Vis. Computing, 1–6 (2008).

Brepsant, C.

M. E. Ambuhl, C. Brepsant, J. Meister, A. B. Verkhovsky, and I. F. Sbalzarini, “High-resolution cell outline segmentation and tracking from phase-contrast microscopy images,” J. Microscopy 245, 161–170 (2012).
[Crossref]

Bull, D.

D. Nam, J. Mantell, D. Bull, P. Verkade, and A. Achim, “A novel framework for segmentation of secretory granules in electron micrographs,” MedIA 18, 411–424(2014).

Bunyak, F.

I. Ersoy, F. Bunyak, K. Palaniappan, M. Sun, and G. Forgacs, “Cell spreading analysis with directed edge profile-guided level set active contours,” MICCAI,  5241376–383 (2008).
[PubMed]

Campbell, P. G.

K. Li, E. D. Miller, L. E. Weiss, P. G. Campbell, and T. Kanade, “Online tracking of migrating and proliferating cells imaged with phase-contrast microscopy,” IEEE CVPR workshop, 65 (2006).

Can, A.

A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, “Rapid automated tracing and feature extraction from retinal fundus images using direct exploratory algorithms,” IEEE Trans. Inf. Tech. Biomed. 3, 125–138 (1999).
[Crossref]

Caselles, V.

V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” IJCV 22, 61–79 (1997).
[Crossref]

Chan, R. H.

M. K. Ng, R. H. Chan, and W. C. Tang, “A fast algorithm for deblurring models with Neuman boundary conditions,” SIAM J. Sci. Comput. 21, 851–8666 (1999).
[Crossref]

Chan, T.

L. Vese and T. Chan, “A multiphase level set framework for image segmentation using the Mumford and Shah model,” IJCV 50, 271–293 (2002).
[Crossref]

Chan, T. F.

T. F. Chan and L. A. Vese, “Active contours without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[Crossref]

Chao, C-Y

S-Y Ho, C-Y Chao, H-L Huang, T-W Chiu, P. Charoenkwan, and E. Hwang, “NeurphologyJ: An automatic neuronal morphology quantification method and its application in pharmacological discovery,” BMC Bioinformatics 12, 1471–2105 (2011).
[Crossref]

Charoenkwan, P.

S-Y Ho, C-Y Chao, H-L Huang, T-W Chiu, P. Charoenkwan, and E. Hwang, “NeurphologyJ: An automatic neuronal morphology quantification method and its application in pharmacological discovery,” BMC Bioinformatics 12, 1471–2105 (2011).
[Crossref]

Chen, M.

Z. Yin, T. Kanade, and M. Chen, “Understanding the phase contrast optics to restore artifact-free microscopy images for segmentation,” MedIA 16, 1047–1062 (2012).

Z. Yin, K. Li, T. Kanade, and M. Chen, “Understanding the Optics to Aid Microscopy Image Segmentation,” MICCAI,  6361209–217 (2010).
[PubMed]

Chiu, T-W

S-Y Ho, C-Y Chao, H-L Huang, T-W Chiu, P. Charoenkwan, and E. Hwang, “NeurphologyJ: An automatic neuronal morphology quantification method and its application in pharmacological discovery,” BMC Bioinformatics 12, 1471–2105 (2011).
[Crossref]

Dahl, A. L.

J. S. Vestergaard, A. L. Dahl, P. Holm, and R. Larsen, “Pipeline for tracking neural progenitor cells,” MICCAI Workshop WCV,  7766155–164 (2012).

Davis, C. S.

C. S. Davis, Statistical Methods for the Analysis of Repeated Measurements(Springer, 2002).

Dehmelt, L.

L. Dehmelt, G. Poplawski, E. Hwang, and S. Halpain, “NeuriteQuant: An open source toolkit for high content screens of neuronal morphogenesis,” BMC Neurosci. 12, 100 (2011).
[Crossref] [PubMed]

Ding, Z.

C. Li, C.Y Kao, J.C. Gore, and Z. Ding, “Minimization of region-scalable fitting energy for image segmentation,” IEEE Trans. Image Process. 17, 1940–1949 (2008).
[Crossref] [PubMed]

Dohi, T.

A. Gooya, H. Liao, K. Matsumiya, K. Masamune, Y. Masutani, and T. Dohi, “A variational method for geometric regularization of vascular segmentation in medical images,” IEEE Trans. Image Process. 17, 1295–1312 (2008).
[Crossref] [PubMed]

Driban, J. B.

J. Pang, J. B. Driban, T. E. McAlindon, J. G. Tamez-Pena, J. Fripp, and E. L. Miller, “On the use of directional edge information and coupled shape priors for segmentation of magnetic resonance images of the knee,” IEEE J. of Biomed. and Health Inform. 19, 1153–1167 (2015).

Eddins, S. L.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing using MATLAB (Gatesmark Publishing, 2009).

Ersoy, I.

I. Ersoy, F. Bunyak, K. Palaniappan, M. Sun, and G. Forgacs, “Cell spreading analysis with directed edge profile-guided level set active contours,” MICCAI,  5241376–383 (2008).
[PubMed]

Faugeras, O. D.

L. M. Lorigo, O. D. Faugeras, W. E. L. Grimson, R. Kervien, R. Kikinis, A. Nabavi, and C. F. Westin, “CURVES: Curve evolution for vessel segmentation,” MedIA 5, 195–206 (2001).

Fedkiw, R. P.

S. J. Osher and R. P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces(Springer-Verlag, 2002).

Forgacs, G.

I. Ersoy, F. Bunyak, K. Palaniappan, M. Sun, and G. Forgacs, “Cell spreading analysis with directed edge profile-guided level set active contours,” MICCAI,  5241376–383 (2008).
[PubMed]

Frangi, R. F.

R. F. Frangi, W. J. Niessen, K. L. Vincken, and M. A. Viergever, “Multiscale vessel enhancement filtring,” MICCAI,  1496130–137 (1998).

Freeman, W.

W. Freeman and E. Adelson, “The design and use of steerable filters,” IEEE Trans. PAMI 13, 891–906 (1991).
[Crossref]

Fripp, J.

J. Pang, J. B. Driban, T. E. McAlindon, J. G. Tamez-Pena, J. Fripp, and E. L. Miller, “On the use of directional edge information and coupled shape priors for segmentation of magnetic resonance images of the knee,” IEEE J. of Biomed. and Health Inform. 19, 1153–1167 (2015).

Gauthier, D.

K. Wu, D. Gauthier, and M. Levine, “Live cell image segmentation,” IEEE Trans. on Biomed. Eng. 42, 1–12 (1995).
[Crossref]

Georgakoudi, I.

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Biomed. Opt. Express (1)

BMC Bioinformatics (1)

S-Y Ho, C-Y Chao, H-L Huang, T-W Chiu, P. Charoenkwan, and E. Hwang, “NeurphologyJ: An automatic neuronal morphology quantification method and its application in pharmacological discovery,” BMC Bioinformatics 12, 1471–2105 (2011).
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BMC Neurosci. (1)

L. Dehmelt, G. Poplawski, E. Hwang, and S. Halpain, “NeuriteQuant: An open source toolkit for high content screens of neuronal morphogenesis,” BMC Neurosci. 12, 100 (2011).
[Crossref] [PubMed]

Brain and Behav. (1)

N. Ozkucur, K. P. Quinn, J. Pang, I. Georgakoudi, E. Miller, M. Levin, and D. L. Kaplan, “Membrane potential depolarization causes alterations in neuron arrangement and connectivity in cocultures,” Brain and Behav. 5, e00295 (2005).

Comput. Vis. Image Und. (1)

K. Krissian, “Model based detection of tubular strucutres in 3D images,” Comput. Vis. Image Und. 80, 130–171 (2000).
[Crossref]

Cytometry Part A (2)

E. Meijering, “Neuron tracing in perspective,” Cytometry Part A 77, 693–704 (2010).
[Crossref]

E. Meijering, M. Jacob, J.-C.F. Sarria, P. Steiner, H. Hirling, and M. Unser, “Design and validation of a tool for neurite tracing and analysis in fluorescence microscopy images,” Cytometry Part A 58A, 167–176 (2004).
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Figures (17)

Fig. 1
Fig. 1 (a) The optical path of phase contrast microscopy; (b) A cropped region of a PCM neuron image. The blue arrows point to somas; the red arrows point to dendrites; the yellow arrows point to places where dendrites and somas are separated by halos; the purple dash arrows point to shade-offs.
Fig. 2
Fig. 2 Illustration of the pipeline for PCM neuron image analysis
Fig. 3
Fig. 3 An Illustration for the localized level set method. The point x is in the evolving curve C and Br(x) is a neighborhood of x. Note: the neighborhood Br(x) is divided into two parts by C .
Fig. 4
Fig. 4 Curve evolution under the geometric tubular structure regularization. From left to right: iteration 0, 20, 40 and 60 respectively.
Fig. 5
Fig. 5 Left: A binary image; Right: The corresponding weight map w(x) with a = 20 and b = 0.5.
Fig. 6
Fig. 6 Effects of the tubular regularization for segmentation.
Fig. 7
Fig. 7 Illustration of the initializations for somas and dendrites. The input image to be processed is in Fig. 2 for curve initializations. (a): local standard deviation (b) Otsu’s thresholding results on (a); (c): closing and hole filling on (b); (d): erosion of (c); (e): steerable filtering enhancement for dendrite initializations; (f) The minimum error thresholding results on (e); (g) The final dendrites initialization.
Fig. 8
Fig. 8 Illustration of morphological operations
Fig. 9
Fig. 9 Synthesized images. (a): a true phase retardation image; (b): the simulated PCM image without noise; (c): the simulated PCM image with Gaussian noise (PSNR = 15dB)
Fig. 10
Fig. 10 Results comparison among Yin’s model [2, 3] and the Single Level Set (SLS) and the Double Level Sets (DLS) model for Fig. 9(b) and (c) above. Columns 1-3: Initializations, reconstruction results and segmentation results. Rows 1-3: Yin’s model, SLS and DLS model (top to bottom) for noise free data. Rows 4-6: Yin’s model, SLS and DLS model (top to bottom) for data with noise.
Fig. 11
Fig. 11 Mean square error of the reconstruction results
Fig. 12
Fig. 12 Segmentation results comparison. Columns: different PCM images. Rows (top to bottom): PCM images, Yin’s reconstruction results, Yin’s segmentation results, final results of our method and manual results respectively.
Fig. 13
Fig. 13 A group of images. Left column: PCM images; Right Column: segmentation results based on our method. Blue areas indicate connected somas, read areas denote isolated somas and yellow lines represent dendrites. Rows (top to bottom): 0, 2, 4, 6 and 24 hour respectively.
Fig. 14
Fig. 14 Quantitative comparison between the proposed method and manual results for somas. Left: ACC results; Right: Dice results.
Fig. 15
Fig. 15 Soma numbers comparison.
Fig. 16
Fig. 16 Quantitative comparison between the proposed method and manual results for dendrites. Left: The x axis represents the manually traced dendrite length; the y axis denotes the segmented dendrites length using the proposed method. Each dot represents a results for each PCM image. Right: Comparison for dendrites at different time points and different groups
Fig. 17
Fig. 17 Ratio of all soma numbers to isolated soma numbers. Left: manual results; Right: proposed method.

Tables (1)

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Algorithm 1 Optimization for energy functional (8) for each iteration

Equations (39)

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E c v ( C ) = Ω 1 [ I ( x ) c 1 ] 2 d x + Ω 1 c [ I ( x ) c 2 ] 2 d x + α | C |
E c v ( ϕ ) = Ω [ I ( x ) c 1 ] 2 H ε ( ϕ ( x ) ) d x + Ω [ I ( x ) c 2 ] 2 [ 1 H ε ( ϕ ( x ) ) ] d x + α Ω | H ε ( ϕ ( x ) ) | d x
H ε ( z ) = 1 2 ( 1 + 2 π arctan ( z ε ) ) ,
H ( z ) = { 1 , z 0 0 , e l s e .
ϕ ( x ) t = δ ε ( ϕ ( x ) ) [ α ( ϕ ( x ) | ϕ ( x ) | ) ( I ( x ) c 1 ) + ( I ( x ) c 2 ) 2 ]
c 1 = Ω I ( x ) H ε ( ϕ ( x ) ) d x Ω H ε ( ϕ ( x ) ) d x , c 2 = Ω I ( x ) [ 1 H ε ( ϕ ( x ) ) ] d x Ω [ 1 H ε ( ϕ ( x ) ) ] d x .
I ( x ) G ( x ) + [ a i r y ( x ) δ ( x ) ] θ ( x )
a i r y ( x ) = J 1 ( k m | x | ) k m | x | r m J 1 ( k m r m | x | ) k m | x | , x = ( x , y ) , x , y N m , , 1 , 0 , 1 , N m ,
θ ( x ) = s 1 H ε ( ϕ 1 ( x ) ) + s 2 H ε ( ϕ 2 ( x ) ) ,
E ( ϕ 1 , ϕ 2 ; s 1 , s 2 ) = E p h y ( ϕ 1 , ϕ 2 ; s 1 , s 2 ) + λ 1 E l o c ( ϕ 1 , r 1 ) + λ 2 E l o c ( ϕ 2 , r 2 ) + λ 3 E w t u b ( ϕ 2 ) .
I ( x ) = a i r y ( x ) θ ( x ) θ ( x ) .
vec ( w ) = [ w 1 , 1 , , w n , 1 , w 1 , 2 , , w n , 2 , , w 1 , m , , w n , m ] T .
a r r a y ( vec ( w ) ) = w , vec ( a r r a y ( w ) ) = w ,
θ = s 1 H ( ϕ 1 ) + s 2 H ( ϕ 2 )
I = H θ ,
E p h y ( ϕ 1 , ϕ 2 ; s 1 , s 2 ) = 1 2 I H θ 2 .
E l o c ( ϕ ; r ) = δ ε ( ϕ ( x ) ) B r ( x , x ) F l o c ( I ( x ) , ϕ ( x ) ) d x d x
B r ( x , x ) = { 1 , x x 2 r 0 , e l s e .
F l o c ( I ( x ) , ϕ ( x ) ) = ( u x v x ) 2 ,
u x = B r ( x , x ) H ε ( ϕ ( x ) ) I ( x ) d x B r ( x , x ) H ε ( ϕ ( x ) ) d x
v x = B r ( x , x ) [ 1 H ε ( ϕ ( x ) ) ] I ( x ) d x B r ( x , x ) [ 1 H ε ( ϕ ( x ) ) ] d x .
E w t u b ( ϕ ) = w ( x ) δ ε ( ϕ ( x ) ) f ( T r M 1 ( x ) ) | ϕ ( x ) | d x
M ( x ) = H ε ( ϕ ( x ) ) B r ( x , x ) ϕ ( x ) t ϕ ( x ) d x ,
w ( x ) = exp ( d i s t ( B W ( x ) ) / a ) + b
E p h y ( ϕ 1 , ϕ 2 ; s 1 , s 2 ) ϕ k = s k δ ε T ( ϕ k ) H T ( I H θ ) ,
E p h y ( ϕ 1 , ϕ 2 ; s 1 , s 2 ) s k = H ε T ( ϕ k ) H T ( I H θ ) .
E p h y ( ϕ 1 , ϕ 2 ; s 1 , s 2 ) ϕ k = a r r a y ( s k δ ε T ( ϕ k ) H T ( I H θ ) )
E l o c ( ϕ 1 , r 1 ) ϕ 1 = δ ε ( ϕ 1 ( x ) ) B r 1 ( x , x ) δ ε ( ϕ 1 ( x ) ) ( ( I ( x ) u x ) 2 A u ( I ( x ) v x ) 2 A v ) d x
A u = B r ( x , x ) H ε ( ϕ 1 ( x ) ) d x A v = B r ( x , x ) [ 1 H ε ( ϕ 1 ( x ) ) ] d x .
E w t u b ( ϕ 2 ) ϕ 2 = w ( x ) δ ε ( ϕ 2 ( x ) ) { f ( T r M 1 ) ( ϕ 2 | ϕ 2 | ) + t f ( T r M 1 ) ϕ 2 | ϕ 2 | + t ϕ 2 L ϕ 2 }
L ( x ) = B ( x , x ) δ ε ( ϕ 2 ( x ) ) | ϕ 2 ( x ) | f ˙ ( x ) M 2 ( x ) d ( x ) ,
H θ = vec ( 1 ( ( a i r y ( x ) δ ( x ) ) ( θ ( x ) ) ) )
( a i r y ( x ) δ ( x ) ) = ( a i r y ( x ) ) 1.
argmin s 1 | H ε T ( ϕ 1 ) H T ( I H θ ) | , s u b j e c t t o , 0 < s 1 < u b .
argmin s 2 | H ε T ( ϕ 2 ) H T ( I H θ ) | , s u b j e c t t o , 0 < s 2 s 1 .
θ ( x ) = s 1 H ε ( ϕ 1 ( x ) ) ,
P S N R = 10 log 10 max ( I c l e a n ) M S E ( I c l e a n , I n o i s y ) ,
A C C = | T P | + | N | | F P | | P | + | N | .
D I C E = 2 | T P | ( | F P | + | T P | ) + ( | T P | + | F N | ) .

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