A method for sparse spectrum reconstruction in photonic integrated interferometry based on compressed sensing

Abstract

This invention provides a sparse spectrum reconstruction method for photonic integrated interferometry imaging based on compressed sensing, comprising the following steps: S1, using compressed sensing to introduce the L1 paradigm of the spatial domain intensity image gradient into the original objective function of photonic integrated interferometry imaging, obtaining an objective function with the introduced L1 paradigm; S2, performing an inverse Fourier transform on the objective function with the introduced L1 paradigm, converting the original objective function from the frequency domain to the spatial domain; S3, converting the spatial domain objective function into an augmented Lagrange multiplier function, and iteratively solving the subproblems of the augmented Lagrange multiplier function to obtain a high-resolution intensity image. This invention can achieve high-resolution image reconstruction at a sampling rate of less than 50%, especially when low-frequency sampling is severely lacking.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of photonic integrated interferometric imaging, and particularly relates to a photonic integrated interferometric imaging sparse spectrum reconstruction method based on compressed sensing. BACKGROUND

[0002] In the field of astronomical observation and remote sensing imaging, in order to improve the imaging resolution, it is usually selected to increase the aperture of the optical system, but with the increase of the aperture of the telescope, the manufacturing cost increases exponentially. The photonic integrated interferometric imaging technology provides a new technical approach to replace the traditional large-aperture optical telescope, in which the photonic integrated chip (PIC) is used to realize the beam combination of the interference array, which can significantly improve the real-time performance, sensitivity, expandability and other capabilities of the system, and effectively enhance the stability and reliability of the system, and the manufacturing process is simple, the process is stable, and the manufacturing cost is effectively reduced. However, the photonic integrated interferometric imaging technology has discrete sparsity in the sampling of the target spectrum due to the discontinuous baseline matching, and the short baseline corresponds to low-frequency information and the long baseline corresponds to high-frequency information, and the sparse sampling of the spectrum often leads to an undesirable imaging result. Therefore, it is an urgent problem to realize high-resolution reconstruction of the image in the sparse spectrum sampling. SUMMARY

[0003] The purpose of the present application is to overcome the problem that the prior art cannot measure the low-frequency information of the target due to the sparse sampling during sampling and the interference process is limited by the baseline length, resulting in that the collected information is only a small part of the target information, which easily causes defects in the high-resolution intensity reconstruction image of the target. A photonic integrated interferometric imaging sparse spectrum reconstruction method based on compressed sensing is proposed, which can realize high-resolution reconstruction of the image at a sampling rate of less than 50%, especially in the case of serious low-frequency sampling loss.

[0004] To achieve the above purpose, the present application adopts the following specific technical scheme:

[0005] The photonic integrated interferometric imaging sparse spectrum reconstruction method based on compressed sensing provided by the present application comprises the following steps:

[0006] S1, introducing the L1 norm of the spatial domain intensity image gradient into the original objective function of the photonic integrated interferometric imaging using the compressed sensing method to obtain the objective function with the L1 norm; wherein the original objective function is:

[0007] argmin |HΨ-V| 2 (1),

[0008] Wherein, H is a discrete sampling matrix, Ψ is a high-resolution spectrum of the target, and V is a complex visibility of the target calculated through the interference process;

[0009] The objective function with the L1 norm is:

[0010]

[0011] where w=Dx represents the intensity image gradient, D={D h , D v} represents the gradient operator, D h is the horizontal gradient operator, D v is the vertical gradient operator, and ||w||1 represents the anisotropic TV regular term.

[0012] S2, inverse Fourier transform is performed on the target function with the L1 norm, so as to convert the original target function from the frequency domain to the spatial domain, and the target function in the spatial domain is represented as:

[0013]

[0014] s.t.x=F -1 HΨ, w=DF -1 Ψ

[0015] where b is the measurement value obtained by performing inverse Fourier transform on the sparse sampling spectrum, w is the intensity image gradient, x is the target high-resolution intensity image, and x=F -1 Ψ, F -1 represents the inverse Fourier transform.

[0016] S3, the target function in the spatial domain is converted into an augmented Lagrange multiplier function, and the sub-problems of the augmented Lagrange multiplier function are iteratively solved, so as to finally obtain the high-resolution intensity image x, wherein,

[0017] The augmented Lagrange multiplier function is:

[0018]

[0019] where the high-resolution intensity image x and the high-resolution spectrum Ψ are in a Fourier transform pair relationship, F is the Fourier transform, y and v are Lagrange multiplier terms, and μ and η are penalty coefficients.

[0020] Preferably, step S3 comprises the following steps:

[0021] S31, the initial values of Ψ, w, y, and v are all set to 0, the image row number and the image column number of x, Ψ, and w are respectively set to m and n, the high-resolution intensity image x, the high-resolution spectrum Ψ, and the high-resolution intensity image gradient w are respectively taken as the sub-problems of the augmented Lagrange multiplier function, and the expressions of the three sub-problems are as follows:

[0022]

[0023]

[0024]

[0025] Wherein, k represents iteration number, usually fixed number or jump out of loop when reaching termination condition, here taking k=100.

[0026] S32, iteration solving of formula (5) is carried out, wherein,

[0027] For solving high-resolution intensity image x sub-problem, derivation of formula (4) to x can obtain:

[0028]

[0029] For solving high-resolution spectrum Ψ sub-problem, derivation of formula (4) to Ψ can obtain:

[0030]

[0031] For solving high-resolution intensity image gradient w sub-problem, derivation of formula (4) to w can obtain:

[0032]

[0033] Updating Lagrange multiplier term y and v:

[0034]

[0035] Wherein, β is update step, y represents high-resolution spectrum term, and v represents high-resolution intensity image gradient term.

[0036] S33, when k=100, or (x k+1 -x k ) 2 / m*n<0.001, terminate iteration, and obtain high-resolution intensity image x according to iteration result.

[0037] The present application can achieve the following technical effects:

[0038] 1, the present application uses compressed sensing method, by introducing L1 norm reconstruction object spectrum, and restores the missing information, avoids the high-resolution image details missing caused by discrete sampling.

[0039] 2, the present application adopts the compressed sensing method of introducing L1 norm of spatial domain intensity image gradient in objective function, can restore high-resolution image under the premise of serious low-frequency sampling loss. BRIEF DESCRIPTION OF DRAWINGS

[0040] Figure 1 It is the flow chart of the method for reconstructing sparse spectrum based on compressed sensing of photonic integrated interferometric imaging provided by the embodiment of the present application.

[0041] Figure 2 is a high-resolution simulation diagram provided by an embodiment of the present application.

[0042] Figure 3 is a spectrum sampling matrix diagram provided by an embodiment of the present application.

[0043] Figure 4 is a high-resolution image diagram after spectrum sampling optimization provided by an embodiment of the present application.

[0044] Figure 5 is a result diagram obtained by directly performing spectrum sampling Fourier inverse transform in a conventional photonic integrated interferometric imaging technology provided by an embodiment of the present application. DETAILED DESCRIPTION

[0045] Hereinafter, embodiments of the present application will be described with reference to the accompanying drawings. In the following description, the same modules are denoted by the same reference numerals. In the case of the same reference numerals, their names and functions are also the same. Therefore, detailed descriptions thereof will not be repeated.

[0046] In order to make the objectives, technical solutions and advantages of the present application clearer, further detailed descriptions will be given to the present application in combination with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application, and do not constitute a limitation on the present application.

[0047] Figure 1 shows a flow of a photonic integrated interferometric imaging sparse spectrum reconstruction method based on compressed sensing provided by an embodiment of the present application, Figure 2 shows a high-resolution simulation image provided by an embodiment of the present application, Figure 3 shows a spectrum sampling matrix provided by an embodiment of the present application.

[0048] The present application uses a sampling mode as shown in Figure 3 to collect a high-resolution image as shown in Figure 2 . As shown in Figure 1 , the photonic integrated interferometric imaging sparse spectrum reconstruction method based on compressed sensing provided by the present application comprises the following steps:

[0049] S1, using a compressed sensing method to introduce an L1 norm of a spatial domain intensity image gradient in an original objective function of photonic integrated interferometric imaging, to obtain an objective function with the L1 norm introduced; wherein the original objective function is:

[0050] argmin|HΨ-V| 2 (1),

[0051] Where H is the discrete sampling matrix, Ψ is the high-resolution spectrum of the object, and V is the complex visibility of the object calculated by the interference process.

[0052] In the reconstruction process, since the acquisition information cannot completely cover the entire spectrum, and the measurement process is a single measurement for the object, and the natural image gradient has sparsity, the missing information of the object can be obtained by solving the gradient, so as to solve the ill-posed problem of the high-resolution spectrum reconstruction process of the object. By using the compressed sensing method, the L1 norm is introduced to reconstruct the target spectrum of the object and recover the missing information.

[0053] The objective function of the L1 norm is:

[0054]

[0055] w = Dx represents the intensity image gradient, D = {D h , D v} represents the gradient operator, D h is the transverse gradient operator, and D v is the longitudinal gradient operator. ||w||1 represents the anisotropic TV regularization term.

[0056] In order to unify formula (2) with the L1 norm, and in the photon integration interference imaging, the obtained image is an intensity image, so there is no phase term, only a real part term.

[0057] S2, Fourier inverse transform is performed on the objective function of the introduced L1 norm, and the original objective function is converted from the frequency domain to the spatial domain, and the objective function in the spatial domain is represented as:

[0058]

[0059] s.t.x = F -1 HΨ, w = DF -1 Ψ

[0060] Where b is the measurement value obtained by performing Fourier inverse transform on the sparsely sampled spectrum, w is the intensity image gradient, x is the high-resolution intensity image of the object, and x = F -1 Ψ, F -1 represents the Fourier inverse transform.

[0061] In formula (2), the intensity image gradient w is a parameter in the spatial domain form, so only the parameters Ψ and V in the objective function of the introduced L1 norm need to be Fourier inverse transformed, and the objective function in the spatial domain can be obtained.

[0062] S3, the objective function in the spatial domain is converted into an augmented Lagrange multiplier function, and the sub-problems of the augmented Lagrange multiplier function are iteratively solved, and finally the high-resolution intensity image is obtained, wherein,

[0063] The augmented Lagrange multiplier function is:

[0064]

[0065] wherein the high-resolution intensity image x and the high-resolution spectrum Ψ are a Fourier transform pair, wherein F is a Fourier transform, y and v are Lagrange multiplier terms, and μ and η are penalty coefficients.

[0066] Step S3 specifically comprises the following steps:

[0067] S31, set the initial values of Ψ, w, y, and v as 0, and set the image row number and the image column number of x, Ψ, and w as m and n respectively, and take the high-resolution intensity image x, the high-resolution spectrum Ψ, and the high-resolution intensity image gradient w as sub-problems of the augmented Lagrange multiplier function respectively, and the expressions of the three sub-problems are as follows:

[0068]

[0069]

[0070]

[0071] wherein k represents the iteration number, which is usually a fixed number or is exited when the termination condition is reached, and here k = 100.

[0072] In order to accelerate the convergence speed and improve the convergence accuracy, the alternating direction multiplier method is used to solve equation (4).

[0073] S32, iteratively solve equation (5), wherein,

[0074] For solving the high-resolution intensity image x sub-problem, the derivative of equation (4) with respect to x is:

[0075]

[0076] For solving the high-resolution spectrum Ψ sub-problem, the derivative of equation (4) with respect to Ψ is:

[0077]

[0078] For solving the high-resolution intensity image gradient w sub-problem, the derivative of equation (4) with respect to w is:

[0079]

[0080] Update the Lagrange multiplier terms y and v:

[0081]

[0082] Wherein, β is an update step, y represents a high-resolution spectrum item, and v represents a high-resolution intensity image gradient item.

[0083] Solving a plurality of sub-problems and sub-variables separately can accelerate convergence and improve convergence accuracy.

[0084] S33、when k=100, or (x k+1 -x k ) 2 / m*n<0.001, terminate iteration, and obtain a high-resolution intensity image x according to the iteration result.

[0085] Figure 4 A high-resolution image obtained by sampling and optimizing a spectrum according to an embodiment of the present application is shown, Figure 5 A result obtained by directly performing inverse Fourier transform on a spectrum sampled in a conventional photonic integrated interferometric imaging technology is shown.

[0086] A high-resolution image finally obtained by the present application is shown. Figure 4 Compared with Figure 5 , Figure 4 On one hand, false information in the image is avoided, and on the other hand, high-resolution information of the target is obtained.

[0087] In the description of the present specification, the description of the terms "one embodiment", "some embodiments", "an example", "a specific example", or "some examples" means that the specific features, structures, materials or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present application. In the present specification, the illustrative description of the above terms does not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in any appropriate manner in any one or more embodiments or examples. In addition, different embodiments or examples described in the present specification and the features of different embodiments or examples can be combined and modified by those skilled in the art without contradiction.

[0088] Although the embodiments of the present application have been shown and described above, it should be understood that the above embodiments are exemplary and should not be construed as limiting the present application, and those skilled in the art can make changes, modifications, replacements and variations to the above embodiments within the scope of the present application.

[0089] The specific embodiments of the present application described above do not constitute a limitation on the scope of protection of the present application. Any various other corresponding changes and modifications made according to the technical concept of the present application should be included in the scope of protection of the claims of the present application.

Claims

1. A compressive sensing based photonic integrated interferometric imaging sparse spectrum reconstruction method, characterized in that, The method comprises the following steps: S1, using a compressed sensing method to introduce an L1 norm of a spatial domain intensity image gradient in a original objective function of a photonic integrated interferometric imaging, to obtain an objective function with the L1 norm introduced; wherein the original objective function is: argmin | HΨ - V | 2 (1); Wherein, H is a discrete sampling matrix, Ψ is a high-resolution spectrum of the target, and V is a complex visibility of the target calculated through an interference process; The objective function with the L1 norm introduced is: Where w = Dx represents the intensity image gradient, and D = {D h D v } represents the gradient operator, D h For the transverse gradient operator, D v For the longitudinal gradient operator, ||w||1 represents the TV regularization term for anisotropy; S2, performing an inverse Fourier transform on the objective function with the L1 norm introduced, to convert the original objective function from a frequency domain to a spatial domain, and the objective function in the spatial domain is expressed as: where b is the measurement obtained by inverse Fourier transform of the sparsely sampled spectrum, w is the intensity image gradient, x is the high resolution intensity image of the object and x = F -1 Ψ, F -1 denotes the inverse Fourier transform; S3, converting the objective function in the spatial domain into an augmented Lagrange multiplier function, and iteratively solving a sub-problem of the augmented Lagrange multiplier function, to finally obtain a high-resolution intensity image x, wherein, The augmented Lagrange multiplier function is: Wherein, the high-resolution intensity image x and the high-resolution spectrum Ψ are in a Fourier transform pair relationship, wherein F is a Fourier transform, y and v are Lagrange multiplier terms, and μ and η are penalty coefficients.

2. The compressive-sensing based photonic integrated interferometric imaging sparse spectrum reconstruction method according to claim 1, characterized in that, The step S3 comprises the following steps: S31, setting initial values of Ψ, w, y, and v as 0, and setting image row numbers and image column numbers of x, Ψ, and w as m and n respectively, taking the high-resolution intensity image x, the high-resolution spectrum Ψ, and the high-resolution intensity image gradient w as sub-problems of the augmented Lagrange multiplier function, and expressions of the three sub-problems are as follows: Wherein, k represents an iteration number, and is usually a fixed number or is exited from a loop when a termination condition is reached, and here k=100 is taken; S32, iteratively solving formula (5), wherein, For solving the high-resolution intensity image x sub-problem, derivation of formula (4) with respect to x can obtain: For solving the high-resolution spectrum Ψ sub-problem, derivation of formula (4) with respect to Ψ can obtain: For solving the high-resolution intensity image gradient w sub-problem, derivation of formula (4) with respect to w can obtain: Updating the Lagrange multiplier terms y and v: Wherein, β is an update step, y represents a high-resolution spectrum term, and v represents a high-resolution intensity image gradient term; S33, when k = 100, or (x k+1 - x k ) 2 / m*n<0.001, terminate iteration, and obtain a high-resolution intensity image x according to the iteration result.

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