A marker-free three-dimensional coherent light field displacement calculation method

Abstract

The present invention discloses a method for calculating the displacement of a label-free three-dimensional coherent light field, which belongs to the field of optical detection technology, and includes: obtaining the two-dimensional light field complex amplitude distribution of the sample before and after the displacement; performing a two-dimensional Fourier transform on the two-dimensional light field complex amplitude distribution to extract and obtain a spectrum; projecting the spectrum onto a three-dimensional Ewald spherical shell, performing a point multiplication operation on the projected spectrum in the frequency domain, and then performing a three-dimensional inverse Fourier transform to generate a cross-correlation function; calculating the three-dimensional displacement according to the coordinates of the main peak of the cross-correlation function to obtain a displacement calculation result. The present invention avoids the problem of strict parameter restrictions in traditional numerical diffraction integrals through the Ewald spherical shell projection technology, and significantly improves the z-axis measurement stroke. Compared with the traditional three-dimensional cross-correlation operation method, the present invention does not need to perform complex diffraction integral calculations on the light field, greatly reduces the computational complexity, and significantly improves the computational efficiency.

Description

Technical Field

[0001] The present invention belongs to the technical field of optical detection, and in particular relates to a method for calculating displacement of a marker-free three-dimensional coherent light field. Background Art

[0002] In the field of optical detection, direct 3D displacement monitoring methods based on optical images can use numerical diffraction integration to holographically reconstruct the light field. However, this requires the use of complex diffraction propagation algorithms to generate axial light field stacks and combine them with 3D cross-correlation operations to calculate displacement. However, this method has the following problems:

[0003] Strict parameter restrictions: The accuracy of the numerical diffraction integral must meet complex inequality constraints between spatial intervals, sampling numbers, and propagation distances, which limits the effective propagation distance and affects the z-axis measurement range.

[0004] High computational complexity: Generating a three-dimensional light field stack requires multiple Fourier transforms and diffraction integrals. This stage belongs to the algorithm preprocessing and has a computational complexity of (M is the number of z-axis samples, N is the number of x-axis or y-axis samples), which is time-consuming and resource-intensive.

[0005] Poor system adaptability: In practical applications, the pixel resolution of the camera (such as 2048×2048) is difficult to adjust, which further limits the flexibility of parameter optimization.

[0006] While existing methods based on speckle correlation or digital holography can partially address the problem of lateral displacement monitoring, 3D displacement measurement still relies on complex parameter calibration and lacks standardized uncertainty traceability. Therefore, an efficient and high-precision 3D displacement monitoring method is urgently needed. Summary of the Invention

[0007] The present invention aims to provide a marker-free three-dimensional coherent light field displacement monitoring method based on three-dimensional Fourier spectrum Ewald sphere projection. By circumventing the parameter limitations of traditional numerical diffraction integration through spectrum dimensionality reduction processing, the z-axis measurement range is significantly improved, the computational complexity is reduced, and high-precision three-dimensional displacement measurement is achieved.

[0008] To achieve the above objectives, the present invention provides a marker-free three-dimensional coherent light field displacement calculation method, comprising:

[0009] Obtain the two-dimensional light field complex amplitude distribution of the sample before and after displacement;

[0010] Performing a two-dimensional Fourier transform on the two-dimensional light field complex amplitude distribution to extract a frequency spectrum;

[0011] Projecting the spectrum onto a three-dimensional Ewald shell, performing a dot product operation on the projected spectrum in the frequency domain, and then performing a three-dimensional inverse Fourier transform to generate a cross-correlation function;

[0012] The three-dimensional displacement is calculated according to the main peak coordinates of the cross-correlation function to obtain a displacement calculation result.

[0013] Preferably, the process of obtaining the two-dimensional light field complex amplitude distribution of the sample before and after the displacement includes:

[0014] The complex amplitude images of the sample before and after displacement are obtained through digital holography, interferometer measurement or phase shift technology.

[0015] Preferably, the process of performing a two-dimensional Fourier transform on the two-dimensional light field complex amplitude distribution to extract and obtain a spectrum includes:

[0016] Performing a two-dimensional Fourier transform on the two-dimensional light field complex amplitude distribution to generate a two-dimensional spectrum;

[0017] The two-dimensional spectrum is projected onto a three-dimensional Ewald shell according to the free-space dispersion relation.

[0018] Preferably, the process of projecting the two-dimensional spectrum onto a three-dimensional Ewald spherical shell according to the free-space dispersion relation comprises:

[0019] Determining the modulus of a three-dimensional spatial frequency vector corresponding to each spatial frequency component in the two-dimensional spectrum;

[0020] Each spatial frequency component in the two-dimensional spectrum is projected onto an Ewald spherical shell with a radius of the modulus in the three-dimensional Fourier space, and the two-dimensional Fourier spectrum is associated with the three-dimensional Fourier spectrum to obtain all three-dimensional Fourier spectrum information under the current sampling conditions.

[0021] Preferably, the free space dispersion relation is expressed as follows:

[0022]

[0023] in, They represent the components of the three-axis directions of the three-dimensional spatial spectrum, n represents the refractive index of the free space species, represents the wavelength, is the radius of the spherical shell.

[0024] Preferably, the process of associating the two-dimensional Fourier spectrum with the three-dimensional Fourier spectrum comprises:

[0025] The light field in free space The three-dimensional Fourier spectrum of and the two-dimensional Fourier spectrum on the z plane The formula is:

[0026]

[0027] in, Represent the components of the three-axis directions of the three-dimensional spatial spectrum, It means projecting the two-dimensional Fourier spectrum onto half of the Ewald sphere, indicating that the two-dimensional Fourier spectrum on the z plane already contains all the values ​​in the three-dimensional Fourier spectrum.

[0028] Preferably, the process of generating the cross-correlation function includes:

[0029] Perform a dot multiplication operation on the projected spectrum of the sample before and after displacement in the frequency domain;

[0030] Perform a three-dimensional inverse Fourier transform on the spectrum after point multiplication to obtain the cross-correlation function.

[0031] Preferably, the formula for performing a three-dimensional inverse Fourier transform on the spectrum after the dot multiplication is:

[0032]

[0033] in, Represent the components of the three-axis directions of the three-dimensional spatial spectrum, , They represent the three-dimensional Fourier spectrum of the coherent light field obtained before and after the three-dimensional displacement of the sample, Represents a 3D inverse Fourier transform operation.

[0034] Preferably, the process of calculating the three-dimensional displacement according to the main peak coordinates of the cross-correlation function includes:

[0035] detecting the main peak coordinates of the cross-correlation function;

[0036] The three-dimensional displacement is calculated based on the offset between the main peak coordinates and the middle position of the parameter space.

[0037] Preferably, the process of calculating and obtaining the three-dimensional displacement according to the offset between the main peak coordinates and the middle position of the parameter space includes:

[0038] Positioning through sub-pixel peak detection algorithm , and output the three-dimensional displacement.

[0039] Compared with the prior art, the present invention has the following advantages and technical effects:

[0040] The present invention avoids the problem of strict parameter restrictions in traditional numerical diffraction integration by using Ewald spherical shell projection technology, and significantly improves the z-axis measurement range.

[0041] Compared with the traditional three-dimensional cross-correlation calculation method, the present invention does not need to perform complex diffraction integral calculations on the light field, greatly reduces the calculation complexity, and significantly improves the calculation efficiency.

[0042] The present invention uses a sub-pixel peak detection algorithm to achieve high-precision three-dimensional displacement calculation, and is suitable for measurement tasks of ultra-short-distance displacement (e.g., within 1 nm) and long-distance propagation.

[0043] The present invention has relatively loose requirements on the parameters of the optical system, can adapt to cameras with different resolutions, and has good system adaptability. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] The accompanying drawings, which constitute part of this application, are intended to provide a further understanding of this application. The exemplary embodiments and descriptions of this application are intended to explain this application and do not constitute an improper limitation on this application. In the accompanying drawings:

[0045] Figure 1 Schematic diagram of the principle flow of conventional coherent illumination displacement calculation based on numerical diffraction integration according to an embodiment of the present invention;

[0046] Figure 2 Schematic diagram of the principle flow of coherent illumination displacement calculation based on three-dimensional Fourier spectrum Ewald sphere projection according to an embodiment of the present invention;

[0047] Figure 3 1 is a comparison chart of numerical calculation results using the diffraction integral method and the Ewald sphere projection method according to an embodiment of the present invention. DETAILED DESCRIPTION

[0048] It should be noted that, in the absence of conflict, the embodiments and features of the embodiments in this application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0049] It should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and that, although a logical order is shown in the flowcharts, in some cases, the steps shown or described can be executed in an order different from that shown here.

[0050] like Figure 1-3 As shown, this embodiment provides a method for calculating displacement of a label-free three-dimensional coherent light field, including:

[0051] Obtain the two-dimensional light field complex amplitude distribution of the sample before and after displacement;

[0052] Perform two-dimensional Fourier transform on the complex amplitude distribution of the two-dimensional light field and extract the spectrum;

[0053] Project the spectrum onto the three-dimensional Ewald shell, perform a dot product operation on the projected spectrum in the frequency domain, and then perform a three-dimensional inverse Fourier transform to generate a cross-correlation function;

[0054] The three-dimensional displacement is calculated according to the main peak coordinates of the cross-correlation function to obtain the displacement calculation result.

[0055] Furthermore, the process of obtaining the two-dimensional light field complex amplitude distribution of the sample before and after the displacement includes:

[0056] The complex amplitude images of the sample before and after displacement are obtained through digital holography, interferometer measurement or phase shift technology.

[0057] Furthermore, the process of performing a two-dimensional Fourier transform on the complex amplitude distribution of the two-dimensional light field and extracting the spectrum includes:

[0058] Performing a two-dimensional Fourier transform on the complex amplitude distribution of the two-dimensional light field to generate a two-dimensional spectrum;

[0059] According to the free-space dispersion relation, the two-dimensional spectrum is projected onto the three-dimensional Ewald shell.

[0060] Furthermore, according to the free-space dispersion relation, the process of projecting the two-dimensional spectrum onto the three-dimensional Ewald shell includes:

[0061] Determine the modulus of the three-dimensional spatial frequency vector corresponding to each spatial frequency component in the two-dimensional spectrum;

[0062] Each spatial frequency component in the two-dimensional spectrum is projected onto an Ewald shell with a radius of the modulus in the three-dimensional Fourier space, and the two-dimensional Fourier spectrum is correlated with the three-dimensional Fourier spectrum to obtain all three-dimensional Fourier spectrum information under the current sampling conditions.

[0063] Furthermore, the characteristics of the coherent light field determine that its propagation field is fixed in three-dimensional space, but the actual numerical calculation is based on discrete Fourier transform, which brings a problem that must be faced, that is, the effectiveness of numerical calculation. Accurate numerical diffraction light field simulation needs to meet the spatial scale of the initial plane and the observation plane according to different algorithms. and , spatial interval and , propagation distance ,wavelength , sampling grid A group of inequalities consisting of equal factors:

[0064]

[0065]

[0066]

[0067]

[0068] Where R represents the radius of the spherical wavefront, which is used for joint phase analysis with the source field and the quadratic phase factor.

[0069] In actual detection tasks, , , and Usually limited, which directly limits the propagation distance Although there are some multi-step algorithms that can calculate the diffracted light field at long distances, such methods often require changing the observation plane. and , which brings a heavy burden to the subsequent cross-correlation calculation. In fact, in order to ensure the effectiveness and consistency of the three-dimensional cross-correlation operation, the light field to be processed along the xyz direction It should be the same.

[0070] Furthermore, however, the coherent light field is highly redundant in its three-dimensional spatial spectrum because its parameter space It is not completely independent, but is related to the dispersion relation in free space. The formula for the free space dispersion relation is:

[0071]

[0072] in, They represent the components of the three-axis directions of the three-dimensional spatial spectrum, n represents the refractive index of the free space species, represents the wavelength, is the radius of the spherical shell.

[0073] Combined with the angular spectrum theory, this means that the modulus length of the spatial frequency vector formed by plane waves in different directions (spatial frequencies) is consistent, and finally they will be distributed in the three-dimensional Fourier space with a radius of The spherical shell is also called the Ewald sphere.

[0074] Therefore, the process of relating a two-dimensional Fourier spectrum to a three-dimensional Fourier spectrum involves:

[0075] The light field in free space The three-dimensional Fourier spectrum of and the two-dimensional Fourier spectrum on the z plane The formula is:

[0076]

[0077] in, Represent the components of the three-axis directions of the three-dimensional spatial spectrum, It means projecting the two-dimensional Fourier spectrum onto half of the Ewald sphere, indicating that the two-dimensional Fourier spectrum on the z plane already contains all the values ​​in the three-dimensional Fourier spectrum.

[0078] Furthermore, the process of generating the cross-correlation function includes:

[0079] Perform a dot multiplication operation on the projected spectrum of the sample before and after displacement in the frequency domain;

[0080] Perform a three-dimensional inverse Fourier transform on the spectrum after point multiplication to obtain the cross-correlation function.

[0081] Furthermore, according to the Fourier convolution theorem, the three-dimensional cross-correlation operation can be converted into the inverse Fourier transform of the dot product of the three-dimensional Fourier spectra of two light fields;

[0082] The formula for performing a three-dimensional inverse Fourier transform on the spectrum after point multiplication is:

[0083]

[0084] in, Represent the components of the three-axis directions of the three-dimensional spatial spectrum, , They represent the three-dimensional Fourier spectrum of the coherent light field obtained before and after the three-dimensional displacement of the sample, Represents a 3D inverse Fourier transform operation.

[0085] Only a light field distribution on the z plane is needed to obtain This shows that it is not necessary to use the diffraction integral to fully calculate the three-dimensional light field outside the z plane, but the information integration of the light field before and after the displacement can be completed directly in the Fourier spectrum space.

[0086] Furthermore, the process of calculating the three-dimensional displacement according to the main peak coordinates of the cross-correlation function includes:

[0087] Detecting the main peak coordinates of the cross-correlation function;

[0088] The three-dimensional displacement is calculated based on the offset between the main peak coordinates and the middle position of the parameter space.

[0089] Furthermore, the process of calculating the three-dimensional displacement according to the offset between the main peak coordinate and the middle position of the parameter space includes:

[0090] Positioning through sub-pixel peak detection algorithm , the output is the three-dimensional displacement; where, They represent the displacement in the x, y, and z directions in the spatial coordinate system respectively.

[0091] Among them, the calculation accuracy of the three-dimensional displacement is related to the sampling accuracy of the two-dimensional light field complex amplitude distribution and the accuracy of the spectrum projection.

[0092] Furthermore, in the detection task, the parameters of the optical image, such as the spatial interval, sampling grid, and wavelength, are often determined by the actual application and are therefore not constrained. Assume that the detection system directly obtains the two-dimensional complex amplitude distribution of the light field before and after the displacement as , the sizes are , the spatial separation of the observation plane is .

[0093] If the traditional three-dimensional cross-correlation operation method is used, Perform diffraction integral calculations, and the size of the diffraction plane is consistent with the spatial interval, that is, the propagation distance At this point, the three-dimensional complex amplitude of the light field can be obtained, and its size is .

[0094] It should be pointed out that represents the maximum distance it can propagate along the optical axis from the initial plane, which also corresponds to the axial vector range of the method proposed in this embodiment. In practice, although a larger M means a greater range, it is not infinite. On the one hand, this increases the computational cost; on the other hand, the distance that can be digitally refocused based on the complex amplitude acquired by holography is also limited, and this limit is determined by the performance of the optical system.

[0095] Figure 1 The schematic diagram of the traditional coherent illumination displacement calculation principle based on numerical diffraction integration is shown. The spectrum dimension reduction processing of Ewald spherical shell projection is used, so there is no need to Performing diffraction integral calculations avoids complex diffraction numerical calculations, and more importantly, avoids the contradiction between the sampling condition constraints brought about by ensuring the accuracy of diffraction numerical calculations and the sampling conditions required by actual detection tasks.

[0096] First, yes Perform a two-dimensional Fourier transform to obtain a two-dimensional Fourier spectrum , size is According to the formula, the three-dimensional Fourier spectra of the two light fields need to be dot-multiplied in the Fourier frequency domain, which requires that the two be aligned in the parameter space of the spatial spectrum. Although the limited sampling of three-dimensional light fields in the real world restricts the maximum spatial spectrum that can be processed: , which means The corresponding Ewald shell is often not a complete hemisphere, because the space between the system It is extremely difficult to achieve wavelength level. The dimensions and spatial spacing of are the same, which means that they have the same Fourier spectrum parameter space, whether it is two-dimensional or three-dimensional. In other words, no additional processing is required. They are inherently aligned in their respective parameter spaces and can be directly obtained by performing a dot product operation , Represents the dot product result of the light field spectrum before and after displacement in the two-dimensional Fourier space spectrum, and the size is still Next, in order to obtain the three-dimensional cross-correlation function, we only need to convert The numerical value of is projected onto the Ewald spherical shell, forming a size of The sparse matrix of is then transformed into the 3D inverse Fourier transform to obtain the cross-correlation function.

[0097] like Figure 2 As shown, compared with the diffraction integral method, the method of this embodiment does not require preprocessing of the three-dimensional light field data, which significantly saves calculation time and space.

[0098] The steps of coherent illumination displacement calculation based on three-dimensional Fourier spectrum Ewald sphere projection in this embodiment can be specifically summarized as follows:

[0099] (1) Obtain complex amplitude images of the sample before and after displacement through digital holography or other common interference systems and .

[0100] (2) Perform two-dimensional FFT on the two images to generate and .

[0101] (3) According to the formula , project the two-dimensional spectrum onto the Ewald spherical shell to generate a sparse three-dimensional spectrum.

[0102] (3) After point multiplication in the frequency domain, perform a three-dimensional inverse FFT to obtain the cross-correlation function .

[0103] (4) Positioning through sub-pixel peak detection algorithm , output the three-dimensional displacement.

[0104] like Figure 3 The figure shows the calculation results of the z-axis after the numerical calculation based on the above description, which is used to compare the calculation accuracy and effective range of the diffraction integral method and the Ewald sphere projection method. The standard free space spherical wave analytical solution is used to calculate the complex amplitude distribution of the light field:

[0105]

[0106] In numerical calculations, the sampling space interval is , the optical magnification is 10 times, the transmission distance nm, wavelength =532 nm, sampling grid As can be seen, for ultra-short-distance displacements (e.g., within 1 nm), both algorithms maintain high accuracy. However, for long-distance propagation, the accuracy of the diffraction integral method fluctuates erratically with increasing propagation distance, while the accuracy of the Ewald sphere projection method remains stable.

[0107] The Ewald spherical shell projection technique of this embodiment utilizes the redundancy of the 3D Fourier spectrum of the light field in free space to project the 2D Fourier spectrum onto the Ewald spherical shell. This technique avoids the complex diffraction integral calculation and solves the problem of strict parameter restrictions in traditional methods.

[0108] The frequency domain dot product and inverse Fourier transform in this embodiment directly obtains the three-dimensional cross-correlation function by performing a dot product operation on the three-dimensional Fourier spectrum before and after the shift in the frequency domain, and then performing a three-dimensional inverse Fourier transform. This process significantly reduces computational complexity and improves computational efficiency.

[0109] The sub-pixel level peak detection algorithm of this embodiment accurately locates the main peak coordinates of the cross-correlation function through the sub-pixel level peak detection algorithm, thereby achieving high-precision three-dimensional displacement calculation.

[0110] In summary, the present invention provides an efficient and high-precision label-free three-dimensional coherent light field displacement calculation method, which is suitable for a variety of practical application scenarios and has broad application prospects.

[0111] The above are merely preferred embodiments of the present application, but the scope of protection of the present application is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in this application should be included in the scope of protection of the present application. Therefore, the scope of protection of the present application should be based on the scope of protection of the claims.

Claims

1. A marker-free three-dimensional coherent light field displacement calculation method, characterized in that: include: Obtain the two-dimensional light field complex amplitude distribution of the sample before and after displacement; Performing a two-dimensional Fourier transform on the two-dimensional light field complex amplitude distribution to extract a frequency spectrum; Projecting the spectrum onto a three-dimensional Ewald shell, performing a dot product operation on the projected spectrum in the frequency domain, and then performing a three-dimensional inverse Fourier transform to generate a cross-correlation function; Calculating the three-dimensional displacement according to the main peak coordinates of the cross-correlation function to obtain a displacement calculation result; The process of generating the cross-correlation function includes: Perform a dot multiplication operation on the projected spectrum of the sample before and after displacement in the frequency domain; Perform a three-dimensional inverse Fourier transform on the spectrum after point multiplication to obtain the cross-correlation function; The formula for performing a three-dimensional inverse Fourier transform on the spectrum after point multiplication is: in, Represent the components of the three-axis directions of the three-dimensional spatial spectrum, , They represent the three-dimensional Fourier spectrum of the coherent light field obtained before and after the three-dimensional displacement of the sample, Represents a three-dimensional inverse Fourier transform operation; The process of calculating the three-dimensional displacement according to the main peak coordinates of the cross-correlation function includes: detecting the main peak coordinates of the cross-correlation function; Calculating the three-dimensional displacement based on the offset between the main peak coordinates and the middle position of the parameter space; The process of calculating the three-dimensional displacement according to the offset between the main peak coordinate and the middle position of the parameter space includes: Positioning through sub-pixel peak detection algorithm , and output the three-dimensional displacement.

2. The method according to claim 1, characterized in that The process of obtaining the two-dimensional complex amplitude distribution of the light field of the sample before and after displacement includes: The complex amplitude images of the sample before and after displacement are obtained through digital holography, interferometer measurement or phase shift technology.

3. The method according to claim 1, characterized in that The process of performing a two-dimensional Fourier transform on the two-dimensional light field complex amplitude distribution to extract a spectrum includes: Performing a two-dimensional Fourier transform on the two-dimensional light field complex amplitude distribution to generate a two-dimensional spectrum; The two-dimensional spectrum is projected onto a three-dimensional Ewald shell according to the free-space dispersion relation.

4. The method according to claim 3, characterized in that According to the free-space dispersion relation, the process of projecting the two-dimensional spectrum onto the three-dimensional Ewald shell includes: Determining the modulus of a three-dimensional spatial frequency vector corresponding to each spatial frequency component in the two-dimensional spectrum; Each spatial frequency component in the two-dimensional spectrum is projected onto an Ewald spherical shell with a radius of the modulus in the three-dimensional Fourier space, and the two-dimensional Fourier spectrum is associated with the three-dimensional Fourier spectrum to obtain all three-dimensional Fourier spectrum information under the current sampling conditions.

5. The method according to claim 3, characterized in that The free-space dispersion relation is expressed as follows: in, They represent the components of the three-axis directions of the three-dimensional spatial spectrum, n represents the refractive index of the free space species, represents the wavelength, is the radius of the spherical shell.

6. The method according to claim 4, characterized in that The process of relating a 2D Fourier spectrum to a 3D Fourier spectrum involves: The light field in free space The three-dimensional Fourier spectrum of and the two-dimensional Fourier spectrum on the z plane The formula is: in, Represent the components of the three-axis directions of the three-dimensional spatial spectrum, It means projecting the two-dimensional Fourier spectrum onto half of the Ewald sphere, indicating that the two-dimensional Fourier spectrum on the z plane already contains all the values ​​in the three-dimensional Fourier spectrum.

Images