CN114693828A - Fourier laminated imaging reconstruction method based on alternating direction multiplier method - Google Patents

Fourier laminated imaging reconstruction method based on alternating direction multiplier method Download PDF

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CN114693828A
CN114693828A CN202210362791.7A CN202210362791A CN114693828A CN 114693828 A CN114693828 A CN 114693828A CN 202210362791 A CN202210362791 A CN 202210362791A CN 114693828 A CN114693828 A CN 114693828A
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潘安
王爱业
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XiAn Institute of Optics and Precision Mechanics of CAS
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Abstract

The invention discloses a Fourier laminated imaging technology based on an Alternating Direction Multiplier Method (ADMM). The fourier stack imaging (FPM) technique can be essentially reduced to a non-convex optimization problem, and high-quality image reconstruction cannot be realized in the background of noise interference, which brings a challenge to further increase of imaging flux. The method introduces an optimization algorithm ADMM into the FPM technology, and finally realizes stable convergence and robust image reconstruction by appropriately deforming and setting appropriate parameters to be matched with a system model of the FPM, thereby providing an example for the migration application of the ADMM algorithm to other fields. Compared with the existing FPM reconstruction algorithm, the method has more excellent anti-noise performance, and is particularly suitable for application scenes of low-cost and low-performance optical imaging devices. In addition, the method has good compatibility and expansibility, and is expected to realize high-efficiency high-resolution image reconstruction of the special biological sample by the FPM.

Description

Fourier laminated imaging reconstruction method based on alternating direction multiplier method
Technical Field
The invention belongs to the technical field of optical information acquisition and processing, and particularly relates to an alternating direction multiplier algorithm for improving the image reconstruction quality of a Fourier laminated imaging technology under a noise background, which can be used for making up the defects of an optical imaging device with low cost and low performance on the imaging quality, and simultaneously, the compatibility of the method also enables the high-resolution imaging of a special biological sample to be possible.
Background
The accurate English name of Fourier transform tomography (FPM) technology is named as Fourier transform graphics (FPM) technology, and the FPM technology is invented by Yang et al, California institute of technology, U.S. A, in 2013 and reported in (Zheng G, Horstmeyer R, Yang C.wide-field, high-resolution Fourier transform graphics [ J)].Nature Photonics,2013,7(9):739-745.]There are many names for chinese translation such as fourier ptychographic imaging, fourier stacked microscopy imaging, fourier stacked imaging, etc., and the term fourier stacked imaging is used herein in a unified way. The FPM technology is a promising new generation of computational optical imaging technology, and improves the original Abbe far-field diffraction limit formula lambda/2 NA into lambda/(NA)illu+NAobj) Wherein NA isilluAnd NAobjThe illumination and objective Numerical Apertures (NA) are represented, respectively, and are written in the fourier optical introduction of the professor Goodman (fourth edition). Compared with the traditional bright field microscopy, the FPM technology combines the optical phase recovery technology and the microwave synthetic aperture technology, can realize high resolution and quantitative phase imaging while realizing a large field of view by using a low NA objective lens, and further can realize higher imaging flux.
FPM is essentially a non-convex optimization problem, and therefore cannot ensure an optimal global solution under noise interference conditions, resulting in a degradation of the reconstruction quality of the image. The reconstruction algorithm occupies an extremely important position in the FPM technology, and the selection of the reconstruction algorithm with high robustness can effectively eliminate adverse effects caused by noise. Thanks to the rapidly developing computing technology, considerable progress has been made in the study of the FPM reconstruction algorithm, the results of which include the differential mapping [ Thibault P, Dierolf M, Bunk O, et al, Probe reconstruction in the statistical coherent genetic differential imaging [ J ]. Ultramicrosphope, 2009,109(4): 338-343., Vickers flow [ Bian L, Suo J, Zheng G, et al, Fouer reconstruction in the statistical flow [ J ] Optic Express, 23(4): 4856-4866 ], Gaussian Newton [ Yeh L ] -H, Dong J, Zhong J, et al, Experimental prediction of the genetic reconstruction [ J ] 207J. ], Gaussian L. -H, Dong J, Experimental prediction J, [ 207J ] (24. J.) ], Gaussian rule J. 20721, rule J. ], Fourier reconstruction in the statistical reconstruction algorithm [ 24. 12 ] 24, J. (24-rule J.) (24) and J. (12. sub-rule J. ], sub. 12. sub. sub.sub. sub. sub., Convex relaxation [ Horstmeyer R, Chen R. Y, Ou X, et al. Solving ptychographic with a covex relaxation [ J ] New Journal of Physics,2015,17(5):053044 ]. The above algorithms can be divided into two categories: global gradient algorithms and sequence gradient algorithms. Generally speaking, the global gradient algorithm has an advantage in algorithm robustness, and the sequence gradient algorithm has a more excellent performance in algorithm operation efficiency. The above algorithms can be further classified into first order algorithms and second order algorithms according to different ways of solving the objective function. The second-order algorithm is based on the second derivative solution of the objective function, so that the algorithm complexity is higher, but the convergence rate is higher, and the algorithm robustness is better. The classification of the typical algorithm for FPM phase recovery is visualized in fig. 2. At present, the most widely used reconstruction algorithm in the FPM technology is the second order sequence gauss-newton method, which can achieve higher quality image reconstruction with lower time cost. In 2017, Maiden et al introduced the momentum concept in the field of machine learning on the basis of the original PIE algorithm [ Rodenburg J.M, and Faulkner H.M.L.A phase retrieval algorithm for shifting the exercise [ J ] Applied Physics Letter,2004,85(20): 4795-.
The Alternative Direction Multiplier Method (ADMM) [ Gabay D and Mercier B.A dual algorithm for the solution of nonlinear variable schemes via fine elements schemes with schemes [ J ] Computers & Mathematics with Applications,1976,2: 17-40 ] originated in the last 50 th century and was formally proposed in the middle of the 70 th century. For half a century, extensive and intensive research has been conducted into the ADMM methodology and variants thereof. Research finds that the ADMM method is very suitable for the distributed convex optimization problem. The ADMM method has been widely used in both fields, because many of the problems to be solved in statistics and machine learning can be generalized under the framework of convex optimization, such as collection and storage of large-scale data sets. The method adopts a special 'decomposition-coordination' structural form, decomposes the original optimization problem into a plurality of subproblems connected by an augmented Lagrange function by introducing new variables, and then coordinates the local solutions of the subproblems, thereby obtaining the global optimal solution of the original optimization problem.
Disclosure of Invention
The invention aims to provide a Fourier laminated imaging reconstruction method based on an alternating direction multiplier method, which has better algorithm robustness in the background of noise interference and can realize higher-quality image reconstruction compared with the existing algorithm.
The technical scheme adopted by the invention is that the relevant theory of a convex optimization algorithm ADMM widely applied in the field of application statistics and machine learning is transferred to an FPM imaging system, and the method is successfully applied to the non-convex optimization problem of FPM phase recovery by selecting proper algorithm parameters.
The ADMM method for the FPM imaging system has many differences from the standard ADMM algorithm, and is mainly embodied in two aspects: 1) the method is actually a variant form of a standard ADMM algorithm, and intermediate variables selected by the standard ADMM algorithm and the ADMM algorithm are different but follow similar structural forms and algorithm ideas; 2) the iteration cutoff condition of the method performs certain equivalent transformation and simplification operation relative to the standard ADMM algorithm, so that the method is more suitable for solving the FPM non-convex optimization problem.
In order to realize the technical task, the invention adopts the following technical scheme to realize:
a Fourier laminated imaging reconstruction method based on an alternating direction multiplier method comprises the following steps:
step 1, introducing a pupil function constraint intermediate variable, and constructing a phase recovery image reconstruction model;
step 2: acquiring the distribution condition of the LED illumination array and a low-resolution intensity image set of the object to be detected through an optical imaging system, and then initializing an object function of the object to be detected and a pupil function of the imaging system;
and step 3: inputting the object function and the pupil function initialized in the step 2 into a phase recovery image reconstruction model, and further solving a pupil constraint value, complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition to realize FPM high-resolution image reconstruction;
and 4, step 4: calculating an original error and a dual error, and measuring the deviation between the FPM high-resolution image reconstruction result and the image to be restored by adopting the original error; evaluating the deviation between the pupil constraint value, the complex amplitude distribution of the image, the pupil function of the optical imaging system, the combination of the pupil constraint optimization conditions and the numerical value obtained by the phase recovery image reconstruction model by adopting dual errors;
and 5: and setting an original error and a dual error judgment tolerance, outputting an image reconstruction result when the error tolerance meets the set requirement, and returning to the step 3 until outputting the image reconstruction meeting the error set requirement when the error tolerance does not meet the set requirement.
Further, step 1 introduces pupil function constraint intermediate variables as:
qj＝diag(p)Qjs
where diag (-) is the mathematical operator of the matrix multiplication operation, p denotes the pupil function of the optical system, QjRepresenting a sampling matrix corresponding to a jth LED illumination position, and s representing an object to be measured;
the phase recovery image reconstruction model is as follows:
Figure BDA0003584614930000051
wherein "is a symbolic representation of" defined as ",
Figure BDA0003584614930000052
representing two-dimensional inverse Fourier transform, | · | | non-conducting phosphor22-norm operation, q, representing a matrixjIntermediate variables are constrained for the pupil function,
Figure BDA0003584614930000053
the image set is a low-resolution intensity image set of the object to be measured, s represents the object to be measured, N represents the number of LEDs in the lighting array, delta is the expression of a plane light field without any phase information in a frequency domain, and gamma is used for measuring the strength of the applied regularization term.
In step 3, the pupil constraint value solving method is as follows:
Figure BDA0003584614930000054
the complex amplitude distribution of the image is solved as follows:
Figure BDA0003584614930000055
the optical imaging system pupil function solving method is as follows:
Figure BDA0003584614930000056
the pupil constraint optimization condition is solved as follows:
Figure BDA0003584614930000057
in the above four formulae, qj，s，p，λjAnd respectively representing a pupil constraint value, the complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition, wherein the carried superscript indicates the current iteration number.
In step 3, the iterative process of pupil constraint value solving is as follows:
Figure BDA0003584614930000061
the complex amplitude distribution of the image is solved by an iterative process:
Figure BDA0003584614930000062
the optical imaging system pupil function solving iterative process:
Figure BDA0003584614930000063
and (3) solving iterative process of pupil constraint optimization conditions:
Figure BDA0003584614930000064
ωj＝λj/α
wherein
Figure BDA0003584614930000065
And
Figure BDA0003584614930000066
respectively representing a two-dimensional fourier transform and its inverse,
Figure BDA0003584614930000067
representing the complex conjugate of the matrix; qjAnd
Figure BDA0003584614930000068
respectively representing the sampling matrix and its inverse corresponding to the jth LED illumination location,
Figure BDA0003584614930000069
for low-resolution intensity image sets of the object to be measured, qj，s，p，ωjRespectively representing a pupil constraint value, complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition, wherein a carried superscript indicates the current iteration number; n denotes the number of LEDs in the illumination array, δ is the representation of the planar light field in the frequency domain without any phase information, and a penalty parameter α is used to describe the pupil constraint intermediate variableThe accuracy of the definition, the introduction of a parameter beta for stabilizing the updating process of the pupil function to prevent extreme values, the parameter gamma for measuring the strength of the applied regularization term, and the parameter eta for the step size of the algorithm updating.
And 4, calculating the original error and the dual error respectively as follows:
Figure BDA00035846149300000610
Figure BDA0003584614930000071
wherein
Figure BDA0003584614930000072
Respectively representing two-dimensional inverse Fourier transform, | ·| non-woven phosphor22-norm operations representing matrices;
Figure BDA0003584614930000073
for low-resolution intensity image sets of the object to be measured, qj，λjRespectively representing a pupil constraint value and a pupil constraint optimization condition; n represents the number of LEDs in the illumination array, a penalty parameter alpha is used for describing the accuracy degree of the definition of a pupil constraint intermediate variable, a parameter eta is the step size of the algorithm updating, and superscripts carried by all variables indicate the current iteration number.
In step 5, the conditions for judging tolerance of the original error and the dual error are set as follows:
Figure BDA0003584614930000074
Figure BDA0003584614930000075
wherein EpAnd EdRespectively representing the original error and the dual error, the superscript indicating the current iteration number, epsilontolIs a preset error tolerance value.
After the initialization of the object function of the object to be detected and the pupil function of the imaging system in the step 2 is completed, the interference of the object function and the pupil function of the object to be detected in crosstalk prevention is also included, and the interference prevention process is as follows:
Figure BDA0003584614930000076
wherein "is a symbol representing" defined as "and" diag (-) is a mathematical operator of a matrix multiplication operation,
Figure BDA0003584614930000077
representing two-dimensional inverse Fourier transform, | · | | non-conducting phosphor22-norm operations representing matrices; p denotes the pupil function of the optical system, QjRepresenting a sampling matrix corresponding to the jth LED illumination location,
Figure BDA0003584614930000078
the image set is a low-resolution intensity image set of the object to be measured, s represents the object to be measured, N represents the number of LEDs in the lighting array, delta is the expression of a plane light field without any phase information in a frequency domain, and gamma is used for measuring the strength of the applied regularization term.
The invention also includes a processor and a memory;
the memory is used for storing;
the processor is used for executing the Fourier laminated imaging reconstruction method based on the alternating direction multiplier method through calling.
The technical solution of the present invention may also be a computer program and/or an instruction to be executed by a processor, where the computer program and/or the instruction to be executed by the processor perform the fourier transform tomography reconstruction method based on the alternating direction multiplier method according to the present invention.
Compared with the prior art, the invention has the following beneficial technical effects:
1) the FPM reconstruction algorithm with algorithm robustness under the noise interference background is provided, and stable image reconstruction and good anti-noise performance can be provided; 2) compared with the traditional global algorithm, the global optimization algorithm with high parallelism is higher in efficiency and can run efficiently on a GPU and a distributed CPU; 3) the method has compatibility and expansibility, provides possibility for high-quality image reconstruction of a special sample, can extend richer variant algorithms, and is expected to further improve the imaging quality of the FPM technology; 4) the method is an example of the ADMM algorithm for solving the optimization problem in other fields, and can provide reference for migration application of the ADMM algorithm to other fields.
Drawings
FIG. 1 is a flow chart of a method of the present invention;
FIG. 2 is a classification of a typical FPM reconstruction algorithm of the background art;
FIG. 3 is a flow chart of an ADMM reconstruction algorithm applied to an FPM technology model;
fig. 4 is a comparison of image reconstruction results under noise-free conditions. (fig. 4a1, fig. 4a2) are plots of the change in amplitude SSIM and phase SSIM, respectively, as a function of the number of iterations of the ADMM algorithm, (fig. 4b 1-fig. 4b3) are the true values of the amplitude, phase and pupil functions, respectively, for the FPM phase recovery simulation experiment, (fig. 4c 1-fig. 4c3), (fig. 4d 1-fig. 4d3) and (fig. 4e 1-fig. 4e3) are the image reconstruction results using the gauss-newton algorithm, the mPIE algorithm and the ADMM algorithm, respectively;
fig. 5 is a comparison of image reconstruction results under gaussian noise conditions. (fig. 5a1, fig. 5a2) are respectively curves of the amplitude SSIM and the phase SSIM of the reconstructed image by the ADMM algorithm varying with gaussian noise of different degrees, (fig. 5b) is a comparison between the amplitude and the phase of the reconstructed image under the condition of 70% gaussian noise, wherein (fig. 5b 1-fig. 5b4) are respectively enlarged partial areas of phase true values and phase recovery results of three reconstruction algorithms;
fig. 6 is a comparison of image reconstruction results under poisson noise conditions. (fig. 6a1, fig. 6a2) are respectively the curves of the amplitude SSIM and the phase SSIM of the reconstructed image by the ADMM algorithm with the poisson noise of different degrees, and (fig. 6b) is that the noise parameter σ is 10-4The contrast between the amplitude and the phase of the reconstructed image under the Poisson noise condition is obtained;
FIG. 7 is a statistical result of the iteration number and the running time of the algorithm in the three sets of simulation experiments;
FIG. 8 is a comparison of USAF resolution plate reconstructed images at different gains, where (FIG. 8a) is 20dB gain and (FIG. 8b) is 30dB gain;
FIG. 9 is a comparison of reconstructed images of epidermal cell sections of onion squash leaves at different gains, wherein (FIG. 9a) shows a gain of 20dB and (FIG. 9b) shows a gain of 30 dB.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
Referring to fig. 1, the present invention provides a fourier stack imaging reconstruction method based on an alternating direction multiplier method, which includes the following steps:
step 1, introducing a pupil function constraint intermediate variable, and constructing a phase recovery image reconstruction model;
step 2: acquiring the distribution condition of the LED illumination array and a low-resolution intensity image set of the object to be detected through an optical imaging system, and initializing an object function of the object to be detected and a pupil function of the imaging system;
and step 3: inputting the object function and the pupil function initialized in the step 2 into a phase recovery image reconstruction model, and further solving a pupil constraint value, complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition to realize FPM high-resolution image reconstruction;
and 4, step 4: calculating an original error and a dual error, and measuring the deviation between the FPM high-resolution image reconstruction result and the image to be restored by adopting the original error; evaluating the deviation between the pupil constraint value, the complex amplitude distribution of the image, the pupil function of the optical imaging system, the combination of the pupil constraint optimization conditions and the numerical value obtained by the phase recovery image reconstruction model by adopting dual errors;
and 5: and (4) setting an original error and a dual error judgment tolerance, outputting an image reconstruction result when the error tolerance meets the set requirement, and returning to the step (3) until outputting the image reconstruction meeting the error set requirement when the error tolerance does not meet the set requirement.
Based on the basic flow of the FPM technology, Fourier transformation of an airspace object is firstly sampled by an illumination matrix, then is constrained by a pupil function, and finally is subjected to inverse Fourier transformation to form a series of low-resolution intensity images in a spatial domain, and the optimization problem of blind phase recovery is solved by initializing an object function of the object to be detected and the pupil function of an imaging system, as shown in formula 1.
Figure BDA0003584614930000101
Wherein "is a symbol representing" defined as "and" diag (-) is a mathematical operator of a matrix multiplication operation,
Figure BDA0003584614930000102
representing two-dimensional inverse Fourier transform, | · | | non-conducting phosphor22-norm operations representing matrices; p denotes the pupil function of the optical system, QjRepresenting a sampling matrix corresponding to the jth LED illumination location,
Figure BDA0003584614930000103
and the low-resolution intensity image set of the object to be detected is obtained, s represents the object to be detected, and N represents the number of LEDs in the lighting array.
In order to prevent crosstalk interference between an object function and a pupil function of an object to be measured, more stable optimization is realized in subsequent operation, and the crosstalk interference prevention interference process is as follows: a regularization term with prior information is added on the basis of the formula 1, as shown in the formula 2.
Figure BDA0003584614930000111
Wherein "is a symbol representing" defined as "and" diag (-) is a mathematical operator of a matrix multiplication operation,
Figure BDA0003584614930000112
representing two-dimensional inverse Fourier transform, | · | | non-conducting phosphor22-norm operations representing matrices; p denotes the pupil function of the optical system, QjRepresenting a sampling matrix corresponding to the jth LED illumination location,
Figure BDA0003584614930000113
the image set is a low-resolution intensity image set of the object to be measured, s represents the object to be measured, N represents the number of LEDs in the lighting array, delta is the expression of a plane light field without any phase information in a frequency domain, and gamma is used for measuring the strength of the applied regularization term. The aim of crosstalk prevention intervention is that for weak phase objects, adding this regularization term can effectively avoid crosstalk between object functions and pupil functions.
The key innovation of the technical scheme of the invention is that a pupil function constraint intermediate variable q is introduced firstlyj＝diag(p)Qjs, thereby transforming the optimization problem to be solved into:
Figure BDA0003584614930000114
wherein is a symbol representation of "defined as",
Figure BDA0003584614930000115
representing two-dimensional inverse Fourier transform, | · | | non-conducting phosphor22-norm operation, q, representing a matrixjIntermediate variables are constrained for the pupil function,
Figure BDA0003584614930000116
the image set is a low-resolution intensity image set of the object to be measured, s represents the object to be measured, N represents the number of LEDs in the lighting array, delta is the expression of a plane light field without any phase information in a frequency domain, and gamma is used for measuring the strength of the applied regularization term.
The technical scheme of the invention introduces an intermediate variable qjThe image reconstruction problem is further converted into a collaborative iteration problem for solving four variables of a pupil constraint value, the complex amplitude distribution of the image, a pupil function of an optical imaging system and a pupil constraint optimization condition, namely, the augmented LagrangeThe function definition is expressed as:
Figure BDA0003584614930000117
Figure BDA0003584614930000121
wherein diag (·) is the mathematical operator of the matrix multiplication operation, | | · | | | computationally22-norm operations representing matrices; p denotes the pupil function of the optical system, QjRepresenting a sampling matrix corresponding to the jth LED illumination location,
Figure BDA0003584614930000122
the image set is a low-resolution intensity image set of the object to be measured, s represents the object to be measured, N represents the number of LEDs in the lighting array, delta is the expression of a plane light field without any phase information in a frequency domain, and gamma is used for measuring the strength of the applied regularization term. The existing image reconstruction is only completed by means of independent and respective iterations of complex amplitude distribution of an image and a pupil function of an optical imaging system.
Further, the pupil constraint value, the complex amplitude distribution of the image, the pupil function of the optical imaging system, the pupil constraint optimization condition referred to in equation 4 are calculated. The specific solving process is as follows:
the pupil constraint value solving method is as follows:
Figure BDA0003584614930000123
solving mode of complex amplitude distribution of image:
Figure BDA0003584614930000124
solving mode of pupil function of the optical imaging system:
Figure BDA0003584614930000125
the pupil constraint optimization condition is solved as follows:
Figure BDA0003584614930000126
in the above four formulae, qj，s，p，λjAnd respectively representing a pupil constraint value, the complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition, wherein the carried superscript indicates the current iteration number.
Let omegaj＝λjAnd/alpha, the iterative process of solving the pupil constraint value in the step 3 is as follows:
Figure BDA0003584614930000127
the complex amplitude distribution of the image is solved by an iterative process:
Figure BDA0003584614930000131
the optical imaging system pupil function solving iterative process:
Figure BDA0003584614930000132
and (3) solving iterative process of pupil constraint optimization conditions:
Figure BDA0003584614930000133
in the above-mentioned four formulas, the expression,
Figure BDA0003584614930000134
and
Figure BDA0003584614930000135
respectively representing a two-dimensional fourier transform and its inverse,
Figure BDA0003584614930000136
representing the complex conjugate of the matrix; qjAnd
Figure BDA0003584614930000137
respectively representing the sampling matrix and its inverse corresponding to the jth LED illumination location,
Figure BDA0003584614930000138
for low-resolution intensity image sets of the object to be measured, qj，s，p，ωjRespectively representing a pupil constraint value, complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition, wherein the carried superscript indicates the current iteration number; n denotes the number of LEDs in the illumination array, δ is the representation of a planar light field in the frequency domain without any phase information, a penalty parameter α is used to describe the accuracy of the pupil constraint intermediate variable definition, the introduction of a parameter β is used to stabilize the update process of the pupil function to prevent the appearance of extreme values, a parameter γ is used to measure the strength of the applied regularization term, and a parameter η is the step size of the algorithm update.
Normalized error matrix E corresponding to original error and dual errork pAnd Ek dAre respectively defined as:
Figure BDA0003584614930000139
Figure BDA00035846149300001310
wherein
Figure BDA00035846149300001311
Respectively representing two-dimensional inverse Fourier transform, | ·| non-woven phosphor22-norm operations representing matrices;
Figure BDA00035846149300001312
for low-resolution intensity image sets of the object to be measured, qj，λjRespectively representing a pupil constraint value and a pupil constraint optimization condition; n denotes the number of LEDs in the illumination array, the penalty parameter a is used to describe the accuracy of the pupil constraint intermediate variable definition, and the parameter η is the step size of the algorithm update. The superscripts carried by all variables indicate the current number of iterations.
The reasonable condition of the iteration is that the original error and the dual error are converged to zero, and the condition is properly relaxed in the FPM, namely, the iteration is stopped when the normalized error matrixes corresponding to the two errors are converged to a certain stable value, and epsilontolAnd expressing the error tolerance which meets the set requirement as follows:
Figure BDA0003584614930000141
Figure BDA0003584614930000142
wherein EpAnd EdRespectively representing the original error and the dual error, the superscript indicating the current iteration number, epsilontolIs a preset error tolerance value.
The ADMM algorithm flow of the FPM of the present invention is shown in brief with reference to FIG. 3. Wherein the proposed values for the given algorithm parameters are: alpha is from [0.5,1 ]]，β＝1000，γ∈[0.1,0.5]，η＝1，εtol0.001. It is emphasized that this method is actually a variant of the standard ADMM method, differing in the choice of intermediate variables. Furthermore, the derivation of the above described ADMM method is strictly applicable to convex optimization problems. Because of the fourier amplitude constraint, the FPM phase recovery problem remains non-convex in nature, and thus the convergence of the ADMM-FPM algorithm is not guaranteed in theory. Despite this uncertainty, the ADMM algorithm is feasible in practical FPM operation by choosing the appropriate parameters when there is sufficient overlap between different illumination areas.
Examples
In order to verify the feasibility and the correctness of the method, the FPM embodiment adopting the ADMM reconstruction algorithm is respectively verified in a simulation group and an experimental group, and the performance of the FPM embodiment is compared with that of the existing two algorithm sequences, namely a Gauss Newton method and an mPIE algorithm.
The simulation group performs simulation experiments on the same sample under three different conditions of no noise, Gaussian noise and Poisson noise. The sample image size was 128 × 128 pixels, taken by a 15 × 15LED array (4 mm pitch) red illumination 4 × 0.1NA objective, the illumination height of the LED was 86mm, and the pixel size of the image sensor was 6.5 μm. The parameters of the ADMM method are set as follows: α is 0.5, β is 1000, γ is 0.3, and η is 1. The sequence Gauss Newton method adopts a strategy of self-adaptive step length, and the initial step length is set as alpha0＝1，β 01. The start iteration time of the pupil function is delayed appropriately in the mPIE algorithm, setting the parameters to: α is 0.1, β is 0.8, γ is 1, ηobj＝0.9,ηpupil＝0.3,Tpupil15. All three reconstruction algorithms described above use the all 1 matrix as an initial intensity guess, and use the NA sequence to update the sub-apertures in the fourier domain, i.e. gradually expanding outward from the center of the fourier domain. Structural Similarity Index (SSIM) [ Wang Z, Bovik A, Sheikh H, et al. image Quality Association: From Error Visibility to Structural Similarity [ J ]]IEEE Transactions on Image Processing,2004,13(4):600–612.]The method is an objective index for evaluating the image reconstruction quality, the value range of the index is between 0 and 1, and the larger the value is, the more similar structural information of the two images is shown. SSIM is defined by the formula:
Figure BDA0003584614930000151
wherein muxAnd σxRespectively representing the mean and standard deviation (square root of variance), σ, of the image vector xxyRepresenting the correlation coefficient between the two image vectors x and y. The addition of the constants C1 and C2 can avoid instability that occurs when these statistics take values very close to zero. Because the bright field image is concentrated to be largePart of the signal power, the dark field image from the high angle illumination is more susceptible to noise. Therefore, different degrees of gaussian noise and poisson noise are set for the dark field low resolution image in the simulation group to simulate photon shot noise and readout noise in the imaging process. Where gaussian noise is defined in percentage terms and poisson noise is determined from the value of the noise parameter σ (which is inversely related to the poisson noise level).
The simulation experiment result under the noise-free condition is shown in fig. 4, and the ADMM algorithm can realize convergence and stabilize reconstructed image information and correct system aberration along with the update of the pupil function. Although the convergence speed is slightly slow, the convergence to a higher reconstruction quality in dozens of iterations is acceptable in practical use.
The simulation experiment result under the gaussian noise condition is shown in fig. 5, and when the interference of the gaussian noise is severe (more than 50%), the ADMM algorithm can provide better image reconstruction quality. The image reconstruction result and the local amplification result of the phase reconstruction image under the background of 70% of Gaussian noise show that the ADMM has a smoothing effect and can effectively remove noise points and information crosstalk in the image background.
As shown in fig. 6, when the value of the noise parameter σ approaches 0.01 to the right, the quality of image reconstruction is substantially equal to that in the noise-free condition. When noise parameter σ<10-4The ADMM method is more advantageous in terms of noise immunity. Noise parameter σ 10-4The image reconstruction result under the poisson noise background of the invention shows that although the image reconstruction quality of the ADMM method is not ideal from the subjective point of view, the ADMM method can provide more excellent numerical results from the point of view of objective evaluation indexes.
The iteration times and the operation time of the algorithm in the three sets of simulation experiments are shown in fig. 7, wherein the noise levels in the two sets of noise experiments respectively correspond to 70% of gaussian noise in fig. 5 and the noise parameter σ of 10 in fig. 6-4Poisson noise of (a). Overall, the time cost of the ADMM method is greater than that of the other two algorithms, mainly due to its more complex algorithm structure. ADMM method as a full with high parallelismThe local algorithm needs to complete the updating of four variables in turn in an iterative cycle, and completes the calculation and judgment of two indexes at the end of the cycle, while the other two algorithms only need to update two variables and judge one index. The ADMM method can sacrifice less time efficiency to obtain higher image reconstruction quality, thereby having the advantages of a sequence gradient algorithm and a global gradient algorithm.
Experiment groups 20dB and 30dB of experiments under different gain conditions are respectively carried out on the resolution ratio plate and the biological samples. The experimental FPM platform used a programmable 32 × 32LED array (adahurit, controlled by Arduino), placed 68mm above the sample, sequentially illuminating the central 9 × 9 LEDs, and collecting experimental data by an 8-bit CCD camera (DMK23G445, Imaging Source inc., Germany, pixel size 3.75 μm,1280 × 960 pixels). In order to simulate a use scene of a low-cost low-performance image sensor, a camera gain is artificially added in the data acquisition process, so that a low-resolution intensity image with higher brightness and more noise is obtained. Meanwhile, the exposure time of the camera should be properly reduced to prevent the occurrence of the overexposure phenomenon.
The reconstruction results of the USAF resolution plate experiments are shown in fig. 8, which shows an image of a 310 × 310 pixel size region extracted from a full-field-of-view image, including all elements of the 6 th to 9 th groups of resolution plates. The reconstruction result of the ADMM method provides uniform and smooth background information as well as line structure. In contrast, the reconstruction results of the other two algorithms are cluttered with dense noise and become increasingly significant as the gain increases (the noise level increases).
The reconstruction result of the biological sample (onion scale leaf epidermal cell) experiment is shown in fig. 9, and the image shown is an area with the size of 210 × 210 pixels extracted from the full-field image, and the experimental phenomenon similar to the resolution plate experiment can be observed. The reconstructed images of the other two algorithms gather more obvious noise points around the cell wall of the sample, and the ADMM method can effectively avoid the reduction of the image reconstruction quality caused by the existence of the noise.
The experiment proves that the method generally shows under the noise-free condition, but has obvious potential in the noise resistance aspect, and can sacrifice less time efficiency to obtain higher-quality image reconstruction. In addition, the method adds a regularization term in the expression of the objective function to improve the stability of the reconstruction result of the weak phase sample. According to the properties of the sample to be reconstructed, the method can be well compatible with other different regularization terms. In further research, it is expected that robust image reconstruction results with faster convergence speed can be achieved by creating new ADMM variant methods or more robust parameter selection strategies.
The invention also provides an electronic device, which comprises a processor and a memory;
the memory is used for storing;
the processor is configured to execute, by invocation, the fourier stack imaging reconstruction method based on the alternating direction multiplier method according to any one of the above embodiments.
The present invention also provides a computer program product comprising a computer program and/or instructions which, when executed by a processor, implement the fourier stack imaging reconstruction method based on the alternating direction multiplier method according to any of the above embodiments.
As will be appreciated by one skilled in the art, embodiments of the present invention may be provided as a method, apparatus, or computer program product. Accordingly, the present invention may take the form of an entirely software embodiment, an entirely hardware embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention is in the form of a computer program product that may be embodied on one or more computer-usable storage media having computer-usable program code embodied therewith. And such computer-usable storage media include, but are not limited to: various media capable of storing program codes, such as a usb disk, a portable hard disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a magnetic disk Memory, a Compact Disc Read-Only Memory (CD-ROM), and an optical Memory.
The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products of the invention. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create a system for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including an instruction system which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks. While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the present invention and its equivalent technology, it is intended that the present invention also include such modifications and variations.

Claims (9)

1. A Fourier laminated imaging reconstruction method based on an alternating direction multiplier method is characterized by comprising the following steps: the method comprises the following steps:
step 1, introducing a pupil function constraint intermediate variable, and constructing a phase recovery image reconstruction model;
step 2: acquiring the distribution condition of the LED illumination array and a low-resolution intensity image set of the object to be detected through an optical imaging system, and then initializing an object function of the object to be detected and a pupil function of the imaging system;
and step 3: inputting the object function and the pupil function initialized in the step 2 into a phase recovery image reconstruction model, and further solving a pupil constraint value, complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition to realize FPM high-resolution image reconstruction;
and 4, step 4: calculating an original error and a dual error, and measuring the deviation between the FPM high-resolution image reconstruction result and the image to be restored by adopting the original error; evaluating the deviation between numerical values obtained by combining a pupil constraint value, the complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition and a phase recovery image reconstruction model by using dual errors;
and 5: and setting an original error and a dual error judgment tolerance, outputting an image reconstruction result when the error tolerance meets the set requirement, and returning to the step 3 until outputting the image reconstruction meeting the error set requirement when the error tolerance does not meet the set requirement.
2. The method of fourier stack imaging reconstruction based on the alternating direction multiplier method of claim 1, wherein: the step 1 introduces pupil function constraint intermediate variables as:
qj＝diag(p)Qjs
where diag (-) is the mathematical operator of the matrix multiplication operation, p denotes the pupil function of the optical system, QjRepresenting a sampling matrix corresponding to a jth LED illumination position, and s representing an object to be measured;
the phase recovery image reconstruction model is as follows:
Figure FDA0003584614920000021
wherein "is a symbolic representation of" defined as ",
Figure FDA0003584614920000022
representing two-dimensional inverse Fourier transform, | · | | non-conducting phosphor22-norm operation, q, representing a matrixjIntermediate variables are constrained for the pupil function,
Figure FDA0003584614920000023
the image set is a low-resolution intensity image set of the object to be measured, s represents the object to be measured, N represents the number of LEDs in the lighting array, delta is the expression of a plane light field without any phase information in a frequency domain, and gamma is used for measuring the strength of the applied regularization term.
3. The method of fourier stack imaging reconstruction based on the alternating direction multiplier method of claim 1, wherein: in step 3, the pupil constraint value solving method is as follows:
Figure FDA0003584614920000024
the complex amplitude distribution of the image is solved as follows:
Figure FDA0003584614920000025
the optical imaging system pupil function solving method is as follows:
Figure FDA0003584614920000026
the pupil constraint optimization condition is solved as follows:
Figure FDA0003584614920000027
in the above four formulas, qj，s，p，λjAnd respectively representing a pupil constraint value, the complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition, wherein the carried superscript indicates the current iteration number.
4. The method of fourier stack imaging reconstruction based on the alternating direction multiplier method of claim 3, wherein: in step 3, the pupil constraint value solving iteration process is as follows:
Figure FDA0003584614920000031
the complex amplitude distribution of the image is solved by an iterative process:
Figure FDA0003584614920000032
the optical imaging system pupil function solving iterative process:
Figure FDA0003584614920000033
and (3) solving iterative process of pupil constraint optimization conditions:
Figure FDA0003584614920000034
ωj＝λj/α
wherein
Figure FDA0003584614920000035
And
Figure FDA0003584614920000036
respectively representing a two-dimensional fourier transform and its inverse,
Figure FDA0003584614920000037
representing the complex conjugate of the matrix; qjAnd
Figure FDA0003584614920000038
respectively representing the sampling matrix and its inverse corresponding to the jth LED illumination location,
Figure FDA0003584614920000039
for low-resolution intensity image sets of the object to be measured, qj，s，p，ωjRespectively representing a pupil constraint value, complex amplitude distribution of an image, a pupil function of an optical imaging system and a pupil constraint optimization condition, wherein a carried superscript indicates the current iteration number; n denotes the number of LEDs in the illumination array, δ is the representation of a planar light field in the frequency domain without any phase information, a penalty parameter α is used to describe the accuracy of the pupil constraint intermediate variable definition, the introduction of a parameter β is used to stabilize the update process of the pupil function to prevent the appearance of extreme values, a parameter γ is used to measure the strength of the applied regularization term, and a parameter η is the step size of the algorithm update.
5. The method of fourier stack imaging reconstruction based on the alternating direction multiplier method of claim 1, wherein: in step 4, the original error and the dual error are calculated as follows:
Figure FDA00035846149200000310
Figure FDA0003584614920000041
wherein
Figure FDA0003584614920000042
Respectively representing two-dimensional inverse Fourier transform, | · |. non-calculation22-norm operations representing matrices;
Figure FDA0003584614920000043
for low-resolution intensity image sets of the object to be measured, qj，λjRespectively representing a pupil constraint value and a pupil constraint optimization condition; n represents the number of LEDs in the illumination array, a penalty parameter alpha is used for describing the accuracy degree of the definition of a pupil constraint intermediate variable, a parameter eta is the step size of the algorithm updating, and superscripts carried by all variables indicate the current iteration number.
6. The method of fourier stack imaging reconstruction based on the alternating direction multiplier method as claimed in claim 1, wherein: in the step 5, the conditions for judging the tolerance of the original error and the dual error are set as follows:
Figure FDA0003584614920000044
Figure FDA0003584614920000045
wherein EpAnd EdRespectively representing the original error and the dual error, the superscript indicating the current iteration number, epsilontolIs a preset error tolerance value.
7. The method for fourier stack imaging reconstruction based on the alternating direction multiplier method as claimed in any one of claims 1 to 6, wherein: after the initialization of the object function of the object to be detected and the pupil function of the imaging system in the step 2 is completed, the interference of the object function and the pupil function of the object to be detected in crosstalk prevention is also included, and the interference prevention process is as follows:
Figure FDA0003584614920000046
wherein "is a symbol representing" defined as "and" diag (-) is a mathematical operator of a matrix multiplication operation,
Figure FDA0003584614920000047
representing two-dimensional inverse Fourier transform, | · | | non-conducting phosphor22-norm operations representing matrices; p denotes the pupil function of the optical system, QjRepresenting a sampling matrix corresponding to the jth LED illumination location,
Figure FDA0003584614920000048
the image set is a low-resolution intensity image set of the object to be measured, s represents the object to be measured, N represents the number of LEDs in the lighting array, delta is the expression of a plane light field without any phase information in a frequency domain, and gamma is used for measuring the strength of the applied regularization term.
8. An electronic device comprising a processor and a memory;
the memory is used for storing;
the processor is used for executing the Fourier stack imaging reconstruction method based on the alternating direction multiplier method according to any one of claims 1-8 through calling.
9. A computer program product comprising a computer program and/or instructions which, when executed by a processor, performs the method of fourier stack imaging reconstruction based on the alternating direction multiplier method of any of claims 1-8.
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