Abstract

In-line digital holography is a simple yet powerful tool to image absorbing and/or phase objects. Nevertheless, the loss of the phase of the complex wavefront on the sensor can be critical in the reconstruction process. The simplicity of the setup must thus be counterbalanced by dedicated reconstruction algorithms, such as inverse approaches, in order to retrieve the object from its hologram. In the case of simple objects for which the diffraction pattern produced in the hologram plane can be modeled using few parameters, a model fitting algorithm is very effective. However, such an approach fails to reconstruct objects with more complex shapes, and an image reconstruction technique is then needed. The improved flexibility of these methods comes at the cost of a possible loss of reconstruction accuracy. In this work, we combine the two approaches (model fitting and regularized reconstruction) to benefit from their respective advantages. The sample to be reconstructed is modeled as the sum of simple parameterized objects and a complex-valued pixelated transmittance plane. These two components jointly scatter the incident illumination, and the resulting interferences contribute to the intensity on the sensor. The proposed hologram reconstruction algorithm is based on alternating a model fitting step and a regularized inversion step. We apply this algorithm in the context of fluid mechanics, where holograms of evaporating droplets are analyzed. In these holograms, the high contrast fringes produced by each droplet tend to mask the diffraction pattern produced by the surrounding vapor wake. With our method, the droplet and the vapor wake can be jointly reconstructed.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

2017 (9)

A. Berdeu, F. Momey, B. Laperrousaz, T. Bordy, X. Gidrol, J.-M. Dinten, N. Picollet-D’hahan, and C. Allier, “Comparative study of fully three-dimensional reconstruction algorithms for lens-free microscopy,” Appl. Opt. 56, 3939–3951 (2017).
[Crossref] [PubMed]

O. Flasseur, C. Fournier, N. Verrier, L. Denis, F. Jolivet, A. Cazier, and T. Lépine, “Self-calibration for lensless color microscopy,” Appl. Opt. 56, F189–F199 (2017).
[Crossref] [PubMed]

C. Fournier, F. Jolivet, L. Denis, N. Verrier, E. Thiébaut, C. Allier, and T. Fournel, “Pixel super-resolution in digital holography by regularized reconstruction,” Appl. Opt. 56, 69–77 (2017).
[Crossref]

S. Vandewiele, F. Strubbe, C. Schreuer, K. Neyts, and F. Beunis, “Low coherence digital holography microscopy based on the lorenz-mie scattering model,” Opt. Express 25, 25853–25866 (2017).
[Crossref] [PubMed]

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. Can Oztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the Lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

M. P. L. Sentis, F. R. A. Onofri, and F. Lamadie, “Photonic jet reconstruction for particle refractive index measurement by digital in-line holography,” Opt. Express 25, 867–873 (2017).
[Crossref] [PubMed]

L. A. Philips, D. B. Ruffner, F. C. Cheong, J. M. Blusewicz, P. Kasimbeg, B. Waisi, J. R. McCutcheon, and D. G. Grier, “Holographic characterization of contaminants in water: Differentiation of suspended particles in heterogeneous dispersions,” Water Res. 122, 431–439 (2017).
[Crossref] [PubMed]

M. Hejna, A. Jorapur, J. S. Song, and R. L. Judson, “High accuracy label-free classification of single-cell kinetic states from holographic cytometry of human melanoma cells,” Sci. Rep. 7, 11943 (2017).
[Crossref] [PubMed]

C. Allier, S. Morel, R. Vincent, L. Ghenim, F. Navarro, M. Menneteau, T. Bordy, L. Hervé, O. Cioni, X. Gidrol, Y. Usson, and J. M. Dinten, “Imaging of dense cell cultures by multiwavelength lens-free video microscopy,” Cytom. Part A 91, 433–442 (2017).
[Crossref]

2016 (3)

C. Wang, X. Zhong, D. B. Ruffner, A. Stutt, L. A. Philips, M. D. Ward, and D. G. Grier, “Holographic Characterization of Protein Aggregates,” J. Pharm. Sci. 105, 1074–1085 (2016).
[Crossref] [PubMed]

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Reports 6, 37862 (2016).
[Crossref]

E. Mathieu, C. D. Paul, R. Stahl, G. Vanmeerbeeck, V. Reumers, C. Liu, K. Konstantopoulos, and L. Lagae, “Time-lapse lens-free imaging of cell migration in diverse physical microenvironments,” Lab on a Chip 16, 3304–3316 (2016).
[Crossref] [PubMed]

2015 (1)

J. F. Jikeli, L. Alvarez, B. M. Friedrich, L. G. Wilson, R. Pascal, R. Colin, M. Pichlo, A. Rennhack, C. Brenker, and U. B. Kaupp, “Sperm navigation along helical paths in 3D chemoattractant landscapes,” Nat. Commun. 6, 7985 (2015).
[Crossref] [PubMed]

2013 (1)

2012 (1)

G. Baffou, P. Bon, J. Savatier, J. Polleux, M. Zhu, M. Merlin, H. Rigneault, and S. Monneret, “Thermal imaging of nanostructures by quantitative optical phase analysis,” Am. Chem. Soc. Nano 6, 2452–2458 (2012).

2011 (1)

2010 (2)

2009 (3)

2007 (4)

2006 (1)

2004 (1)

S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. 21, 737–750 (2004).
[Crossref]

2002 (2)

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, 85–101 (2002).
[Crossref]

E. Thiébaut, “Optimization issues in blind deconvolution algorithms,” Proc. SPIE 4847, 174–184 (2002).
[Crossref]

1997 (1)

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” Trans. Image Process. 6, 298–311 (1997).
[Crossref]

1995 (1)

1994 (1)

1992 (1)

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

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref] [PubMed]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Berlin) 330, 377–445 (1908).
[Crossref]

Allier, C.

A. Berdeu, B. Laperrousaz, T. Bordy, O. Mandula, S. Morales, X. Gidrol, N. Picollet-D’hahan, and C. Allier, “Lens-free microscopy for 3D + time acquisitions of 3D cell culture,” Sci. Rep. 8, 16135 (2018).
[Crossref]

C. Allier, S. Morel, R. Vincent, L. Ghenim, F. Navarro, M. Menneteau, T. Bordy, L. Hervé, O. Cioni, X. Gidrol, Y. Usson, and J. M. Dinten, “Imaging of dense cell cultures by multiwavelength lens-free video microscopy,” Cytom. Part A 91, 433–442 (2017).
[Crossref]

C. Fournier, F. Jolivet, L. Denis, N. Verrier, E. Thiébaut, C. Allier, and T. Fournel, “Pixel super-resolution in digital holography by regularized reconstruction,” Appl. Opt. 56, 69–77 (2017).
[Crossref]

A. Berdeu, F. Momey, B. Laperrousaz, T. Bordy, X. Gidrol, J.-M. Dinten, N. Picollet-D’hahan, and C. Allier, “Comparative study of fully three-dimensional reconstruction algorithms for lens-free microscopy,” Appl. Opt. 56, 3939–3951 (2017).
[Crossref] [PubMed]

Alvarez, L.

J. F. Jikeli, L. Alvarez, B. M. Friedrich, L. G. Wilson, R. Pascal, R. Colin, M. Pichlo, A. Rennhack, C. Brenker, and U. B. Kaupp, “Sperm navigation along helical paths in 3D chemoattractant landscapes,” Nat. Commun. 6, 7985 (2015).
[Crossref] [PubMed]

Angelini, E.

Atlan, M.

Aubert, G.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” Trans. Image Process. 6, 298–311 (1997).
[Crossref]

Baffou, G.

G. Baffou, P. Bon, J. Savatier, J. Polleux, M. Zhu, M. Merlin, H. Rigneault, and S. Monneret, “Thermal imaging of nanostructures by quantitative optical phase analysis,” Am. Chem. Soc. Nano 6, 2452–2458 (2012).

Barbier, B.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. Can Oztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the Lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

Barlaud, M.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” Trans. Image Process. 6, 298–311 (1997).
[Crossref]

Berdeu, A.

A. Berdeu, B. Laperrousaz, T. Bordy, O. Mandula, S. Morales, X. Gidrol, N. Picollet-D’hahan, and C. Allier, “Lens-free microscopy for 3D + time acquisitions of 3D cell culture,” Sci. Rep. 8, 16135 (2018).
[Crossref]

A. Berdeu, F. Momey, B. Laperrousaz, T. Bordy, X. Gidrol, J.-M. Dinten, N. Picollet-D’hahan, and C. Allier, “Comparative study of fully three-dimensional reconstruction algorithms for lens-free microscopy,” Appl. Opt. 56, 3939–3951 (2017).
[Crossref] [PubMed]

Beunis, F.

Blanc-Feraud, L.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” Trans. Image Process. 6, 298–311 (1997).
[Crossref]

Blusewicz, J. M.

L. A. Philips, D. B. Ruffner, F. C. Cheong, J. M. Blusewicz, P. Kasimbeg, B. Waisi, J. R. McCutcheon, and D. G. Grier, “Holographic characterization of contaminants in water: Differentiation of suspended particles in heterogeneous dispersions,” Water Res. 122, 431–439 (2017).
[Crossref] [PubMed]

Bon, P.

G. Baffou, P. Bon, J. Savatier, J. Polleux, M. Zhu, M. Merlin, H. Rigneault, and S. Monneret, “Thermal imaging of nanostructures by quantitative optical phase analysis,” Am. Chem. Soc. Nano 6, 2452–2458 (2012).

Bordy, T.

A. Berdeu, B. Laperrousaz, T. Bordy, O. Mandula, S. Morales, X. Gidrol, N. Picollet-D’hahan, and C. Allier, “Lens-free microscopy for 3D + time acquisitions of 3D cell culture,” Sci. Rep. 8, 16135 (2018).
[Crossref]

C. Allier, S. Morel, R. Vincent, L. Ghenim, F. Navarro, M. Menneteau, T. Bordy, L. Hervé, O. Cioni, X. Gidrol, Y. Usson, and J. M. Dinten, “Imaging of dense cell cultures by multiwavelength lens-free video microscopy,” Cytom. Part A 91, 433–442 (2017).
[Crossref]

A. Berdeu, F. Momey, B. Laperrousaz, T. Bordy, X. Gidrol, J.-M. Dinten, N. Picollet-D’hahan, and C. Allier, “Comparative study of fully three-dimensional reconstruction algorithms for lens-free microscopy,” Appl. Opt. 56, 3939–3951 (2017).
[Crossref] [PubMed]

Brady, D. J.

Brenker, C.

J. F. Jikeli, L. Alvarez, B. M. Friedrich, L. G. Wilson, R. Pascal, R. Colin, M. Pichlo, A. Rennhack, C. Brenker, and U. B. Kaupp, “Sperm navigation along helical paths in 3D chemoattractant landscapes,” Nat. Commun. 6, 7985 (2015).
[Crossref] [PubMed]

Can Oztürk, O.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. Can Oztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the Lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

Cazier, A.

Charbonnier, P.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” Trans. Image Process. 6, 298–311 (1997).
[Crossref]

Chareyron, D.

Cheong, F. C.

L. A. Philips, D. B. Ruffner, F. C. Cheong, J. M. Blusewicz, P. Kasimbeg, B. Waisi, J. R. McCutcheon, and D. G. Grier, “Holographic characterization of contaminants in water: Differentiation of suspended particles in heterogeneous dispersions,” Water Res. 122, 431–439 (2017).
[Crossref] [PubMed]

Choi, K.

Cioni, O.

C. Allier, S. Morel, R. Vincent, L. Ghenim, F. Navarro, M. Menneteau, T. Bordy, L. Hervé, O. Cioni, X. Gidrol, Y. Usson, and J. M. Dinten, “Imaging of dense cell cultures by multiwavelength lens-free video microscopy,” Cytom. Part A 91, 433–442 (2017).
[Crossref]

Colin, R.

J. F. Jikeli, L. Alvarez, B. M. Friedrich, L. G. Wilson, R. Pascal, R. Colin, M. Pichlo, A. Rennhack, C. Brenker, and U. B. Kaupp, “Sperm navigation along helical paths in 3D chemoattractant landscapes,” Nat. Commun. 6, 7985 (2015).
[Crossref] [PubMed]

Daneshpanah, M.

Denis, L.

Desbiolles, P.

Dinten, J. M.

C. Allier, S. Morel, R. Vincent, L. Ghenim, F. Navarro, M. Menneteau, T. Bordy, L. Hervé, O. Cioni, X. Gidrol, Y. Usson, and J. M. Dinten, “Imaging of dense cell cultures by multiwavelength lens-free video microscopy,” Cytom. Part A 91, 433–442 (2017).
[Crossref]

Dinten, J.-M.

Donati, L.

M. Unser, E. Soubies, F. Soulez, M. McCann, and L. Donati, “GlobalBioIm: A Unifying Computational Framework for Solving Inverse Problems,” Imaging Appl. Opt. CTu1B.1 (2017).

Fatemi, E.

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

Faure, N.

Feizi, A.

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Reports 6, 37862 (2016).
[Crossref]

Fessler, J. A.

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Supplementary Material (1)

NameDescription
» Visualization 1       Reconstruction of 208 holograms of evaporating droplets with a joint reconstruction algorithm combining parametric and non-parametric approaches.

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Figures (5)

Fig. 1
Fig. 1 (a) General scheme of in-line holography: a coherent light source produces a coherent incident wavefront Uinc that is scattered by the sample, producing a diffracted wavefront Udif. These wavefronts interfere on the sensor plane that records the resulting intensity Itot = |Uinc + Udif|2. Note that optional beam shaping optics or imaging optics to magnify the image are not depicted. (b) Scheme of the proposed model in the case of an evaporating droplet: the object to be reconstructed is composed of two subparts. The wave diffracted by the spherical droplet is given by the Mie model, which is an analytic parametric solution of the diffraction equations. The vapor wake is described by a phase transmittance plane whose diffracted wave is propagated according to the Rayleigh-Sommerfeld theory. Based on the linearity of the equations of the diffraction, these two subparts interfere to create the total wave diffracted by the object: U dif = U dif p + U dif np.
Fig. 2
Fig. 2 Reconstruction of a simulated hologram with the proposed method. (a) Simulated intensity produced by the synthetic object. (b) Theoretical projection of the phase shift (radians) induced by the synthetic object along the line of sight. The droplets are masked. (c) Intensity of the hologram due to the reconstructed parametric part of the model. (d) Phase shift (radians) reconstructed by the non-parametric part of the model. (e) Residues IdcIs (u, v) of the model. (f) Evolution of the first (full line) and second (dashed line) droplet parameters along the iterations: the position (xd, yd), the distance (zd) and the radius (rd). The parameter values χ are normalized by their initial and final values: χ ˜ = χ χ f χ i χ f. After the 10th iteration, they stop evolving. Insets (a): pixelated representation of the droplets reconstructed by the parametric model. Insets (b): zooms on the framed regions of interest. Insets (c): zooms on the difference between the reconstructed (d) and theoretical (b) phases (radians) in the two regions of interest. Insets (d): zooms on the framed regions of interest. Insets (e): zooms on the framed regions of interest.
Fig. 3
Fig. 3 Reconstruction of an in-line hologram of evaporating droplets (see Visualization 1). (a) Residues IdcIs (u, v) of the reconstruction. (b) Absorption of the sample reconstructed by the parametric part of the model. (c) Phase shift induced by the vapor wake and the surrounding flow (radians) reconstructed by the non-parametric part of the model. (d) Original raw hologram. (e,h) Intensity and phase (radians) predicted by the parametric part of the model. (f,i) Intensity and phase (radians) predicted by the non-parametric part of the model. (g) Retrieved phase (radians) on the sensor plane. (j,k,l,m,n,o) are zooms on the regions of interest framed in (a,d,b,e,c,f).
Fig. 4
Fig. 4 Comparison of a multi-z reconstruction with the proposed method (a,c,e,f) and a mono-z reconstruction (b,d,g,h). (a,b) Reconstructed phase shifts (radians). (c,d) Residues IdcIs (u, v). (e,f) Zooms on the regions of interest framed in red and green in (a). (g,h) Zooms on the regions of interest framed in red and green in (b).
Fig. 5
Fig. 5 Scheme of the multi-z propagation model. (a,b) v 1 = 2 π λ 1 contains the global phase shift produced by the 3D sample according to the optically thin medium hypothesis as well as the first droplet shadow footprint. (a,c) The Nd − 1 other patches are placed at their corresponding zd and are null except in the vicinity of the droplet concerned, on the support of W r d. (a,d) The resulting phase shift introduced by the sample is: v tot = d = 1 N d 2 π λ d. (e) Weighting matrix d = 1 N d W r d that makes it possible to combine edge-preserving regularization around the droplets (w = 1, white) and smoothing regularization on the background (w = 0, black). In the transition areas, the two regularizations are combined. (f) Profile of the weighting coefficients drawn along the dashed line of (e) on the first droplet. The droplet radius r1 is emphasized as are the boundaries defined by ηinr1 and ηoutr1, linked by a power law.

Tables (3)

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Algorithm 1 Hologram reconstruction by a joint diffraction model

Tables Icon

Table 1 Percentage errors on the two droplets parameters ud compared to their theoritical values u d th relative to the pixel pitch p = 20 μm for the positions xd and yd and the radii rd and relative to the theoretical values z d th for the distances. In red, error on the first estimate u d i. In blue error on the final estimate u d f.

Tables Icon

Table 2 Table of the parameters u d f of the eight droplets at the final iteration as well as the difference in their estimation u d i at the initial iteration of the proposed method.

Equations (31)

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I s ( u , v ) = | U inc + U dif p ( u ) + U dif np ( v ) | 2 .
( u ^ , v ^ ) = argmin u , v 𝒞 data ( I d , I s ( u , v ) , W ) ,
𝒞 data ( I d , I s , W ) = min c i , j [ W ] i , j ( [ I d ] i , j c [ I s ] i , j ) 2 ,
v ^ = argmin v 𝒟 ( v ) 𝒞 data ( I d , I s ( u , v ) , W ) + r μ r 𝒞 reg r ( v ) ,
n cloud ( r ) n air = ( n s n air ) e r r d σ ,
I sim = c th ( I th + ) .
U inc ( z ) = e i k z ,
E dif θ = E 0 k r n = 0 c n p w [ a n ξ n ( k r ) τ n ( cos θ ) cos φ i b n ξ n ( k r ) π n ( cos θ ) sin φ ] ,
U dif p ( u ) = d = 1 N d U dif Mie ( u d ) ,
u argmin u = { u d } d [ [ 1 , N d ] ] 𝒞 data ( I d , | 1 + U dif np ( v ) + d = 1 N d U dif Mie ( u d ) | 2 , W ) .
U dif RS ( t 2 D , z ) = h z ( x , y ) * [ U inc ( z ) × ( t 2 D ( x , y ) 1 ) ] = e i k z h z ( x , y ) * ( t 2 D ( x , y ) 1 ) ,
h z ( r ) = 1 i λ z r ( 1 1 i k r ) e i k r r ,
𝔪 ( ) = 0 ( i , j ) , [ ] i , j .
𝒞 reg 1 ( M , W ) = i , j [ W ] i , j [ ( [ M ] i + 1 , j [ M ] i , j ) 2 + ( [ M ] i , j + 1 [ M ] i , j ) 2 ] ,
𝒞 reg 2 ( M , W , ) = i , j [ W ] i , j ( [ M ] i + 1 , j [ M ] i , j ) 2 + ( [ M ] i , j + 1 [ M ] i , j ) 2 + 2 ,
U dif np ( v = { v d } d [ [ 1 , N d ] ] ) = d = 1 N d U dif RS ( e i v d , z d ) ,
v argmin v 𝒟 ( v = { v d } d [ [ 1 , N d ] ] ) 𝒞 data ( I d , | 1 + U dif p ( u ) + d = 1 N d U dif RS ( e i v d , z d ) | 2 , W ) + μ 1 𝒞 reg 1 ( d = 1 N d v d , 1 d = 1 N d W r d ) + μ 2 𝒞 reg 2 ( d = 1 N d v d , d = 1 N d W r d , ) ,
𝒟 ( v ) = { v = { v d } d [ [ 1 , N d ] ] / ( i , j ) , { d , [ v d ] i , j d > 1 , [ v d ] i , j = 0 if [ W r d ] i , j = 0 } ,
c ^ = argmin c i , j [ W ] i , j ( [ I d ] i , j c [ I s ] i , j ) 2 = i , j [ W ] i , j [ I d ] i , j [ I s ] i , j i , j [ W ] i , j [ I s ] i , j 2 .
[ I d ] d N p ; c [ I s ( u , v ) ] m ( v ) N p ; { v d } d [ [ 1 , N d ] ] v N d × N z ,
i , j [ W ] i , j ( [ I d ] i , j c [ I s ] i , j ) 2 𝒞 ( v ) ( m ( v ) d ) t W ( m ( v ) d ) .
𝒞 ( v ) = a ( | b ( v ) | 2 ) d W 2 = ( a ( | b ( v ) | 2 ) d ) t W ( a ( | b ( v ) | 2 ) d ) ,
𝒞 ( v + δ v ) = a ( | b ( v + δ v ) | 2 ) d W 2 a ( | b ( v ) + B δ v | 2 ) d W 2 a ( | b ( v ) | 2 + 2 𝔢 ( diag ( b ¯ ( v ) ) B δ v ) ) d W 2 m ( v ) + A 2 𝔢 ( diag ( b ¯ ( v ) ) B δ v ) d W 2 𝒞 ( v ) + 2 ( m ( v ) d ) t W ( A 2 𝔢 ( diag ( b ¯ ( v ) ) B δ v ) ) 𝒞 ( v ) + 4 𝔢 ( ( m ( v ) d ) t W A diag ( b ¯ ( v ) ) BM ) δ v ,
v = ( 𝔢 ( v ) 𝔪 ( v ) ) and M = ( 𝕀 n i 𝕀 n ) .
𝒞 ( v ) = 4 [ 𝔢 ( ( m ( v ) d ) t W A diag ( b ¯ ( v ) ) BM ) ] t = 4 𝔢 ( [ ( m ( v ) d ) t W A diag ( b ¯ ( v ) ) BM ] ) = 4 𝔢 ( M B diag ( b ( v ) ) A t W ( m ( v ) d ) ) .
𝔢 ( M z ) = 𝔢 ( ( 𝕀 n i 𝕀 n ) ( 𝔢 ( z ) + i 𝔪 ( z ) ) ) = ( 𝔢 ( z ) 𝔪 ( z ) ) .
{ 𝒞 ( v ) | 𝔢 ( v ) = 𝔢 ( 4 B diag ( b ( v ) ) A t W ( m ( v ) d ) ) 𝒞 ( v ) | 𝔪 ( v ) = 𝔪 ( 4 B diag ( b ( v ) ) A t W ( m ( v ) d ) ) .
𝒞 ( v ) = 4 𝒥 b ( v ) × diag ( b ( v ) ) × 𝒥 a t ( | b ( v ) | 2 ) × W × ( m ( v ) d ) .
a = c ^ 𝕀 N p x n , 𝒥 a t ( x ) = c ^ 𝕀 N p .
b ( v = { v d } d [ [ 1 , N d ] ] ) = 1 + U dif p ( u ) + d = 1 N d e i k z d h z d * ( e i v d 1 ) ,
𝒥 b ( v ) | d ( P ) = i e i k z d e i v d ¯ . ( h z d ¯ * P ) ,