Method for eliminating off-axis digital holographic zero-order item

Abstract

The invention relates to a method for eliminating an off-axis digital hologram zero-order item, which comprises the following steps of: acquiring hologram data of off-axis digital hologram, and acquiring frequency spectrum information through Fourier transform; pixel points with the intensity higher than a certain value in the frequency spectrum are selected as initial data objects, clustering categories of the initial data objects are divided through a K-means clustering method, and a clustering center is calculated. Taking the clustering center as an initial mean value of the GMM, calculating the covariance of each cluster, calculating the mixing coefficient of the GMM according to the proportion of the data points in each cluster, calculating the posterior probability that each data point belongs to each Gaussian distribution through the Bayesian theorem, and updating the GMM model to obtain a final clustering result. And eliminating the clustering result at the central position of the frequency spectrum to eliminate zero-order frequency spectrum information. Compared with the prior art, the method has the advantages of high accuracy, high universality and the like.

Description

Technical Field

[0001] The invention relates to the technical field of digital holography, and in particular to a method for eliminating off-axis digital holography zero-order terms. Background Art

[0002] Digital holography has been widely used in the fields of quantitative imaging of living cells, microscopic particle tracking, MEMS detection, etc. According to the tilt angle between the object beam and the reference beam, digital holography can be divided into on-axis, off-axis and micro-off-axis digital holography. Off-axis digital holography reconstruction usually involves a spatial-spectral filtering process. In the Fourier spectrum domain of the off-axis hologram, both the positive first-order spectrum and the negative first-order spectrum carry the spatial frequency of the object, and their centers are separated from the center of the zero-order spectrum. By filtering and intercepting any of the positive first-order spectrum and the negative first-order spectrum, the reconstruction imaging of the off-axis hologram can be achieved. However, due to the existence of the zero-order spectrum, the filtering interception area of ​​the first-order spectrum is limited, which means that the effective spatial bandwidth for reconstructing the complex amplitude is limited, which affects the resolution of the reconstructed image. However, if the intercepted spectrum filtering area is too large, some zero-order spectra may be included, resulting in some striped noise in the reconstructed image.

[0003] In optical imaging and digital image processing, the effective spatial bandwidth product of the system is an important parameter affecting the imaging quality. Some multi-frame methods can improve the spatial bandwidth product of the system, thereby improving the imaging resolution or field of view. However, these methods are not always suitable for off-axis digital holographic reconstruction. For off-axis digital holographic imaging, the resolution and quality of the reconstructed image can be significantly improved by suppressing the zero-order spectrum. In online digital holography technology, the phase shift method can effectively eliminate the zero-order term and conjugate term of the hologram. Since the phase shift method usually requires a precise displacement device or a complex motion estimation algorithm, it is not used in off-axis digital holographic reconstruction. By adjusting the reflector on the object arm, a phase shift can be introduced into the object wave, thereby suppressing the zero-order term in the hologram, but this method requires at least two holograms to be collected, which is not suitable for real-time imaging. The use of nonlinear filters can effectively suppress the zero-order term and conjugate term of the off-axis hologram, but the amplitude of the reference wave is required to be much larger than the amplitude of the object wave. In summary, it is necessary to design a digital holographic zero-order term elimination method that is efficient and highly versatile. Summary of the invention

[0004] The purpose of the present invention is to provide a method for eliminating the zero-order term of off-axis digital holography in order to overcome the defects of low versatility and low precision in the above-mentioned prior art.

[0005] The purpose of the present invention can be achieved by the following technical solutions:

[0006] A method for eliminating zero-order terms of off-axis digital holography comprises the following steps:

[0007] S1: Run the off-axis digital holographic optical path system to obtain the off-axis hologram to be processed;

[0008] S2: Perform Fourier transform on the off-axis hologram to be processed to obtain a spectrum diagram of the off-axis hologram;

[0009] S3: According to the spectrum graph, select the pixel points with intensity higher than the threshold as the initial data object;

[0010] S4: Select the center point of the spectrum graph as the initial cluster center of the zero-order spectrum, and randomly select two points as the cluster center of the positive first-order spectrum and the cluster center of the negative first-order spectrum respectively;

[0011] S5: For each pixel in the initial data object, calculate its Euclidean distance to all cluster centers and classify it into the cluster center closest to it; recalculate the mean of all data points in each cluster to update the center of each cluster;

[0012] S6: Repeat S5 until the cluster center no longer changes significantly or the set number of iterations is reached;

[0013] S7: Use the cluster center in the result of S6 as the initial mean of GMM. According to the clustering result in S6, calculate the distribution of each cluster data point, calculate the covariance of each cluster, calculate the proportion of the number of data points contained in each cluster to the total number of data points, and initialize the mixing coefficient of GMM.

[0014] S8: Calculate the posterior probability that each data point in S6 belongs to each Gaussian distribution through Bayesian theorem. According to the posterior probability, recalculate the mean, covariance matrix and mixing coefficient of each Gaussian distribution to maximize the likelihood function of the Gaussian mixture model.

[0015] S9: Repeat S8 until the parameters of the model no longer change significantly, or the increment of the likelihood function is less than a preset threshold;

[0016] S10: GMM calculates the probability of each pixel point belonging to each cluster, thereby obtaining the final clustering result, and sets all clustering results located at the center of the spectrum to 0, thereby obtaining the result of finally eliminating zero-order spectrum information.

[0017] Furthermore, the parameter expression of the Gaussian mixture model is:

[0018]

[0019] Where k is the number of mixed models, α k is the mixing weight, 0<α k <1, is the Gaussian model of the kth mixture component, μ k and represents the mean and covariance matrix.

[0020] Furthermore, the following steps are also included: S11: extracting the positive first-order spectrum information and centering it, and reconstructing the intensity and phase of the object to be measured based on the angular spectrum method.

[0021] Furthermore, the calculation expression of the angular spectrum method is:

[0022]

[0023] I(x,y,z)=|U(x,y,z)| 2

[0024]

[0025] In the formula, and denote Fourier transform and inverse Fourier transform respectively, is a filter for extracting the +1 or -1 order spectrum, G(f x ,f y ,z) is the transfer function of the angular spectrum diffraction method, f x and f y is the frequency domain coordinate, λ is the wavelength of light; z is the reconstruction distance, I(x,y,z) and are the reconstructed intensity and phase of the object to be measured, respectively.

[0026] Furthermore, the calculation expression of Euclidean distance is:

[0027]

[0028] In the formula, v i represents the sample, center represents the cluster center coordinates, and n represents the observation value.

[0029] Furthermore, the calculation expression of the mixing coefficient is:

[0030]

[0031] Where k is the number of mixed models, α k is the mixing weight, 0<α k <1, is the Gaussian model of the kth mixture component, μ k and represents the mean and covariance matrix.

[0032] Furthermore, the calculation expression of the posterior probability is:

[0033]

[0034] In the formula, α k is the mixing weight, is the Gaussian model of the kth mixture component,

[0035] Furthermore, the calculation expressions for recalculating the mean, covariance matrix and mixing coefficient of each Gaussian distribution are:

[0036]

[0037]

[0038] In the formula, α k is the mixing weight, N represents the number of samples, γ jk represents the posterior probability that the i-th sample belongs to the k-th cluster, μ k is the mean of the kth Gaussian distribution, indicating the center position of the cluster, X n is the data point of the nth sample, represents the covariance matrix of the k-th Gaussian distribution, describing the expansion direction and shape of the cluster, (X n -μ k )(X n -μ k ) T It is the outer product of the difference between the nth sample and the mean, indicating the degree of dispersion of the sample and the mean.

[0039] Furthermore, the calculation expression of the likelihood function is:

[0040]

[0041] In the formula, L(θ|X) is the likelihood function value, N is the total number of samples, k is the number of Gaussian distributions, and α is k is the mixing weight, Represents the probability density function of the k-th Gaussian distribution.

[0042] Furthermore, the calculation expression for updating the cluster center is:

[0043] Center n =((∑(v i )) / p)

[0044] In the formula, v i represents a sample, and p represents the number of samples in this class after clustering.

[0045] Compared with the prior art, the present invention has the following beneficial effects:

[0046] 1) The present invention eliminates zero-order terms by combining K-means and GMM. After K-means is used for clustering, GMM further subdivides the difference between the zero-order spectrum and other spectra by calculating the posterior probability of the Gaussian distribution, and obtains accurate clustering results. Especially when the spectrum distribution is more complex, the interference of the zero-order spectrum can be accurately eliminated.

[0047] 2) The combined method of K-means and GMM can reduce the need for manual adjustment of filter parameters through automated clustering and modeling when processing large amounts of hologram data. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a schematic diagram of the structure of an off-axis digital holographic system in a specific embodiment.

[0049] Figure 2 It is a flow chart of a method for eliminating the zero-order term of off-axis digital holography provided in an embodiment of the present invention.

[0050] Figure 3 is the original hologram collected in the embodiment of the present invention.

[0051] Figure 4 It is a spectrum diagram of the original hologram after K-means clustering in the embodiment of the present invention.

[0052] Figure 5 It is a spectrum distribution diagram after GMM clustering in an embodiment of the present invention.

[0053] Figure 6 It is a spectrum diagram after eliminating the zero-order spectrum in an embodiment of the present invention.

[0054] Figure 7 It is a phase diagram reconstructed by the angular spectrum method in an embodiment of the present invention.

[0055] Explanation of the marks in the figure: 1. light source, 2. beam expander collimator, 3. beam splitter, 4. first reflector, 5. second reflector, 6. measurement object, 7. beam splitter, 8. CCD camera, 9. computer. DETAILED DESCRIPTION

[0056] The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0057] Example 1

[0058] The present invention is a method for eliminating the zero-order term of off-axis digital holography, such as Figure 2 shown.

[0059] The specific steps include:

[0060] S1: Run the off-axis digital holographic optical path system to obtain the off-axis hologram to be processed;

[0061] The experimental environment of the present invention is as follows Figure 1 As shown, Figure 1 The figure is a structural diagram of the off-axis digital holography device. The light source 1 is a helium-neon laser with a wavelength λ equal to 632.8nm. The light beam emitted by the light source 1 is divided into two beams by the beam splitter 3 after the beam expansion and collimation 2. One beam is the object light, which is reflected by the second reflector 5 and then illuminates the object 6. The other beam is the reference light, which is reflected by the first reflector 4 and then combined with the object light on the beam splitter 7. The interference angle is adjusted by adjusting the first reflector 4 or the second reflector 5. Finally, the object light and the reference light interfere on the CCD camera 9 to form an off-axis hologram, where the pixel size of the CCD camera is 4.65um×4.65um. The digitized hologram is input into the computer 10.

[0062] S2: Perform Fourier transform on the off-axis hologram to be processed to obtain spectrum information and spectrum matrix of the off-axis hologram;

[0063] S3: According to the spectrum information, select the pixel points with intensity higher than the threshold as the initial data object;

[0064] S4: Select the center point of the spectrum matrix as the initial cluster center of the zero-order spectrum, and randomly select two points as the cluster center of the positive first-order spectrum and the cluster center of the negative first-order spectrum respectively;

[0065] S5: For each pixel in the initial data object, calculate the Euclidean distance from it to all cluster centers, and classify it into the cluster center closest to it; then for each cluster, reselect its center according to the mean value of all pixels classified into this class;

[0066] The calculation expression of Euclidean distance is:

[0067]

[0068] In the formula, v i represents the sample, center represents the cluster center coordinates, and n represents the observation value.

[0069] Based on the minimum distance principle, the samples are divided into corresponding clusters mindis(n)=1,2,…k, where k represents the number of clusters. Update the cluster center Center n =((∑(v i )) / p), where v i represents a sample, and p represents the number of samples in this class after clustering.

[0070] S6: Repeat S5 until the cluster center no longer changes significantly or the set number of iterations is reached;

[0071] S7: Use the cluster center obtained in S6 as the initial mean of the Gaussian mixture model, calculate the covariance of each cluster, and calculate the mixing coefficient of the Gaussian mixture model according to the proportion of data points in each cluster; the calculation expression of the mixing coefficient is:

[0072]

[0073] Where k is the number of mixed models, α k is the mixing weight, 0<α k <1, is the Gaussian model of the kth mixture component, μ k and represents the mean and covariance matrix.

[0074] S8: Calculate the posterior probability of each data point belonging to each Gaussian distribution through Bayesian theorem. According to the posterior probability, recalculate the mean, covariance matrix and mixing coefficient of each Gaussian distribution to maximize the likelihood function of the Gaussian mixture model.

[0075] Based on the given The calculation expression of the posterior probability is:

[0076]

[0077] In the formula, α k is the mixing weight, is the Gaussian model of the kth mixture component,

[0078] The calculation expressions for recalculating the mean, covariance matrix and mixing coefficient of each Gaussian distribution are:

[0079]

[0080]

[0081] In the formula, α k is the mixing weight, N represents the number of samples, γ jk μ represents the posterior probability that the i-th sample belongs to the k-th cluster. k is the mean of the kth Gaussian distribution, indicating the center position of the cluster, X n is the data point of the nth sample. represents the covariance matrix of the kth Gaussian distribution, describing the expansion direction and shape of the cluster. (X n -μ k )(X n -μ k )T It is the outer product of the difference between the nth sample and the mean, indicating the degree of dispersion of the sample and the mean.

[0082] S9: Repeat S8 until the parameters of the model no longer change significantly, or the increment of the likelihood function is less than a preset threshold;

[0083] The calculation expression of the likelihood function is:

[0084]

[0085] In the formula, L(θ|X) is the likelihood function value, N is the total number of samples, k is the number of Gaussian distributions, and α is k is the mixing weight, Represents the probability density function of the k-th Gaussian distribution.

[0086] S10: The Gaussian mixture model calculates the probability of each pixel belonging to each cluster, thereby obtaining the final clustering result, and setting all clustering results at the center of the spectrum to 0, thereby obtaining the result of eliminating the zero-order spectrum information.

[0087] In this example, the object to be measured is a resolution test target. The original hologram obtained is as follows: Figure 3 As shown. Perform Fourier transform on the original hologram to obtain the frequency information of the original hologram, that is, obtain the holographic image spectrum. In the spectrum, select pixels with intensities higher than 0.003 times the maximum spectrum intensity as initial data objects. The coordinate positions (u, v) of these pixels are selected as the input data for clustering. Each data object contains two dimensions, namely two-dimensional coordinates. The center point of the spectrum is selected as the initial clustering center of the zero-order spectrum, and the other two clustering centers are randomly selected. Calculate the Euclidean distance of each data point to all clustering centers, and classify them into the cluster center closest to it. For each cluster, recalculate its center until the cluster center no longer changes significantly or reaches the set number of iterations. The result after K-means preliminary clustering is as follows. Figure 4 As shown. Figure 4 It can be seen that K-means can successfully distinguish the three types of spectral information in the spectrum. Use the cluster center in the K-means result as the initial mean of GMM. According to the clustering results of K-means, calculate the distribution of data points in each cluster and obtain the covariance of each cluster. Initialize the mixing coefficient of GMM according to the ratio of the number of data points contained in each cluster in K-means to the total number of data points. Calculate the posterior probability that each data point belongs to each Gaussian distribution through Bayes' theorem. According to the posterior probability, recalculate the mean, covariance matrix and mixing coefficient of each Gaussian distribution. The final result of GMM clustering is as follows: Figure 5As shown in the figure, it can be seen that GMM divides the spectrum into three types of Gaussian distribution spectrum information, and then sets the cluster data at the center position to 0, and finally obtains the spectrum information after eliminating the zero-order term as shown in Figure 6 Then the phase of the resolution plate is reconstructed based on the angular spectrum method.

[0088] In order to verify the result of the present invention, the following steps are also included:

[0089] S11: extract the positive first-order spectrum information and center it, and reconstruct the intensity and phase of the object to be measured based on the angular spectrum method.

[0090] The calculation expression of the angular spectrum method is:

[0091]

[0092] I(x,y,z)=|U(x,y,z)| 2

[0093]

[0094] In the formula, and represent Fourier transform and inverse Fourier transform respectively. is a filter for extracting the +1 or -1 order spectrum, G(f x ,f y ,z) is the transfer function of the angular spectrum diffraction method, f x and f y is the frequency domain coordinate, λ is the wavelength of light; z is the reconstruction distance, I(x,y,z) and are the reconstructed intensity and phase of the object to be measured, respectively.

[0095] The reconstructed image in this embodiment is as follows Figure 7 As shown, the local enlarged image shows the 8th and 9th groups of elements in the resolution plate. It can be seen that the reconstructed phase image obtained by the present invention has the advantages of clear contours, no artifacts and high resolution. From the experimental results, the method can better restore the information of the object light, which proves the effectiveness of the present invention.

[0096] The preferred specific embodiments of the present invention are described in detail above. It should be understood that a person skilled in the art can make many modifications and changes based on the concept of the present invention without creative work. Therefore, any technical solution that can be obtained by a person skilled in the art through logical analysis, reasoning or limited experiments based on the concept of the present invention on the basis of the prior art should be within the scope of protection determined by the claims.

Claims

1. A method for eliminating the zero-order term of off-axis digital holography, characterized in that: The following steps are involved: S1: Run the off-axis digital holographic optical path system to obtain the off-axis hologram to be processed; S2: Perform Fourier transform on the off-axis hologram to be processed to obtain a spectrum diagram of the off-axis hologram; S3: According to the spectrum graph, select the pixel points with intensity higher than the threshold as the initial data object; S4: Select the center point of the spectrum graph as the initial cluster center of the zero-order spectrum, and randomly select two points as the cluster center of the positive first-order spectrum and the cluster center of the negative first-order spectrum respectively; S5: For each pixel in the initial data object, calculate its Euclidean distance to all cluster centers and classify it into the cluster center closest to it; recalculate the mean of all data points in each cluster to update the center of each cluster; S6: Repeat S5 until the cluster center no longer changes significantly or the set number of iterations is reached; S7: Use the cluster center in the result of S6 as the initial mean of GMM. According to the clustering result in S6, calculate the distribution of each cluster data point, calculate the covariance of each cluster, calculate the proportion of the number of data points contained in each cluster to the total number of data points, and initialize the mixing coefficient of GMM. S8: Calculate the posterior probability that each data point in S6 belongs to each Gaussian distribution through Bayesian theorem. According to the posterior probability, recalculate the mean, covariance matrix and mixing coefficient of each Gaussian distribution to maximize the likelihood function of the Gaussian mixture model. S9: Repeat S8 until the parameters of the model no longer change significantly, or the increment of the likelihood function is less than a preset threshold; S10: GMM calculates the probability of each pixel point belonging to each cluster, thereby obtaining the final clustering result, and sets all clustering results located at the center of the spectrum to 0, thereby obtaining the result of finally eliminating zero-order spectrum information.

2. The method for eliminating the zero-order term of off-axis digital holography according to claim 1, characterized in that: The parameter expression of the Gaussian mixture model is: Where k is the number of mixed models, α k is the mixing weight, 0<α k <1, is the Gaussian model of the kth mixture component, μ k and represents the mean and covariance matrix.

3. The method for eliminating the zero-order term of off-axis digital holography according to claim 1, characterized in that: The method also includes the following steps: S11: extracting the positive first-order spectrum information and centering it, and reconstructing the intensity and phase of the object to be measured based on the angular spectrum method.

4. The method for eliminating the zero-order term of off-axis digital holography according to claim 3, characterized in that: The calculation expression of the angular spectrum method is: I(x,y,z)=|U(x,y,z) 2 In the formula, and denote Fourier transform and inverse Fourier transform respectively, is a filter for extracting the +1 or -1 order spectrum, G(f x ,f y ,z) is the transfer function of the angular spectrum diffraction method, f x and f y is the frequency domain coordinate, λ is the wavelength of light; z is the reconstruction distance, I(x,y,z) and are the reconstructed intensity and phase of the object to be measured, respectively.

5. The method for eliminating the zero-order term of off-axis digital holography according to claim 1, characterized in that: The calculation expression of the Euclidean distance is: In the formula, v i represents the sample, center represents the cluster center coordinates, and n represents the observation value.

6. The method for eliminating the zero-order term of off-axis digital holography according to claim 1, characterized in that: The calculation expression of the mixing coefficient is: Where k is the number of mixed models, α k is the mixing weight, 0<α k <1, is the Gaussian model of the kth mixture component, μ k and represents the mean and covariance matrix.

7. The method for eliminating the zero-order term of off-axis digital holography according to claim 1, characterized in that: The calculation expression of the posterior probability is: In the formula, α k is the mixing weight, is the Gaussian model of the kth mixture component, 8. The method for eliminating the zero-order term of off-axis digital holography according to claim 1, characterized in that: The calculation expression for recalculating the mean, covariance matrix and mixing coefficient of each Gaussian distribution is: In the formula, α k is the mixing weight, N represents the number of samples, γ jk represents the posterior probability that the i-th sample belongs to the k-th cluster, μ k is the mean of the kth Gaussian distribution, indicating the center position of the cluster, X n is the data point of the nth sample, represents the covariance matrix of the k-th Gaussian distribution, describing the expansion direction and shape of the cluster, (X n -μ k )(X n -μ k ) T It is the outer product of the difference between the nth sample and the mean, indicating the degree of dispersion of the sample and the mean.

9. The method for eliminating the zero-order term of off-axis digital holography according to claim 1, characterized in that: The calculation expression of the likelihood function is: In the formula, L(θ|X) is the likelihood function value, N is the total number of samples, k is the number of Gaussian distributions, and α is k is the mixing weight, Represents the probability density function of the k-th Gaussian distribution.

10. The method for eliminating the zero-order term of off-axis digital holography according to claim 1, characterized in that: The calculation expression for updating the cluster center is: Center n =((∑(v i )) / p) In the formula, v i represents a sample, and p represents the number of samples in this class after clustering.

Images