Abstract

Conventional deconvolution methods assume that the microscopy system is spatially invariant, introducing considerable errors. We developed a method to more precisely estimate space-variant point-spread functions from sparse measurements. To this end, a space-variant version of deblurring algorithm was developed and combined with a total-variation regularization. Validation with both simulation and real data showed that our PSF model is more accurate than the piecewise-invariant model and the blending model. Comparing with the orthogonal basis decomposition based PSF model, our proposed model also performed with a considerable improvement. We also evaluated the proposed deblurring algorithm. Our new deblurring algorithm showed a significantly better signal-to-noise ratio and higher image quality than those of the conventional space-invariant algorithm.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

D. Sage, L. Donati, F. Soulez, D. Fortun, G. Schmit, A. Seitz, R. Guiet, C. Vonesch, and M. Unser, “Deconvolutionlab2: An open-source software for deconvolution microscopy,” Methods 115, 28–41 (2017).
[Crossref] [PubMed]

2016 (4)

N. Patwary, S. V. King, G. Saavedra, and C. Preza, “Reducing effects of aberration in 3d fluorescence imaging using wavefront coding with a radially symmetric phase mask,” Opt. Express 24, 12905–12921 (2016).
[Crossref] [PubMed]

B. Kim and T. Naemura, “Blind deconvolution of 3d fluorescence microscopy using depth-variant asymmetric psf,” Microsc. Res. Tech. 79, 480–494 (2016).
[Crossref] [PubMed]

C. Roider, R. Heintzmann, R. Piestun, and A. Jesacher, “Deconvolution approach for 3d scanning microscopy with helical phase engineering,” Opt. Express 24, 15456–15467 (2016).
[Crossref] [PubMed]

M. Chen, Y. Li, M. Yang, X. Chen, Y. Chen, F. Yang, S. Lu, S. Yao, T. Zhou, and J. Liu, “A new method for quantifying mitochondrial axonal transport,” Protein & Cell 7, 804–819 (2016).
[Crossref]

2015 (3)

N. Patwary and C. Preza, “Image restoration for three-dimensional fluorescence microscopy using an orthonormal basis for efficient representation of depth-variant point-spread functions,” Biomed. Opt. Express 6, 3826–3841 (2015).
[Crossref] [PubMed]

B. Kim and T. Naemura, “Blind depth-variant deconvolution of 3d data in wide-field fluorescence microscopy,” Sci. Reports 5, 9894 (2015).
[Crossref]

A. Wong, X. Y. Wang, and M. Gorbet, “Bayesian-based deconvolution fluorescence microscopy using dynamically updated nonstationary expectation estimates,” Sci. Reports 5, 10849 (2015).
[Crossref]

2014 (4)

Z. Chen and H. Chen, “The parameter estimation method of gaussian point spread function in microscopic images,” J. Biomed. Eng. 31, 53 (2014).

T. H. Besseling, J. Jose, and A. Van Blaaderen, “Methods to calibrate and scale axial distances in confocal microscopy as a function of refractive index,” J. Microsc. 257, 142–150 (2014).
[Crossref] [PubMed]

M. Castello, A. Diaspro, and G. Vicidomini, “Multi-images deconvolution improves signal-to-noise ratio on gated stimulated emission depletion microscopy,” Appl. Phys. Lett. 105, 234106 (2014).
[Crossref]

A. Kumar, Y. Wu, R. Christensen, P. Chandris, W. Gandler, E. Mccreedy, A. Bokinsky, D. A. Colonramos, Z. Bao, and M. Mcauliffe, “Dual-view plane illumination microscopy for rapid and spatially isotropic imaging,” Nat. Protoc. 9, 2555 (2014).
[Crossref] [PubMed]

2013 (5)

Y. Wu, P. Wawrzusin, J. Senseney, R. S. Fischer, R. Christensen, A. Santella, A. G. York, P. W. Winter, C. M. Waterman, and Z. Bao, “Spatially isotropic four-dimensional imaging with dual-view plane illumination microscopy,” Nat. Biotechnol. 31, 1032–1038 (2013).
[Crossref] [PubMed]

J. Kim, S. An, S. Ahn, and B. Kim, “Depth-variant deconvolution of 3d widefield fluorescence microscopy using the penalized maximum likelihood estimation method,” Opt. Express 21, 27668 (2013).
[Crossref]

A. Matakos, S. Ramani, and J. A. Fessler, “Accelerated edge-preserving image restoration without boundary artifacts,” IEEE Transactions on Image Process.  22, 2019–2029 (2013).
[Crossref]

S. Liu, J. Li, Z. Zhang, Z. Wang, Z. Tian, G. Wang, and D. Pang, “Fast and high-accuracy localization for three-dimensional single-particle tracking,” Sci. Reports 3, 2462 (2013).
[Crossref]

H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3-d psf fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13 (2013).
[Crossref]

2012 (1)

M. Temerinac-Ott, O. Ronneberger, P. Ochs, W. Driever, T. Brox, and H. Burkhardt, “Multiview deblurring for 3-d images from light-sheet-based fluorescence microscopy,” IEEE Transactions on Image Process.  21, 1863–1873 (2012).
[Crossref]

2011 (4)

M. Laasmaa, M. Vendelin, and P. Peterson, “Application of regularized richardson-lucy algorithm for deconvolution of confocal microscopy images,” J. Microsc. 243, 124–140 (2011).
[Crossref] [PubMed]

S. Yuan and C. Preza, “Point-spread function engineering to reduce the impact of spherical aberration on 3d computational fluorescence microscopy imaging,” Opt. Express 19, 23298–23314 (2011).
[Crossref] [PubMed]

E. Maalouf, B. Colicchio, and A. Dieterlen, “Fluorescence microscopy three-dimensional depth variant point spread function interpolation using zernike moments,” J. The Opt. Soc. Am. A-optics Image Sci. Vis. 28, 1864–1870 (2011).
[Crossref]

M. A. Model, J. Fang, P. Yuvaraj, Y. Chen, and B. M. Z. Newby, “3d deconvolution of spherically aberrated images using commercial software,” J. Microsc. 241, 94–100 (2011).
[Crossref]

2010 (1)

2007 (1)

2006 (4)

G. Vicidomini, P. P. Mondal, and A. Diaspro, “Fuzzy logic and maximum a posteriori-based image restoration for confocal microscopy,” Opt. Lett. 31, 3582–3584 (2006).
[Crossref] [PubMed]

P. Sarder and A. Nehorai, “Deconvolution methods for 3-d fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[Crossref]

N. Dey, L. Blancferaud, C. Zimmer, P. Roux, Z. Kam, J. Olivomarin, and J. Zerubia, “Richardson-lucy algorithm with total variation regularization for 3d confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[Crossref] [PubMed]

C. J. Engelbrecht and E. H. K. Stelzer, “Resolution enhancement in a light-sheet-based microscope (spim),” Opt. Lett. 31, 1477–1479 (2006).
[Crossref] [PubMed]

2004 (1)

C. Preza and J. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J Opt Soc Am A Opt Image Sci Vis 21, 1593–1601 (2004).
[Crossref] [PubMed]

2003 (1)

C. Preza and J. A. Conchello, “Image estimation accounting for point-spread function depth cariation in three-dimensional fluorescence microscopy,” Proc. SPIE - The Int. Soc. for Opt. Eng. 4964, 135–142 (2003).

2000 (2)

G. M. Van Kempen and L. J. Van Vliet, “The influence of the regularization parameter and the first estimate on the performance of tikhonov regularized non-linear image restoration algorithms,” J. Microsc. 198 (Pt 1), 63–75 (2000).
[Crossref] [PubMed]

G. M. Van Kempen and L. J. Van Vliet, “Background estimation in nonlinear image restoration,” J. Opt. Soc. Am. A-optics Image Sci. Vis. 17, 425 (2000).
[Crossref]

1996 (1)

J. A. Conchello, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” Proc. SPIE - The Int. Soc. for Opt. Eng. 23, 511–513 (1996).

1995 (1)

1993 (1)

1992 (1)

S. F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. The Opt. Soc. Am. A-optics Image Sci. Vis. 9, 154–166 (1992).
[Crossref]

1963 (2)

P. A. Bello, “Characterization of randomly time-variant linear channels,” IEEE Transactions on Commun. 11, 360–393 (1963).
[Crossref]

D. W. Marquardt, “An algorithm for least square estimation of non-linear parameters,” J. Soc. for Ind. & Appl. Math. 11, 431–441 (1963).
[Crossref]

1959 (1)

B. Richards and E. E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” Proc. The Royal Soc. A: Math. Phys. Eng. Sci. 253, 358–379 (1959).
[Crossref]

1944 (1)

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” J. Hear. & Lung Transplantation Off. Publ. Int. Soc. for Hear. Transplantation 31, 436–438 (1944).

Agard, D. A.

Aguet, F.

H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3-d psf fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13 (2013).
[Crossref]

Ahn, S.

An, S.

Arigovindan, M.

Bao, Z.

A. Kumar, Y. Wu, R. Christensen, P. Chandris, W. Gandler, E. Mccreedy, A. Bokinsky, D. A. Colonramos, Z. Bao, and M. Mcauliffe, “Dual-view plane illumination microscopy for rapid and spatially isotropic imaging,” Nat. Protoc. 9, 2555 (2014).
[Crossref] [PubMed]

Y. Wu, P. Wawrzusin, J. Senseney, R. S. Fischer, R. Christensen, A. Santella, A. G. York, P. W. Winter, C. M. Waterman, and Z. Bao, “Spatially isotropic four-dimensional imaging with dual-view plane illumination microscopy,” Nat. Biotechnol. 31, 1032–1038 (2013).
[Crossref] [PubMed]

Bello, P. A.

P. A. Bello, “Characterization of randomly time-variant linear channels,” IEEE Transactions on Commun. 11, 360–393 (1963).
[Crossref]

Besseling, T. H.

T. H. Besseling, J. Jose, and A. Van Blaaderen, “Methods to calibrate and scale axial distances in confocal microscopy as a function of refractive index,” J. Microsc. 257, 142–150 (2014).
[Crossref] [PubMed]

Blancferaud, L.

N. Dey, L. Blancferaud, C. Zimmer, P. Roux, Z. Kam, J. Olivomarin, and J. Zerubia, “Richardson-lucy algorithm with total variation regularization for 3d confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[Crossref] [PubMed]

R. Cavicchioli, C. Chaux, L. Blancferaud, and L. Zanni, “Ml estimation of wavelet regularization hyperparameters in inverse problems,” in “international conference on acoustics, speech, and signal processing,” (2013), pp. 1553–1557.

S. B. E. Hadj, L. Blancferaud, E. Maalouf, B. Colicchio, and A. Dieterlen, “Depth-variant image restoration in 3d fluorescence microscopy: Two approaches under gaussian and poissonian noise conditions,” in “international symposium on biomedical imaging,” (2012), pp. 1671–1674.

Bokinsky, A.

A. Kumar, Y. Wu, R. Christensen, P. Chandris, W. Gandler, E. Mccreedy, A. Bokinsky, D. A. Colonramos, Z. Bao, and M. Mcauliffe, “Dual-view plane illumination microscopy for rapid and spatially isotropic imaging,” Nat. Protoc. 9, 2555 (2014).
[Crossref] [PubMed]

Brox, T.

M. Temerinac-Ott, O. Ronneberger, P. Ochs, W. Driever, T. Brox, and H. Burkhardt, “Multiview deblurring for 3-d images from light-sheet-based fluorescence microscopy,” IEEE Transactions on Image Process.  21, 1863–1873 (2012).
[Crossref]

Burkhardt, H.

M. Temerinac-Ott, O. Ronneberger, P. Ochs, W. Driever, T. Brox, and H. Burkhardt, “Multiview deblurring for 3-d images from light-sheet-based fluorescence microscopy,” IEEE Transactions on Image Process.  21, 1863–1873 (2012).
[Crossref]

Castello, M.

M. Castello, A. Diaspro, and G. Vicidomini, “Multi-images deconvolution improves signal-to-noise ratio on gated stimulated emission depletion microscopy,” Appl. Phys. Lett. 105, 234106 (2014).
[Crossref]

Cavicchioli, R.

R. Cavicchioli, C. Chaux, L. Blancferaud, and L. Zanni, “Ml estimation of wavelet regularization hyperparameters in inverse problems,” in “international conference on acoustics, speech, and signal processing,” (2013), pp. 1553–1557.

Chandris, P.

A. Kumar, Y. Wu, R. Christensen, P. Chandris, W. Gandler, E. Mccreedy, A. Bokinsky, D. A. Colonramos, Z. Bao, and M. Mcauliffe, “Dual-view plane illumination microscopy for rapid and spatially isotropic imaging,” Nat. Protoc. 9, 2555 (2014).
[Crossref] [PubMed]

Chaux, C.

R. Cavicchioli, C. Chaux, L. Blancferaud, and L. Zanni, “Ml estimation of wavelet regularization hyperparameters in inverse problems,” in “international conference on acoustics, speech, and signal processing,” (2013), pp. 1553–1557.

Chen, H.

Z. Chen and H. Chen, “The parameter estimation method of gaussian point spread function in microscopic images,” J. Biomed. Eng. 31, 53 (2014).

Chen, M.

M. Chen, Y. Li, M. Yang, X. Chen, Y. Chen, F. Yang, S. Lu, S. Yao, T. Zhou, and J. Liu, “A new method for quantifying mitochondrial axonal transport,” Protein & Cell 7, 804–819 (2016).
[Crossref]

Chen, X.

M. Chen, Y. Li, M. Yang, X. Chen, Y. Chen, F. Yang, S. Lu, S. Yao, T. Zhou, and J. Liu, “A new method for quantifying mitochondrial axonal transport,” Protein & Cell 7, 804–819 (2016).
[Crossref]

Chen, Y.

M. Chen, Y. Li, M. Yang, X. Chen, Y. Chen, F. Yang, S. Lu, S. Yao, T. Zhou, and J. Liu, “A new method for quantifying mitochondrial axonal transport,” Protein & Cell 7, 804–819 (2016).
[Crossref]

M. A. Model, J. Fang, P. Yuvaraj, Y. Chen, and B. M. Z. Newby, “3d deconvolution of spherically aberrated images using commercial software,” J. Microsc. 241, 94–100 (2011).
[Crossref]

Chen, Z.

Z. Chen and H. Chen, “The parameter estimation method of gaussian point spread function in microscopic images,” J. Biomed. Eng. 31, 53 (2014).

Christensen, R.

A. Kumar, Y. Wu, R. Christensen, P. Chandris, W. Gandler, E. Mccreedy, A. Bokinsky, D. A. Colonramos, Z. Bao, and M. Mcauliffe, “Dual-view plane illumination microscopy for rapid and spatially isotropic imaging,” Nat. Protoc. 9, 2555 (2014).
[Crossref] [PubMed]

Y. Wu, P. Wawrzusin, J. Senseney, R. S. Fischer, R. Christensen, A. Santella, A. G. York, P. W. Winter, C. M. Waterman, and Z. Bao, “Spatially isotropic four-dimensional imaging with dual-view plane illumination microscopy,” Nat. Biotechnol. 31, 1032–1038 (2013).
[Crossref] [PubMed]

Colicchio, B.

E. Maalouf, B. Colicchio, and A. Dieterlen, “Fluorescence microscopy three-dimensional depth variant point spread function interpolation using zernike moments,” J. The Opt. Soc. Am. A-optics Image Sci. Vis. 28, 1864–1870 (2011).
[Crossref]

S. B. E. Hadj, L. Blancferaud, E. Maalouf, B. Colicchio, and A. Dieterlen, “Depth-variant image restoration in 3d fluorescence microscopy: Two approaches under gaussian and poissonian noise conditions,” in “international symposium on biomedical imaging,” (2012), pp. 1671–1674.

Colonramos, D. A.

A. Kumar, Y. Wu, R. Christensen, P. Chandris, W. Gandler, E. Mccreedy, A. Bokinsky, D. A. Colonramos, Z. Bao, and M. Mcauliffe, “Dual-view plane illumination microscopy for rapid and spatially isotropic imaging,” Nat. Protoc. 9, 2555 (2014).
[Crossref] [PubMed]

Conchello, J.

C. Preza and J. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J Opt Soc Am A Opt Image Sci Vis 21, 1593–1601 (2004).
[Crossref] [PubMed]

Conchello, J. A.

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Figures (12)

Fig. 1
Fig. 1 The optical path difference (OPD) for the point source located at different depths under the same condition in WFM.
Fig. 2
Fig. 2 A framework of proposed SV deblurring for fluorescence microscopy images. Sparsely distributed fluorescent beads were recorded and used to generate a dense estimation of the SV-PSF. The PSF model considered the axial and radial variation. The proposed space-variant restoration algorithm considered the matching object point with its corresponding PSF by using an alternating optimization scheme.
Fig. 3
Fig. 3 A diagram illustrating the process of SV-PSF estimation. The PSF at any specific depth is defined as a weighted average of all its approximations from measurements using the similarity transformation. Stage drift was also considered in most cases.
Fig. 4
Fig. 4 PSF model evaluation with synthetic data (a)–(c), (g)–(i) and experimental data (d)–(f). (a)(d) Coefficients of similarity transformation with different relative depths. (b)(e) MSE between PSFs with different relative depths. (c)(f) Comparison of our proposed similarity transformation based PSF model (STM) with the piecewise-invariant model and the blending model. PSFs in sub-region 1 are approximated with the PSF of zp = 1000nm and PSFs in sub-region 2 are approximated with the PSF of zp = 3500nm (zp refers to the axial location of the object point). The synthetic data was generated by PSF Generator, the ImageJ plugin with acquisition parameters: the emission wavelength λ=530nm, the numerical aperture of the microscope NA = 1.4, the refractive index of the oil-immersion layer ni = 1.515, the refractive index of the specimen layer ns = 1.3, the working distance ti = 150µm and the voxel size 25 × 25 × 100nm3. Besides, the particle position is set from 1000 nm to 3500 nm whose step is 50 nm. S.T. refers to the acronym of similarity transformation. The real data was acquired by Leica SP8 microscope system and a NA 1.4 oil-immersion objective under wide-field mode with the same λ, ni and voxel size as the synthetic data. (g) Slopes for coefficients of similarity transformation along with different depths. (h-i) Comparison between our proposed similarity transformation based PSF model (STM) with the PCA based model of approximating PSFs at 1 − 3.5 µm with step equal to 100 nm using six synthetic PSFs at 1, 1.5, 2, 2.5, 3, 3.5 µm for parameter fitting.
Fig. 5
Fig. 5 One example of PSF estimation using different methods on synthetic data (a) and real data (b). The ground truth of PSF located at the specific depth (first column), the estimation result with the piecewise-invariant PSF model (second column), the estimation result with the blending PSF model (third column) and the estimation result with the proposed SV similarity transformation based PSF model (fourth column) are shown in the corresponding panels.
Fig. 6
Fig. 6 Influence of stage-drift. (a) & (c) Stage drift paths along the y-axis and x-axis. Blue lines represent the drift paths of the PSFs extracted from one bead image, and red lines are the drift paths of PSFs extracted from another bead image. For observability, the stage drift is amplified 100 times. (b) & (d) Meridional cross-section of the PSF before (upper) and after (lower) stage drift correction along the y-axis (b) and x-axis (d). Red lines refer to the projection of the drift path along the y-axis (b) and x-axis (d). The PSF size is 83 × 83 × 61, and the voxel size is 25 × 25 × 100nm3.
Fig. 7
Fig. 7 Spatial variation for the PSF in LSFM, wherein the system PSF is estimated using the pointwise product of the intrinsic PSF and the intensity distribution of its excitation beam.
Fig. 8
Fig. 8 Simulations for deconvolution of synthetic data using the space-invariant and space-variant methods in widefield system. (a)–(c) The raw image (upper-left), the degraded image blurred by a series of space-variant PSFs and Poisson noise (lower-left), the restored image using SI-MLTV (upper-right) and the restored image using SV-MLTV (lower-right). Dotted lines in the XZ section (upper) show where the XY sections are taken. The lower-left XY section is taken where the upper dotted line is located, and the lower-right XY section is taken where the lower dotted line is located. The image size is 199 × 199 × 99, and the voxel size is 25 × 25 100nm3. (a) Deconvolution of three balls, where the intensity of the raw image is 255 and the background is 1. The PSNR is 21.2872. (b) Deconvolution of a cavity, where the intensities of a light spot inside, at the central portion, shell and background are 255, 240, 200 and 1, respectively. The PSNR is 23.2761. (c) Deconvolution of the multi-objects with different intensities: 255 for the ball, 211 for the cube, 180 for the cylinder, 225 for the triangle, 150 for cross and 1 for the background. The PSNR is 24.8370.
Fig. 9
Fig. 9 Simulation for deconvolution of complicated images of neuronal axons in primary neuronal cultures acquired using a Leica SP8 confocal system: (a) the z-stack of the degraded image blurred by space-variant PSFs and Poisson noise with PSNR of 21.5493, (b) the z-stack of the restored image using SV-MLTV, (c) the z-projection (upper) and its partial views (lower) of the restored image using SI-MLTV, (d) the z-projection (upper) and its partial views (lower) of the restored image using SV-MLTV. The raw image size is 601 × 601 × 8, and the voxel size is 50 × 50 × 200nm3. Before image degradation, the raw image is scaled into 601 × 601 × 40 using the software ImageJ by setting the option named “Z Scale” equal to 5. PSF size is 33 × 33 × 25 and the voxel size is 50 × 50 × 40nm3.
Fig. 10
Fig. 10 Deconvolution of the complicated image where the raw sample is the measured confocal image acquired with the Leica SP8. (a) Lateral cross-sections of the raw sample image. (b) Lateral cross-sections of the degraded image blurred by space-variant PSFs and Poisson noise with PSNR of 21.5493. (c) Lateral cross-sections of the restored image using SI-MLTV. (d) Lateral cross-sections of the restored image using SV-MLTV. The raw image size is 601 × 601 × 8, and the voxel size is 50 × 50 × 200nm3.
Fig. 11
Fig. 11 Deconvolution of real data acquired using a Leica SP8 system in wide-field mode: (a) the z-stack of the degraded image, (b) the z-stack of the restored image using SV-MLTV, (c) the z-projection (upper) and its partial views (lower) of the restored image using SI-MLTV, (d) the z-projection (upper) and its partial views (lower) of the restored image using SV-MLTV.
Fig. 12
Fig. 12 Deconvolution of the synthetic image of mitochondria: (a) the raw images, (b) the degraded images blurred by space-variant PSFs and Poisson noise with PSNR of 29.2674, (c) the restored images using SI-MLTV, and (d) the restored images using SV-MLTV. The meridional cross-sections of the PSF along the y-axis (upper panels) and x-axis (lower panels). The image size is 199 × 199 × 29, and the voxel size is 50 × 50 200nm3. SI-MLTV: space-invariant ML algorithm with TV regularization; SV-MLTV: space-variant ML algorithm with TV regularization.

Tables (3)

Tables Icon

Table 1 Derivative expressions for similarity-transformation based PSF model.

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Table 2 Quantitative analyses of particle detection.

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Table 3 Quantitative analyses of deconvolution with different methods.

Equations (20)

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h j ( s ) = T ( h i ( s ) )
h j ( s ) T ( h i ( s ) ; θ i j ) = h i ( θ i j s )
θ i j = d i a g ( α i j , α i j , β i j )
C ( θ i j ) = 1 2 h i ( θ i j s ) h j ( s ) 2
θ ^ i j = arg min θ i j C ( θ i j )
h ^ ( s ; s 0 ) = A i = 1 N ω i T ( h i ( s ) ; θ i )
ω i = { 1 / | z z i | , | z z i | Δ z t h r e s h 0 , otherwise
S H ( s ; x 0 , z 0 ) = I H ( s ; z 0 ) * I D ( s ; x 0 )
( f h ) ( s ) = m = 1 M f m ( s ) * h m ( s )
f m ( s ) = { f ( s ) for z = m 0 otherwise
( f h ) ( s ) = A i = 1 N m = 1 M ω i ( f m ( s ) * h i ( θ i m s ) ) = A i = 1 N m = 1 M ω i F 1 ( F ( f m ( s ) ) F ( h i ( θ i m s ) ) ) A i = 1 N m = 1 M ω i | θ i m | F 1 ( F ( f m ( s ) ) F ( h i ( s ) ) )
f ( k + 1 ) ( s ) = m = 1 M { [ g ( s ) ( f ( k ) h ) ( s ) ] * h m ( s ) } f m ( k ) ( s )
J ( f ) = s S { g ( s ) ln [ ( f h ) ( s ) ] + ( f h ) ( s ) }
z ( j + 1 ) = z ( j ) [ J ( f , z ) z / 2 J ( f , z ) z 2 ] | z = z ( j )
J ( f , z ) z = s S [ f ( s ) * ( 1 g ( s ) ( f h ) ( s ) ) ] h ( s ; s 0 ) z
2 J ( f , z ) z 2 = s S { [ f ( s ) * ( 1 g ( s ) ( f h ) ( s ) ) ] g ( s ) f ( s ) [ ( f h ) ( s ) ] 2 [ h ( s ; s 0 ) z ] 2 + [ f ( s ) * ( 1 g ( s ) ( f h ) ( s ) ) ] 2 h ( s ; s 0 ) z 2 }
h ( s ; s 0 ) z = A i = 1 N ω i [ ( k r ( α u ) u + k r ( α v ) v + k z ( β w ) w ) h i ( θ i s ) ]
2 h ( s ; s 0 ) z 2 = A i = 1 N ω i [ ( k r ( α u ) u + k r ( α v ) v + k z ( β w ) w ) h i ( θ i s ) ]
J ( f ) + J T V ( f ) = s S { g ( s ) ln [ ( f h ) ( s ) ] + ( f h ) ( s ) } + λ T V s S | f ( s ) |
f ( k + 1 ) ( s ) = m = 1 M f m ( k ) ( s 0 ) 1 λ T V d i v ( f ( k ) ( s ) | f ( k ) ( s ) | ) { [ g ( s ) ( f ( k ) h ) ( s ) ] * h m ( s ) }

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