Unmarked three-dimensional coherent light field displacement calculation method

Abstract

The invention discloses an unmarked three-dimensional coherent light field displacement calculation method, which belongs to the technical field of optical detection, and comprises the following steps: obtaining two-dimensional light field complex amplitude distribution of a sample before and after displacement; performing two-dimensional Fourier transform on the two-dimensional light field complex amplitude distribution, and extracting to obtain a frequency spectrum; projecting the frequency spectrum to a three-dimensional Ewald spherical shell, performing point multiplication operation on the projected frequency spectrum in a frequency domain, and then executing three-dimensional inverse Fourier transform to generate a cross-correlation function; and calculating a three-dimensional displacement according to the main peak coordinates of the cross-correlation function to obtain a displacement calculation result. Through the Ewald spherical shell projection technology, the problem of strict parameter limitation in traditional numerical diffraction integration is avoided, and the z-axis measurement stroke is remarkably improved. Compared with a traditional three-dimensional cross-correlation operation mode, complicated diffraction integral calculation does not need to be carried out on the light field, the calculation complexity is greatly reduced, and the calculation efficiency is remarkably improved.

Description

Technical Field

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

[0002] In the field of optical detection, the direct three-dimensional displacement monitoring method based on optical images can use numerical diffraction integration to perform holographic reconstruction of the light field. However, this requires using a complex diffraction propagation algorithm to generate an axial light field stack and combining three-dimensional cross-correlation operations to calculate the displacement. However, such methods have the following problems:

[0003] Strict parameter limitations: The accuracy of numerical diffraction integration needs to satisfy complex inequality constraints among the spatial interval, the number of samples, and the propagation distance, resulting in a limited effective propagation distance and affecting the measurement range of the z-axis.

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

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

[0006] In the prior art, methods based on speckle correlation or digital holography can partially solve the problem of lateral displacement monitoring, but the measurement of three-dimensional displacement still relies on complex parameter calibration and it is difficult to unify the uncertainty traceability. Therefore, there is an urgent need for an efficient and high-precision three-dimensional displacement monitoring method. Summary of the Invention

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

[0008] To achieve the above object, the present invention provides a method for calculating the displacement of a label-free three-dimensional coherent light field, including:

[0009] Obtaining the two-dimensional light field complex amplitude distributions 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 and obtain the spectrum;

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

[0012] Calculate the three-dimensional displacement based on the coordinates of the main peak of the cross-correlation function to obtain the displacement calculation result.

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

[0014] Obtain the complex amplitude images of the sample before and after displacement through digital holography, interferometer measurement or phase-shifting technology respectively.

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

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

[0017] According to the free-space dispersion relation, project the two-dimensional spectrum onto a three-dimensional Ewald sphere shell.

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

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

[0020] Project each spatial frequency component in the two-dimensional spectrum onto the Ewald sphere shell with a radius equal to the modulus length in the three-dimensional Fourier space, associate the two-dimensional Fourier spectrum with the three-dimensional Fourier spectrum, and obtain all three-dimensional Fourier spectrum information under the current sampling conditions.

[0021] Preferably, the formula expression of the free-space dispersion relation is:

[0022]

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

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

[0025] Associate the three-dimensional Fourier spectrum of the light field in free space with the two-dimensional Fourier spectrum on the z plane; the formula expression is:

[0026]

[0027] Among them, respectively represent the components in the three-axis directions of the three-dimensional spatial spectrum, represents the projection of the two-dimensional Fourier spectrum on a semi-Ewald spherical shell, indicating that the two-dimensional Fourier spectrum located 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] Performing a point multiplication operation on the projection spectra of the sample before and after displacement in the frequency domain;

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

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

[0032]

[0033] Among them, respectively represent the components in the three-axis directions of the three-dimensional spatial spectrum, , respectively represent the three-dimensional Fourier spectra of the coherent light fields obtained before and after the three-dimensional displacement of the sample, represents the three-dimensional inverse Fourier transform operation.

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

[0035] Detecting the main peak coordinate of the cross-correlation function;

[0036] Calculating and obtaining the three-dimensional displacement amount according to the offset between the main peak coordinate and the middle position of the parameter space.

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

[0038] Locating through a sub-pixel level peak detection algorithm , and outputting and obtaining the three-dimensional displacement amount.

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

[0040] Through the Ewald spherical shell projection technology, the present invention avoids the problem of strict parameter limitations in traditional numerical diffraction integration, and significantly improves the z-axis measurement stroke.

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

[0042] Through the sub-pixel level peak detection algorithm, the present invention can achieve high-precision three-dimensional displacement calculation, and is applicable to measurement tasks of ultra-short distance displacement (such as within 1 nm) and long-distance propagation.

[0043] The present invention has relatively loose requirements for 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 drawings constituting a part of this application are used to provide a further understanding of this application. The schematic embodiments of this application and their descriptions are used to explain this application and do not constitute an improper limitation to this application. In the drawings:

[0045] Figure 1 It is a schematic flow chart of the traditional coherent illumination displacement calculation principle based on numerical diffraction integral for the embodiment of the present invention;

[0046] Figure 2 It is a schematic flow chart of the coherent illumination displacement calculation principle based on the three-dimensional Fourier spectrum Ewald sphere projection for the embodiment of the present invention;

[0047] Figure 3 It is a comparison diagram of the numerical calculation results of applying the diffraction integral method and the Ewald sphere projection method for the embodiment of the present invention. Detailed Embodiment

[0048] It should be noted that, without conflict, the embodiments in this application and the features in the embodiments can be combined with each other. The following will refer to the drawings and combine with the embodiments to detail this application.

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

[0050] As Figures 1-3 shown, in this embodiment, a method for calculating the displacement of a label-free three-dimensional coherent light field is provided, including:

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

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

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

[0054] Calculate the three-dimensional displacement based on the coordinates of the main peak of the cross-correlation function to obtain the displacement calculation result.

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

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

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

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

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

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

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

[0062] Project each spatial frequency component in the two-dimensional spectrum onto the Ewald sphere shell with a radius equal to the modulus length in the three-dimensional Fourier space, associate the two-dimensional Fourier spectrum with the three-dimensional Fourier spectrum, and obtain all the three-dimensional Fourier spectrum information under the current sampling conditions.

[0063] Even further, the characteristics of the coherent optical field determine that its propagation field is determined in three-dimensional space, but the actual numerical calculation is based on the discrete Fourier transform, thus bringing a problem that has to be faced, that is, the effectiveness of the numerical calculation. Accurate numerical simulation of the diffracted optical field needs to satisfy an inequality group composed of factors such as the spatial scales of the initial plane and the observation plane and , spatial intervals and , propagation distance , wavelength , sampling grid points and so on:

[0064]

[0065]

[0066]

[0067]

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

[0069] In actual detection tasks, , , and are usually limited, which directly restricts the propagation distance . Although there are some multi-step algorithms to calculate the diffraction light field at a long distance, such methods often require changing the and of the observation plane, which brings a very heavy burden to the subsequent cross-correlation calculation. In fact, to ensure the effectiveness and consistency of the three-dimensional cross-correlation operation, the of the light field to be processed along the xyz directions should be the same.

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

[0071]

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

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

[0074] Therefore, the process of correlating the two-dimensional Fourier spectrum with the three-dimensional Fourier spectrum includes:

[0075] Correlating the three-dimensional Fourier spectrum of the light field in free space with the two-dimensional Fourier spectrum on the z-plane; the formula expression is:

[0076]

[0077] Where, respectively represent the components in the three axial directions of the three-dimensional spatial spectrum, It means that the two-dimensional Fourier spectrum is projected onto a half Ewald sphere shell, indicating that the two-dimensional Fourier spectrum located 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] Performing a point multiplication operation on the projected spectra of the sample before and after displacement in the frequency domain;

[0080] Performing 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 point multiplication of the three-dimensional Fourier spectra of two optical fields;

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

[0083]

[0084] Where, respectively represent the components in the three axial directions of the three-dimensional spatial spectrum, , respectively represent the three-dimensional Fourier spectra of the coherent optical fields obtained before and after the three-dimensional displacement of the sample, represents the three-dimensional inverse Fourier transform operation.

[0085] Only the optical field distribution on one z-plane is needed to obtain all the values, which indicates that it is not necessary to use the diffraction integral to completely calculate the three-dimensional optical field outside the z-plane, but the information integration of the optical fields before and after displacement can be directly completed in the Fourier spectrum space.

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

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

[0088] Calculating and obtaining the three-dimensional displacement amount according to the offset between the main peak coordinate and the middle position of the parameter space.

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

[0090] Locating through a sub-pixel level peak detection algorithm , and outputting to obtain the three-dimensional displacement amount; where, respectively represent the displacement amounts in the x, y, and z directions in the spatial coordinate system.

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

[0092] Furthermore, in the detection task, parameters such as the spatial interval, sampling grid points, and wavelength of the optical image are often determined by actual applications, so they are not restricted. Assume that the two-dimensional complex amplitude distributions of the optical field before and after displacement directly obtained by the detection system are , and their sizes are both , and the spatial interval of the observation plane is .

[0093] If the traditional three-dimensional cross-correlation operation method is adopted, it is necessary to perform diffraction integral calculation on , and the size of the diffraction plane is consistent with the spatial interval, that is, the propagation distance . Thus, the three-dimensional spatial complex amplitude of the optical field can be obtained, and its size is .

[0094] It should be noted that represents the farthest distance propagated along the optical axis from the initial plane, and this distance also corresponds to the axial vector range of the method proposed in this embodiment. In fact, although a larger M means a larger range, this is not infinite. On the one hand, this increases the calculation cost; on the other hand, the distance that can be digitally refocused based on the complex amplitude obtained by holography also has a limit, and this limit is determined by the performance of the optical system.

[0095] Figure 1 shows the schematic diagram of the principle flow of the traditional coherent illumination displacement calculation based on numerical diffraction integral. By adopting the spectral dimension reduction processing of the Ewald spherical shell projection, it is not necessary to perform diffraction integral calculation on , which avoids complex diffraction numerical calculations. More importantly, it avoids the contradiction between the sampling condition constraints for ensuring the accuracy of diffraction numerical calculations and the sampling conditions required by the actual detection task.

[0096] First, perform two-dimensional Fourier transform on to obtain the two-dimensional Fourier spectrum , and its size is . According to the formula, the three-dimensional Fourier spectra of the two optical fields need to perform a dot product operation in the Fourier frequency domain space, which requires them to be aligned in the parameter space of the spatial spectrum. Although the finite sampling of the three-dimensional spatial optical field in the real world restricts the maximum spatial spectrum that can be processed: , which means the corresponding Ewald spherical shell is often not a complete hemisphere because it is extremely difficult to make the spatial interval of the system reach the wavelength level. However The size and spatial interval are the same, which means they have the same Fourier spectrum parameter space, whether two-dimensional or three-dimensional. In other words, without additional processing, they are inherently aligned in their respective parameter spaces and can be directly dot-multiplied to obtain , which represents the dot-product result of the optical field spectra 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, only the value of needs to be projected onto the Ewald sphere shell according to the formula to form a sparse matrix with a size of , and then the three-dimensional inverse Fourier transform is used to obtain the cross-correlation function.

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

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

[0099] (1) Obtain the 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 sphere shell to generate a sparse three-dimensional spectrum.

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

[0103] (4) Locate through the sub-pixel peak detection algorithm and output the three-dimensional displacement amount.

[0104] As Figure 3 shown, it shows the calculation result of the z-axis after numerical calculation based on the above description, 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 for calculating the complex amplitude distribution of the optical field:

[0105]

[0106] In the numerical calculation, the sampling space interval is , with an optical magnification of 10 times and a propagation distance nm, wavelength = 532 nm, sampling grid points . It can be seen that for ultra-short distance displacements (such as within 1 nm), both algorithms can maintain high accuracy. However, for long-distance propagation, the accuracy rate of the diffraction integral method fluctuates irregularly as the propagation distance increases; the accuracy rate of the Ewald sphere projection method remains stable.

[0107] The Ewald sphere shell projection technology of this embodiment utilizes the three-dimensional Fourier spectrum redundancy characteristic of the optical field in free space to project the two-dimensional Fourier spectrum onto the Ewald sphere shell. This technology avoids complex diffraction integral calculations and at the same time solves the problem of strict parameter limitations in traditional methods.

[0108] The frequency-domain dot product and inverse Fourier transform of this embodiment directly obtain the three-dimensional cross-correlation function by performing a dot product operation on the three-dimensional Fourier spectra before and after displacement in the frequency domain and then performing a three-dimensional inverse Fourier transform. This process significantly reduces the computational complexity and improves the 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 realizing high-precision three-dimensional displacement calculation.

[0110] In summary, the present invention provides an efficient and high-precision method for calculating the displacement of a non-labeled three-dimensional coherent optical field, which is applicable to a variety of actual application scenarios and has a wide range of application prospects.

[0111] The above is only a preferred specific embodiment of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed in the present application should be covered by the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A method for calculating the displacement of a non - marked three - dimensional coherent light field, characterized in that, Including: Obtain the two-dimensional optical field complex amplitude distributions of the sample before and after displacement; Perform two-dimensional Fourier transform on the two-dimensional optical field complex amplitude distribution, and extract the obtained spectrum; Project the spectrum onto a three-dimensional Ewald sphere shell, perform a dot product operation on the projected spectrum in the frequency domain, and then perform three-dimensional inverse Fourier transform to generate a cross-correlation function; Calculate the three-dimensional displacement amount according to the main peak coordinate of the cross-correlation function to obtain the displacement calculation result.

2. The method according to claim 1, wherein: The process of obtaining the two-dimensional optical field complex amplitude distributions of the sample before and after displacement includes: Obtain the complex amplitude images of the sample before and after displacement through digital holography, interferometer measurement, or phase-shift technology respectively.

3. The method according to claim 1, wherein: The process of performing two-dimensional Fourier transform on the two-dimensional optical field complex amplitude distribution and extracting the obtained spectrum includes: Perform two-dimensional Fourier transform on the two-dimensional optical field complex amplitude distribution to generate a two-dimensional spectrum; Project the two-dimensional spectrum onto a three-dimensional Ewald sphere shell according to the free-space dispersion relation.

4. The method according to claim 3, wherein: The process of projecting the two-dimensional spectrum onto a three-dimensional Ewald sphere shell according to the free-space dispersion relation includes: Determine the modulus length of the three-dimensional spatial frequency vector corresponding to each spatial frequency component in the two-dimensional spectrum; Project each spatial frequency component in the two-dimensional spectrum onto the Ewald sphere shell with a radius equal to the modulus length in the three-dimensional Fourier space, associate the two-dimensional Fourier spectrum with the three-dimensional Fourier spectrum, and obtain all three-dimensional Fourier spectrum information under the current sampling conditions.

5. The method according to claim 3, wherein: The formula expression of the free-space dispersion relation is: Among them, respectively represent the components in the three axial directions of the three-dimensional spatial spectrum, n represents the refractive index in free space, represents the wavelength, is the radius of the spherical shell.

6. The method according to claim 4, wherein: The process of associating the two-dimensional Fourier spectrum with the three-dimensional Fourier spectrum includes: The three-dimensional Fourier spectrum of the optical field in free space is associated with the two-dimensional Fourier spectrum on the z-plane ; the formula expression is as follows: Among them, respectively represent the components in the three-axis directions of the three-dimensional spatial spectrum, represents the projection of the two-dimensional Fourier spectrum on a semi-Ewald spherical shell, indicating that the two-dimensional Fourier spectrum located on the z-plane already contains all the values in the three-dimensional Fourier spectrum.

7. The method according to claim 1, wherein: The process of generating the cross-correlation function includes: Perform a dot product operation on the projected spectra of the sample before and after displacement in the frequency domain; Perform three-dimensional inverse Fourier transform on the spectrum after dot product to obtain the cross-correlation function.

8. The method according to claim 1, wherein: The formula expression for performing three-dimensional inverse Fourier transform on the spectrum after dot product is: Among them, respectively represent the components in the three axial directions of the three-dimensional spatial spectrum, , respectively represent the three-dimensional Fourier spectra of the coherent optical fields obtained before and after the three-dimensional displacement of the sample, represents the three-dimensional inverse Fourier transform operation.

9. The method according to claim 1, wherein: The process of calculating the three-dimensional displacement amount according to the main peak coordinate of the cross-correlation function includes: Detect the main peak coordinate of the cross-correlation function; Calculate the three-dimensional displacement amount according to the offset between the main peak coordinate and the middle position of the parameter space.

10. The method according to claim 9, wherein: The process of calculating the three-dimensional displacement amount according to the offset between the main peak coordinate and the middle position of the parameter space includes: Locate by sub-pixel peak detection algorithm , and output to obtain the three-dimensional displacement

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