Single-pixel compressed sensing phase shift holographic reconstruction method based on KAN-CAFM network

Abstract

The invention belongs to the technical field of optical information processing, and particularly relates to a single-pixel compressed sensing phase shift holographic reconstruction method based on a KAN-CAFM network, and the method comprises the steps: introducing a Kolmogorov-ArnoldNetwork (KAN) and a convolution-attention fusion module (CAFM), and constructing a high-efficiency mapping frame from one-dimensional compressed data to a multi-phase shift hologram: a. A KAN module: based on the Kolmogorov-ArnoldNetwork theorem (KART), carrying out the one-dimensional compressed data to the multi-phase shift hologram; high-dimensional nonlinear mapping is achieved through learnable univariate function combination, a linear weight matrix in a traditional MLP is replaced, and the modeling capacity of the network for high-dimensional phase shift features is remarkably improved. And b, a CAFM module which realizes cooperative enhancement of local feature extraction and global dependence modeling through double-branch parallel design (local branch + global branch) in combination with convolution operation and a self-attention mechanism.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of optical information processing, and particularly relates to a single-pixel compressed sensing phase shift holographic reconstruction method based on a KAN-CAFM network. BACKGROUND

[0002] Digital holography is a non-invasive quantitative phase imaging technology based on the principle of optical interference. By recording the interference hologram of the object light and the reference light and combining with the numerical reconstruction algorithm, the complete information of the amplitude and phase distribution of the sample can be obtained. It shows significant potential in real-time monitoring of biological cells, micro-nano device topography measurement and other fields. However, (1) DH faces the bottleneck of time resolution and data processing efficiency in dynamic monitoring: the existing technology relies on multi-frame hologram reconstruction, which limits the real-time imaging capability; dynamic measurement is limited; (2) single-exposure phase shift technology improves the time resolution, but requires processing of a large amount of single-frame holographic data, which puts higher requirements on storage and algorithm, such as data redundancy; in addition, (3) the detection system based on CCD / CMOS is limited by the physical arrangement of the pixel array, and the sensitivity decreases in the infrared or terahertz waveband, and some phase shift schemes rely on additional devices, which increases the system complexity and restricts the imaging rate; the system complexity is high.

[0003] Single-pixel imaging is a computational imaging technology based on spatial pattern coding. Its principle is derived from quantum ghost imaging and compressed sensing (CS) theory. It projects the target through structured illumination and synchronously records the projection signals using a single-pixel detector. Finally, the spatial information of the target scene is recovered through a reconstruction algorithm. Compared with traditional array detectors, SPI replaces high-cost CCD modules with single-pixel detectors, has stronger anti-noise ability in low-light environments, and overcomes the sensitivity limitations of array sensors in special wavebands such as infrared or terahertz. However, traditional SPI is limited by the Nyquist sampling theorem, requiring a large number of projection signals to achieve stable reconstruction, resulting in low time resolution and limited applicability to dynamic scenes; the time resolution is low. To break this bottleneck, under-sampling SPI algorithms; such as partial sampling Fourier SPI (FSPI) and partial sampling Hadamard SPI (OHSPI) focus on the sparse distribution characteristics of the image in the Fourier spectrum or Hadamard spectrum, and only sample the low-frequency region to achieve preliminary reconstruction. Compressed sensing SPI (CSSPI) further introduces CS theory, which can complete high-quality reconstruction at sub-Nyquist sampling rate, but still faces problems such as strong subjectivity in measurement matrix design, slow convergence speed in iterative algorithm, and high data volume required for reconstruction; the performance is poor at extremely low sampling rate. Traditional compression reconstruction algorithms rely on artificially designed measurement matrices; such as Hadamard basis and Fourier basis), lack of adaptability, and high computational complexity in the iterative process, making it difficult to meet real-time requirements; the reconstruction quality depends on artificial design.

[0004] With the rise of deep learning (DL) in the field of computational imaging, its powerful nonlinear mapping ability provides a new path to break through the technical bottleneck of digital holography (DH) and single-pixel imaging (SPI). By constructing a multi-layer neural network model, DL can extract high-dimensional features from massive data and establish a complex mapping relationship between the input signal and the target output, significantly improving the computational efficiency of the imaging system. In recent years, phase reconstruction methods based on deep learning have gradually replaced traditional multi-frame phase-shifting algorithms. Only a single hologram is needed to reconstruct holograms with different phase shifts through neural network backpropagation, without relying on mechanical phase shifters or multiple data acquisitions to obtain three-dimensional information of the sample, effectively solving the problem of the dependence of traditional phase-shifting technology on multiple frames of data and physical devices. The integration of DL and computational imaging has given birth to a variety of deep single-pixel imaging (DLSPI) methods, such as SPI based on convolutional neural network (CNN), SPI based on recurrent neural network (RNN), and SPI based on Transformer, which have shown significant advantages in end-to-end reconstruction, dynamic scene analysis, and complex texture recovery tasks, respectively. Compared with traditional partial sampling Fourier SPI (FSPI), partial sampling Hadamard SPI (OHSPI), and compressed sensing SPI (CSSPI), DLSPI avoids the subjective limitations of artificial design through data-driven measurement matrix optimization, breaks through the convergence speed bottleneck of traditional iterative algorithms using the forward inference characteristics of neural networks, and significantly reduces the amount of data required for reconstruction by learning the sparse representation of the high-level feature space through nonlinear mapping.

[0005] Qimnan Zhang et al., "Phase-shifting interferometry from single frame in-line interferogram using deep learning phase-shifting technology," Optics Communications 498; 2021 127226. proposed a deep learning-based phase-shifting network PSN; for the first time, end-to-end mapping from single-frame interferogram to multi-frame phase-shifted interferogram is realized, which uses a standard convolutional neural network (CNN) to directly map single-frame interferogram to multi-frame phase-shifted interferogram without mechanical phase shifter or multiple data acquisition. However, the independent multi-channel design lacks cross-channel feature fusion mechanism, resulting in insufficient consistency between the generated phase-shifted interferograms; the cross-channel feature consistency is insufficient; the basic convolutional structure only uses standard convolutional layers without introducing attention mechanisms or global dependency modeling, making it difficult to capture the complex spatial relationships of high-dimensional phase-shifted features; the local feature extraction capability is limited; although the PSN technology reduces the dependence on mechanical phase shifters, it still needs to be based on CCD / CMOS array detectors to realize high-resolution interferogram acquisition, resulting in high hardware cost and difficulty in adapting to special spectral bands; such as infrared or terahertz bands; the application cost and adaptability are limited; therefore, the present application proposes a single-pixel compressive sensing phase-shifting holographic reconstruction method based on KAN-CAFM network. SUMMARY

[0006] The purpose of the present application is to provide a single-pixel compressive sensing phase-shifting holographic reconstruction method based on KAN-CAFM network, which can solve the technical bottlenecks of traditional digital holography (DH) in dynamic monitoring, data processing efficiency and hardware dependency, and single-pixel imaging (SPI) in time resolution, low sampling rate performance and reconstruction quality dependent on artificial design of measurement matrix. The core technical scheme of the method includes data acquisition and preprocessing, network structure design, and training and optimization.

[0007] The technical scheme adopted by the present application is as follows:

[0008] The single-pixel compressive sensing phase-shifting holographic reconstruction method based on KAN-CAFM network comprises the following steps:

[0009] Step 1. Build a single-pixel interference sampling system hardware: use a 633 nm frequency stabilized helium-neon laser, a polarizing beam splitter PBS and a beam splitter BS, load target information through DMD1, and construct a single-pixel interference sampling system; generate a hologram: the BS combines the reference light and the DMD1 modulated object light through a 1 / 4 wave plate, and the experimental hologram obtained by interference of the object light carrying object information and the unmodulated reference light can be expressed as:

[0010]

[0011] Generally, the incident reference light R is a plane wave with constant amplitude A R , which is independent of coordinates. Therefore, it can be expressed as where A R and represent the amplitude and phase of the reference light, respectively. O(x H , y H ) represents the object light on the object plane propagating to the recording plane H through Fresnel diffraction to form an object light field.

[0012] Step 2. Compressed sensing data acquisition: DMD2 loads a random binary measurement matrix Ψ, and converts the holographic image pixel intensity I H into low-dimensional data. The formula is:

[0013]

[0014] In the formula, <·> represents the inner product operation, y m is the mth row measurement value, is the state of the mth row of micro-mirrors of DMD;

[0015] Repeat the acquisition M times to obtain one-dimensional compressed data:

[0016] Y = [y1 y2…y m …y M ] = <Ψ, I H > (3);

[0017] In the formula, Ψ ∈ {0, 1} M×N is a random binary matrix generated by the compressed sensing algorithm, and N is the total number of pixels;

[0018] Step 3. Construct MNIST dataset: based on the MNIST handwritten digit set, collect 4 frames of phase shift holograms; the phase shift amounts are 0, π / 2, π, and 3π / 2, and the one-dimensional data is compressed at a sampling rate of 20%; divide it into 80% training set, 10% validation set, and 10% test set;

[0019] Step 4. Design KAN-CAFM network architecture: design an encoding-decoding structure network, which includes an encoder and a decoder. The encoder includes an up-sampling transpose convolution block + CAFM module + down-sampling KA-C block; the decoder includes a multi-branch copy + CAFM module + KA-DC block + transpose convolution layer, which preserves details through U-shaped jump connection and outputs four frames of phase shift holograms. The formula is:

[0020] H′ k = φ KAFE-PSN (Y), k ∈ {1, 2, 3, 4} (4);

[0021] Step 5: input data processing;

[0022] Step 6. The encoder upsampling part uses a ReLU activation + a transpose convolution block with a 4x4 kernel and a stride of 2 to upsample the 52430x1x1 one-dimensional data to a 128x512x512 three-dimensional feature space. Subsequently, the network introduces a convolution-attention fusion module (CAFM) to realize the synergistic enhancement of local features and global dependencies through the fusion of convolution operations and self-attention mechanisms.

[0023] Step 7. Implementing encoder downsampling: the downsampling part uses a 4x4 convolution kernel combined with three KA-C blocks to extract features into a 512x8x8 compact representation; the stride of the 4x4 convolution kernel is 2, the LeakyReLU activation is α=0.2; the three KA-C blocks contain PatchEmbed projection, KAN linear layer, and residual connection;

[0024] Step 8. Decoder structure design, realizing the decoder multi-branch structure: the decoder copies the 512x8x8 features output by the encoder into four independent branches; corresponding to four phase-shifted graphs; each branch is enhanced by the CAFM module, three KA-DC blocks with residual connection, and three upsampled transpose convolution layers to restore to 1x512x512, obtaining four interference graph outputs H' with specific phase shifts k ,k∈{1,2,3,4};

[0025] Step 9. Designing the CAFM module: using the convolution-attention fusion module (CAFM), the local branch extracts local features through 1x1 convolution, channel shuffle, and 3x3 convolution; the global branch captures long-distance dependencies through the self-attention mechanism, and the fused output enhances the features;

[0026] Step 10. Designing the KAN module: based on the Kolmogorov-Arnold theorem, the KAN block uses SiLU basis functions and B-spline functions to replace the MLP linear layer; through residual connection, the non-linear mapping ability is improved;

[0027] Step 11. Defining the L1 loss function: using the L1 loss function to calculate the pixel-level difference between the predicted value and the true value of the four-frame phase-shifted hologram, the formula is the sum of absolute errors normalized by the total number of pixels, guiding the network parameter optimization;

[0028] Step 12. Configuring network training parameters: learning rate 0.0001, iterating 300 times to complete network training;

[0029] Step 13. Synthesizing four-frame phase-shifted holograms: the network outputs four-frame interference graphs, completing the mapping from single-frame input to four-frame phase-shifted graphs;

[0030] Step 14. Fresnel diffraction reconstruction target image: the complex-valued hologram calculated by the four-step phase shift algorithm is converted into a real target image by the Fresnel diffraction formula, and high-fidelity reconstruction is realized.

[0031] The technical effects achieved by the present application are:

[0032] The present application reduces the dependence of traditional digital holography (DH) and single-pixel imaging (SPI) on phase shift and area array acquisition equipment. The present application solves the problem of "traditional DH relying on mechanical phase shifter devices such as phase shifters to obtain multiple interference patterns step by step, resulting in high system complexity and limited dynamic measurement"; the present application also solves the problem of "traditional SPI being limited by the Nyquist sampling theorem, requiring a large number of projection signals to achieve stable reconstruction, and having low time resolution".

[0033] The present application introduces Kolmogorov-Arnold Networks (KAN) and Convolution-Attention Fusion Module (CAFM) to construct an efficient mapping framework from one-dimensional compressed data to multi-phase shift holograms. At a sampling rate of 20%, the present application can generate high-quality holograms with an average SSIM of 0.8742 and an average PSNR of 28.081 dB. The average SSIM of the multi-hologram output by the network remains above 0.85 under 50% Gaussian noise or 0.05% sampling rate. This method does not require a mechanical phase shifter or multiple data acquisitions, breaking through the resolution and field of view limitations of traditional array sensors, and providing a new solution for real-time and lightweight applications in industrial detection, biomedical imaging, and optical information encryption.

[0034] The present application has high-precision phase reconstruction: at a low sampling rate of 20%, the phase shift hologram generated by the network has an average structural similarity SSIM of 0.8742 and a peak signal-to-noise ratio PSNR of 28.081, and the generated phase shift hologram is highly consistent with the label in terms of shape and gray scale distribution.

[0035] The present application has robustness at extremely low sampling rates: at an extremely low sampling rate of 0.05%, the KAFE-SPN can still maintain stable reconstruction performance. At a sampling rate of 0.05%, the average SSIM of the hologram is 0.8592, and the peak signal-to-noise ratio is 26.684. BRIEF DESCRIPTION OF DRAWINGS

[0036] Figure 1 is a flowchart of the single-pixel compressed sensing phase shift holographic reconstruction method of the present application based on KAN-CAFM network;

[0037] Figure 2 is a data acquisition optical path diagram of the present application;

[0038] Figure 3 is a single-pixel phase shift network structure diagram of the present application;

[0039] Figure 4 (a-d) holograms with different phase shift amounts as labels; (e-h) holograms with different phase shift amounts generated by KAFE-SPN; (i-l) cross-sectional curves corresponding to holograms with different phase shift amounts;

[0040] Figure 5 (a) SSIM and (b) PSNR of the average of holograms and reconstruction results generated by the network in the application. DETAILED DESCRIPTION

[0041] In order to make the objects and advantages of the present application clearer, the present application will be specifically described below in conjunction with embodiments. It should be understood that the following description is only used to describe one or several specific embodiments of the present application, and does not strictly limit the specific protection scope requested by the present application.

[0042] As shown in Figure 1 , the present application relates to the field of optical information processing, and specifically aims at the application of fusion of Single-Pixel Imaging (SPI) and Digital Holography (DH). The core goal is to solve the problems of multi-frame data acquisition, mechanical phase shifter dependence and low computational efficiency in traditional holographic imaging through a deep learning method, and to realize efficient, lightweight and real-time holographic image reconstruction.

[0043] The single-pixel compressive sensing phase shift holographic reconstruction method based on the KAN-CAFM network comprises the following steps:

[0044] Step 1. Build a single-pixel interference sampling system hardware: use a 633 nm frequency stabilized helium-neon laser, a polarizing beam splitter PBS and a beam splitter prism BS, load target information through a DMD1, and construct a single-pixel interference sampling system; generate a hologram: the BS combines the reference light and the DMD1 modulated object light together through a 1 / 4 wave plate, and the experimental hologram obtained by interference of the object light carrying object information and the unmodulated reference light can be expressed as:

[0045]

[0046] Generally, the incident reference light R is generally a plane wave with a constant amplitude A R , which is independent of coordinates. Therefore, it can be expressed as where A R and represent the amplitude and phase of the reference light, respectively. O(x H ,y H ) represents the object light on the object plane propagating to the recording plane H through Fresnel diffraction to form an object light field.

[0047] Step 2. Compressed sensing data acquisition: DMD2 loads a random binary measurement matrix Ψ, and the holographic image pixel intensity I H The conversion to low-dimensional data formula is:

[0048]

[0049] In the formula, <·> represents the inner product operation, y m is the mth row of the measurement value, is the state of the micro-mirror of the mth row of DMD;

[0050] Repeat M times to obtain one-dimensional compressed data:

[0051] Y = [y1 y2…y m …y M ] = <Ψ, I H > (3);

[0052] In the formula, Ψ ∈ {0, 1} M×N is a random binary matrix generated by the compressed sensing algorithm, and N is the total number of pixels;

[0053] Step 3. Construct the MNIST dataset: Based on the MNIST handwritten digit set, 4 frames of phase shift holograms are collected; the phase shift is 0, π / 2, π, 3π / 2, and the one-dimensional data is compressed to a sampling rate of 20%; 80% of the training set, 10% of the validation set, and 10% of the test set are divided;

[0054] Step 4. Design the KAN-CAFM network architecture: design an encoding-decoding structure network, including an encoder and a decoder, the encoder includes an up-sampling transpose convolution block + CAFM module + down-sampling KA-C block; the decoder includes a multi-branch copy + CAFM module + KA-DC block + transpose convolution layer, which preserves details through U-shaped jump connection, and outputs four frames of phase shift holograms formula:

[0055] H′ k = φ KAFE-PSN (Y), k ∈ {1, 2, 3, 4} (4);

[0056] Step 5: input data processing;

[0057] Step 6. The up-sampling part of the encoder uses ReLU activation + a transpose convolution block with a 4×4 transpose convolution kernel and a step size of 2 to up-sample the one-dimensional data of 52430×1×1 to a three-dimensional feature space of 128×512×512. Subsequently, the network introduces a convolution-attention fusion module CAFM module, which realizes the collaborative enhancement of local features and global dependencies through the fusion of convolution operation and self-attention mechanism.

[0058] Step 7. Realize the encoder downsampling: the downsampling part adopts a 4x4 convolution kernel combined with three KA-C blocks to extract features into a compact representation of 512x8x8; the 4x4 convolution kernel has a step of 2, LeakyReLU activation, and alpha=0.2; the three KA-C blocks contain PatchEmbed projection, KAN linear layer, and residual connection;

[0059] Step 8. Design the decoder structure to realize the decoder multi-branch structure: the decoder copies the 512x8x8 features output by the encoder into four independent branches; corresponding to four phase shift graphs; each branch is enhanced through a CAFM module, three KA-DC blocks with residual connection, and three up-sampling transpose convolution layers to restore to 1x512x512, and outputs four frames of interference graph formula:

[0060] H' k =φ KAFE-PSN (Y),k∈{1,2,3,4} (4);

[0061] Step 9. Design the CAFM module: adopt the convolution-attention fusion module CAFM, the local branch extracts local features through 1x1 convolution, channel shuffle and 3x3 convolution; the global branch captures long-distance dependence through self-attention mechanism, and outputs enhanced features after fusion;

[0062] Step 10. Design the KAN module: design the KAN block based on the Kolmogorov-Arnold theorem, replace the MLP linear layer with SiLU basis function and B-spline function; improve the non-linear mapping ability through residual connection;

[0063] Step 11. Define the L1 loss function: use the L1 loss function to calculate the pixel-level difference between the predicted value and the true value of the four frames of phase shift hologram, the formula is the sum of absolute errors normalized by the total number of pixels, which guides the optimization of network parameters;

[0064] Step 12. Configure network training parameters: learning rate 0.0001, iterate 300 times to complete network training;

[0065] Step 13. Synthesize four frames of phase shift hologram: the network outputs four frames of interference graph, which completes the mapping from single frame input to four frames of phase shift graph;

[0066] Step 14. Fresnel diffraction reconstructs the target image: through the Fresnel diffraction formula, the complex-valued hologram calculated by the four-step phase shift algorithm is converted into a real target image, realizing high-fidelity reconstruction

[0067] In the application, specifically, in step 1, specifically: a Mach-Zehnder interference system based on single-pixel sampling modification is constructed to complete the sampling of one-dimensional single-pixel data; the optical path diagram is as follows Figure 2As shown; using a 633nm frequency-stabilized helium-neon laser as the light source, the incident light is split into a reference beam and an object beam using a polarization beam splitter (PBS); the PBS combines the reference beam and the object beam modulated by DMD1 and passes them together through a quarter-wave plate. The experimental hologram obtained by the interference of the object beam carrying object information with the unmodulated reference beam can be expressed as:

[0068]

[0069] In step 2, specifically, DMD2 subsequently loads a random binary measurement matrix Ψ; the random binary measurement matrix Ψ is generated by a compressed sensing algorithm, and the holographic pixel intensity I... H Convert to low-dimensional compressed data y m The formula is:

[0070]

[0071] The final modulated compressed hologram is acquired by a single-pixel detector, and this process is repeated M times to obtain the compressed one-dimensional data.

[0072] Y = [y1 y2…y m …y M ]=<Ψ,I>(3).

[0073] Step 3. Construct the MNIST dataset: Collect 4 frames of phase-shifted holograms based on the MNIST handwritten digit set; the phase shifts are 0, π / 2, π, and 3π / 2, compressed into one-dimensional data with a 20% sampling rate, and divided into 80% training set, 10% validation set, and 10% test set to ensure the model's generalization ability;

[0074] Specifically, in step 4, after collecting the dataset, a network combining the Kolmogorov-Arnold network and an attention mechanism is designed for learning. 1212 holograms were collected as the training set, 100 as the test set, and 100 as the validation set. By combining the optimized network model, only a single frame of hologram is needed as input to complete the phase-shifting operation, thus obtaining four high-fidelity holograms with specific phase shifts. The designed single-pixel phase-shifting network structure is as follows: Figure 3 As shown, a framework with an encoder-decoder structure is designed, comprising an encoder, a CAFM module (convolutional-attention fusion), a KAN module, and a decoder; the CAFM module includes convolutional-attention fusion; the network input is one-dimensional compressed data Y, and the output is four frames of high-fidelity real-time holograms H′. k k∈{1,2,3,4}, realize the mapping from low-dimensional data to high-dimensional features:

[0075] H′ k =φ KAFE-PSN(Y), k e {1, 2, 3, 4} (4).

[0076] The step 5 is specifically, input data processing, the input is 52430 one-dimensional data, because it is one-dimensional data, therefore the feature dimension of the input is set to [batch_size, 52430, 1, 1]; here batch_size represents the number of samples input into the network at one time, 52429 represents 52429 channels, and 1x1 is the spatial dimension of the image; these preprocessed features serve as initial data entering the phase shift network;

[0077] The step 6 is specifically, the processing of the up-sampling part in the encoder, the input features are first processed by the up-sampling part in the encoder, which is composed of a series of transpose convolution blocks, each of which is composed of a Relu activation function and a 4x4 transpose convolution kernel, and the convolution kernel step size is 2; the transpose convolution block up-samples the original one-dimensional signal 52430x1x1 to a three-dimensional feature space of 128x512x512; then the network introduces a convolution-attention fusion module to realize the synergistic enhancement of local features and global dependencies through the fusion of convolution operation and self-attention mechanism.

[0078] The step 7 is specifically, the processing of the down-sampling part in the encoder, in which a 4x4 convolution kernel is used, the convolution kernel step size is 2, and the convolution layer is followed by an activation function layer and a normalization layer, and the activation function uses LeakyRelu:

[0079]

[0080] wherein a is set to 0.2; on this basis, the down-sampling stage of the encoder includes three standard convolution blocks and three KA-C blocks, forming a hierarchical feature extraction network; in the standard convolution block, the feature map is gradually extracted as 512x64x64;

[0081] Then the three KA-C blocks further process the features, project the features to the shape adapted by the KAN linear layer through PatchEmbed, process through the KAN, then normalize, then connect the output of the KAN linear layer with the input in residual connection to ensure the stability of the deep network, and finally further down-sample through the convolution layer; after processing by the three KA-C blocks, the encoder finally extracts the features as 512x8x8 compact representation;

[0082] The step 8 is specifically, the processing of the decoder part, the decoder adopts a U-shaped architecture with dense skip connection, which splices the high-resolution features of the encoder with the low-resolution features of the decoder to strengthen the model's ability to retain high-resolution details in the input data; at the same time, a multi-branch structure is adopted, which copies the 512x8x8 feature map output by the encoder into four independent branches H k, k e {1, 2, 3, 4}, each branch corresponds to a specific phase shift control, and the sharing of the encoder features is achieved through the parameter copy mechanism; each decoder branch first applies

[0083] The CAFM attention module enhances the features, and then gradually restores the feature map size through a six-layer upsampling process; it is worth noting that, like the encoder downsampling stage, the decoder upsampling process also designs two-stage upsampling modules, that is, first through three KA-DC blocks, and then through three transpose convolution layers to finally expand to 1x512x512, obtaining four interference outputs H' with specific phase shifts k , k e {1, 2, 3, 4}.

[0084] The step 9 specifically, the CAFM block, the CAFM block adopts a double-branch parallel design, and the module includes two important branches of local and global; the local branch uses convolution and channel rearrangement operation to extract local features, while the global branch captures long-distance dependency through self-attention mechanism, and the two complement each other in the feature space; in the local branch, first, a convolution with a kernel of 1x1 is used to adjust the channel dimension to enhance the information integration across channels, then a channel shuffling operation is performed, and finally a convolution with a kernel size of 3x3 is used to extract features, the output F conv of the local branch can be represented as:

[0085] F conv = W 3×3×3 (CS(W 1×1 (X)))(6)

[0086] In the formula, W 1×1 and W 3×3×3 represent 1x1 and 3x3 convolution kernels, and CS represents a channel shuffling operation;

[0087] In the global branch, first, a depth convolution with a kernel of 1x1 and 3x3 is used to generate queries (Q), keys (K) and values (V), obtaining three tensors with a shape of , then the attention weight matrix is calculated through and

[0088]

[0089] In the formula, a is a learnable parameter used to control the amplitude of the product. The output of the global branch can be represented as:

[0090]

[0091] ​To further enhance the stability of the network and promote deep feature learning, the output results of the local branch and the global branch are fused; the final output is:

[0092] F CAFM = F conv + F att (9) ;

[0093] The step 10 is specific, KAN block, the design of KAN block is based on Kolmogorov-Arnold representation theorem, which represents that any multivariate continuous function can be represented as a combination of a series of single variable functions, that is, for the input vector x p , KAN is represented as:

[0094]

[0095] Where Φ q and φ q,p are single variable functions, representing the qth layer of the entire KAN network; each KAN layer has n in dimensional input and n out dimensional output, so Φ q contains n in ×n out learnable activation functions φ:

[0096] Φ q = {φ q,p}, p = 1, 2, …, n in , q = 1, 2, …, n out (11)

[0097] In the KAFE-SPN network, the linear weight matrix in the traditional MLP is replaced by a learnable single variable function, and the non-linear expression ability is improved by using the single variable function combination characteristics of KAN:

[0098] φ(x) = w b ·b(x) + w s ·spline(x) (12)

[0099] Where b(x) is a Sigmoid Linear Unit (SiLU) basis function, and spline(x) is a parameterized B-spline function; therefore, in the KAFE-SPN network designed by us, the KAN linear layer linearly combines the outputs of the basis function branch and the B-spline branch:

[0100] KANLinear(x) = Linear(SiLU(x)·w b ) + Linear(bases(x)·w s ) (13)

[0101] Here, bases(x) is the combination result of B-spline basis functions, w b and w s are the learnable weights of basis function branches and spline branches, respectively; to ensure the training stability of the deep network, the KAN block adopts a residual connection mechanism, and the final output is:

[0102] F KAN = Norm(KANLinear(x)) + shortcut(x) (14)

[0103] Where shortcut(x) is the projection layer of the input feature.

[0104] The step 11 is specifically, in the definition of the loss function, in the KAFE-SPN network, the L1 loss function is used as the loss function of network training, and its core advantage lies in the robustness to optical path noise and outliers, and the ability to preserve detailed features in the generation process of phase shift interferograms; the mathematical expression of the L1 loss function is:

[0105]

[0106] Where y i represents the true interference intensity value of the i-th pixel, f(x i ) represents the predicted interference intensity value of the corresponding position network, and n is the total number of pixels; in the KAFE-SPN network, N = 4 x 512 x 512, corresponding to the total number of pixels of four frames of phase shift interferograms, the L1 loss function is directly applied to the output section of the encoder-decoder structure, by comparing the pixel-level difference between the predicted phase shift interferogram and the true interferogram, the network parameters are optimized; for the four-channel phase shift output of the KAFE-SPN network, the L1 loss function can be extended to

[0107]

[0108] Where y i (c,h,w) represents the true interference intensity value of the c-th phase shift channel, the h-th row and the w-th column, f(x i ) (c,h,w) represents the predicted interference intensity value of the corresponding position network;

[0109] The step 12 is specifically, all programs are executed in the Python3.9 environment using the PyTorch command, and NVIDIA GeForce GTX4090Ti GPU is used for acceleration operation, the Adam optimizer is used, the learning rate is set to 0.0001 for training, and the total iteration step is 300.

[0110] Specifically, in step 13, after obtaining the four-frame phase-shifted holograms output by the network, the phase-shifting technique can be used to record and reconstruct complex-valued holograms of the same object using the four-frame phase-shifted holograms; the network generates a hologram with a phase shift θ. k The form of a single-frame hologram (k = 1, 2, 3, 4) is as follows:

[0111] I k =|A(x,y)| 2 +|R(x,y)| 2 +2A(x,y)R(x,y)cos(φ+θ k (17)

[0112] Equation (5) is used to obtain a four-frame phase-shifted composite hologram:

[0113] I F (x,y)=(I1-jI2-I3+jI4) / 4 (18)

[0114] Specifically, step 14 involves phase-shift synthesis of hologram I. F The portions corresponding to the conjugate image and the zeroth-order image of the reconstructed light field were eliminated in (x,y), thus the light wave at the reconstructed distance Z was simulated using the Fresnel diffraction formula. r The image is reconstructed by propagation at a certain point; the specific formula is as follows:

[0115]

[0116] This invention introduces Kolmogorov-Arnold Networks (KAN) and a Convolutional-Attention Fusion Module (CAFM) to construct an efficient mapping framework from one-dimensional compressed data to multi-phase-shift holograms: a. KAN module: Based on the Kolmogorov-Arnold theorem (KART), it achieves high-dimensional nonlinear mapping through learnable univariate function combinations, replacing the linear weight matrix in traditional MLPs and significantly improving the network's ability to model high-dimensional phase-shift features. b. CAFM module: Through a dual-branch parallel design (local branch + global branch), combined with convolutional operations and self-attention mechanisms, it achieves synergistic enhancement of local feature extraction and global dependency modeling.

[0117] This invention addresses the phase shift interference caused by the independent design of multiple channels in the traditional PSN method by utilizing the cross-channel feature fusion mechanism of the CAFM module. Figure 1 To address the issue of insufficient consistency, the local branch extracts local features through convolution and channel rearrangement operations, while the global branch captures long-range dependencies through a self-attention mechanism. The fusion of these two methods outputs a more consistent multi-frame phase-shifted interferogram. The decoder employs a SIMO (Single-Input Multiple-Output) architecture, simultaneously generating four phase-shifted holograms with phase shifts of 0, π / 2, π, and 3π / 2, eliminating the need for multiple data acquisitions or mechanical phase shifters.

[0118] The application improves the stability and feature expression ability of a deep network through hierarchical design (standard convolution block + KA-C block) of an encoder-decoder and a residual connection mechanism.

[0119] The application adopts a single-pixel detector to replace a traditional CCD / CMOS array, combines a compressed sensing (CS) theory, reduces hardware cost, and adapts to special spectral bands, such as infrared / terahertz. A random binary measurement matrix is loaded by a DMD2, and one-dimensional data is generated by a CS algorithm to compress pixel intensity of a holographic image. The pixel detector only needs to record the light intensity signal of a single pixel, avoiding the high cost and band limitation of a traditional array sensor.

[0120] The application has high-precision phase reconstruction: under the condition of a low sampling rate such as 20%, the average structural similarity (SSIM) of the phase shift hologram generated by the network reaches 0.8742, and the peak signal-to-noise ratio (PSNR) reaches 28.081, and the generated phase shift hologram is highly consistent with the label in shape and gray distribution, as shown in Figure 4 .

[0121] The application has robustness under an extremely low sampling rate: under the condition of an extremely low sampling rate such as 0.05%, the KAFE-SPN can still maintain stable reconstruction performance. Under the condition of 0.05 sampling rate, the average SSIM of the hologram reaches 0.8592, and the peak signal-to-noise ratio reaches 26.684.

[0122] Experimental cases of the application:

[0123] Example 1: high-precision hologram reconstruction under a 20% sampling rate

[0124] Test scheme:

[0125] Input data: a single-frame hologram of a handwritten number in an MNIST handwritten number data set is compressed into one-dimensional data with a 20% sampling rate, 52429 channels, and 1x1 spatial dimensions.

[0126] Network configuration: the network uses default parameters, a learning rate of 0.0001, an Adam optimizer, 300 iterations, and generates four frames of phase shift holograms with phase shifts of 0, π / 2, π, and 3π / 2.

[0127] Reconstruction algorithm: a four-step phase shift algorithm is used to recover the phase of the generated four frames of phase shift holograms.

[0128] Test effect:

[0129] Reconstruction quality: the SSIM average of the four-frame phase-shift hologram is 0.8742, and the PSNR average is 28.081 dB, as shown in Figure 5 . The SSIM of the reconstructed target image is 0.8616, and the PSNR is 25.168 dB, as shown in Figure 5 .

[0130] Advantages: the structural similarity SSIM and signal-to-noise ratio PSNR of the hologram generated by the network reach the practical level under the condition of using only 20% of the original data amount, verifying the high efficiency of the network under low sampling rate.

[0131] Example 2: robustness test under a low sampling rate of 0.05%;

[0132] Test scheme:

[0133] Input data: compress the same set of single-frame holograms of handwritten digits into one-dimensional data with a sampling rate of 10%, 5%, 0.05%, and 0.01%.

[0134] Network configuration: the network parameters are the same as in Example 1.

[0135] Reconstruction algorithm: perform phase recovery on the generated four-frame phase-shift hologram based on the four-step phase-shift algorithm.

[0136] Reconstruction quality:

[0137] Under a sampling rate of 10%, the SSIM average of the four-frame phase-shift hologram is 0.8687, and the PSNR average is 27.434 dB, as shown in Figure 5 . The SSIM of the reconstructed target image is 0.8495, and the PSNR is 24.873 dB, as shown in Figure 5 .

[0138] Under a sampling rate of 5%, the SSIM average of the four-frame phase-shift hologram is 0.8652, and the PSNR average is 27.140 dB, as shown in Figure 5 . The SSIM of the reconstructed target image is 0.8356, and the PSNR is 24.679 dB (see Figure 5 ).

[0139] Under a sampling rate of 0.05%, the SSIM average of the four-frame phase-shift hologram is 0.8592, and the PSNR average is 26.684 dB, as shown in Figure 5 . The SSIM of the reconstructed target image is 0.8222, and the PSNR is 24.208 dB, as shown in Figure 5 .

[0140] Under a sampling rate of 0.01%, the SSIM average of the four-frame phase-shift hologram is 0.7672, and the PSNR average is 21.530 dB, as shown inFigure 5 The reconstructed target image SSIM is 0.6102 and the PSNR is 21.926 dB, as shown in Table 1. ​

[0141] Advantages:

[0142] At an extremely low sampling rate, such as 0.05%, a high SSIM (>0.85) and PSNR (>25 dB) can still be maintained, proving the robustness of the method to extremely low sampling rates.

[0143] The above description is only preferred embodiments of the present application, and it should be pointed out that, for those skilled in the art, without departing from the principles of the present application, a number of improvements and refinements can be made, which should also be considered as the protection scope of the present application. The structures, devices and operation methods not specifically described and explained in the present application, such as without special description and limitation, are implemented according to the conventional means in the art.​

Claims

1. A single-pixel compressed sensing phase-shift holographic reconstruction method based on KAN-CAFM network, characterized in that: Includes the following steps: Step 1. Hardware Construction of a Single-Pixel Interference Sampling System: Using a 633nm frequency-stabilized helium-neon laser, a polarization beam splitter (PBS), and a beam splitter (BS), target information is loaded through a DMD1 to construct a single-pixel interference sampling system; Hologram Generation: The BS combines the reference light and the object light modulated by the DMD1 and passes them through a quarter-wave plate. The experimental interference hologram obtained by the interference of the object light carrying object information with the unmodulated reference light can be expressed as: Typically, the incident reference light R is taken to have an amplitude of constant A. R A plane wave, independent of coordinates; therefore, it can be represented as Where A R and O(x) represents the amplitude and phase of the reference light, respectively; H ,y H This indicates that the object light on the object plane propagates to the recording plane H through Fresnel diffraction, forming the object light field; Step 2. Compressed Sensing Data Acquisition: The DMD2 loads a random binary measurement matrix Ψ, and the holographic pixel intensity I... H The formula for converting to low-dimensional data is: In the formula, <·> represents the inner product operation, and y m For the measurement value of the m-th row, This represents the micromirror state in the m-th row of the DMD; One-dimensional compressed data is obtained by repeatedly collecting data M times: Y=[y1 y2 … y m … yes M ]=<Ψ,I H > (3); In the formula, Ψ∈{0,1} M×N It is represented as a random binary matrix generated by the compressed sensing algorithm, where N is the total number of pixels; Step 3. Construct the MNIST dataset: Collect 4 frames of phase-shifted holograms based on the MNIST handwritten digit set; the phase shifts are 0, π / 2, π, and 3π / 2, compressed into one-dimensional data with a sampling rate of 20%, and divided into 80% training set, 10% validation set, and 10% test set. Step 4. Design the KAN-CAFM network architecture: Design an encoder-decoder network, including an encoder and a decoder. The encoder includes an upsampled transposed convolutional block + CAFM module + downsampled KA-C block; the decoder includes multi-branch replication + CAFM module + KA-DC block + transposed convolutional layer. Details are preserved through U-shaped skip connections, and the output is a four-frame phase-shifted hologram. H′ k =φ KAFE-PSN (Y),k∈{1,2,3,4} (4); Step 5: Input data processing; Step 6. Implement encoder upsampling: The encoder upsampling part uses ReLU activation + 4×4 transposed convolution kernel and transposed convolution block with stride 2 to upsample the 52430×1×1 one-dimensional data to a 128×512×512 three-dimensional feature space. Then the network introduces the convolution-attention fusion module CAFM module, which achieves the synergistic enhancement of local features and global dependencies through the fusion of convolution operation and self-attention mechanism. Step 7. Implement encoder downsampling: The downsampling part uses a 4×4 convolutional kernel combined with three layers of KA-C blocks to extract features into a compact representation of 512×8×8; the stride of the 4×4 convolutional kernel is 2, LeakyReLU activation, α=0.2; the three layers of KA-C blocks contain PatchEmbed projection, KAN linear layer and residual connection; Step 8. Decoder structure design, implementing a multi-branch structure for the decoder: The decoder copies the 512×8×8 features output by the encoder into 4 independent branches; The corresponding four-frame phase-shifted graphs are obtained; each branch is enhanced by a CAFM module, restored to 1×512×512 by three layers of KA-DC blocks with residual connections and three layers of upsampled transposed convolutional layers, resulting in four interferogram outputs H′ with specific phase shifts. k k∈{1,2,3,4}; Step 9. Design the CAFM module: The convolution-attention fusion module CAFM is adopted. Local branches extract local features through 1×1 convolution, channel shuffling and 3×3 convolution. The global branch captures long-range dependencies through a self-attention mechanism, and outputs enhanced features after fusion. Step 10. Design the KAN module: Design the KAN block based on the Kolmogorov-Arnold theorem, replace the MLP linear layer with SiLU basis functions and B spline functions; improve the nonlinear mapping capability through residual connections; Step 11. Define the L1 loss function: The L1 loss function is used to calculate the pixel-level difference between the predicted and true values ​​of the four-frame phase-shifted holograms. The formula is the sum of the absolute errors normalized to the total number of pixels, which guides the optimization of network parameters. Step 12. Configure network training parameters: learning rate 0.0001, 300 iterations to complete network training; Step 13. Synthesize four-frame phase-shifted holograms: The network outputs four-frame interferograms, completing the mapping from a single-frame input to four-frame phase-shifted holograms; Step 14. Fresnel diffraction reconstruction of the target image: Using the Fresnel diffraction formula, the complex-valued hologram calculated by the four-step phase-shifting algorithm is converted into a real target image to achieve high-fidelity reconstruction.

2. The single-pixel compressed sensing phase-shift holographic reconstruction method based on KAN-CAFM network according to claim 1, characterized in that: In step 1, specifically: a Mach-Zehnder interferometer system based on single-pixel sampling modification was constructed to sample one-dimensional single-pixel data; a 633nm frequency-stabilized helium-neon laser was used as the light source, and a polarization beam splitter (PBS) was used to split the incident light into a reference beam and an object beam; the PBS combined the reference beam and the object beam modulated by DMD1 and passed them through a quarter-wave plate. The experimental hologram obtained by the interference of the object beam carrying object information with the unmodulated reference beam can be expressed as: In step 2, specifically, DMD2 subsequently loads a random binary measurement matrix Ψ; the random binary measurement matrix Ψ is generated by a compressed sensing algorithm, and the holographic pixel intensity I... H Convert to low-dimensional compressed data y m The formula is: The final modulated compressed hologram is acquired by a single-pixel detector, and this process is repeated M times to obtain the compressed one-dimensional data. Y=[y1 y2 …y m …y M ]=<Ψ,I H > (3)。 3. The single-pixel compressed sensing phase-shift holographic reconstruction method based on KAN-CAFM network according to claim 1, characterized in that: In step 3, specifically, the MNIST handwritten digit dataset is used as the training basis. Four phase-shifted holograms of 0, π / 2, π, and 3π / 2 are collected through an interferometric system and compressed into one-dimensional data with a sampling rate of 20%. The dataset is divided into an 80% training set, a 10% validation set, and a 10% test set. Specifically, in step 4, after completing the dataset collection, a network combining the Kolmogorov-Arnold network and an attention mechanism is designed for learning. 1212 holograms were collected as the training set, 100 holograms as the test set, and 100 holograms as the validation set. First, a framework with an encoder-decoder structure is designed, including an encoder, a CAFM module (convolutional-attention fusion), a KAN module, and a decoder. The CAFM module includes convolutional-attention fusion. The network input is one-dimensional compressed data Y, and the output is four frames of high-fidelity shift holograms H′. k k∈{1,2,3,4}, realize the mapping from low-dimensional data to high-dimensional features: H′ k =φ KAFE-PSN (Y),k∈{1,2,3,4} (4)。 4. The single-pixel compressed sensing phase-shift holographic reconstruction method based on KAN-CAFM network according to claim 1, characterized in that: Specifically, step 5 involves input data processing. The input is a one-dimensional data set of 52430. Since it is one-dimensional data, the feature dimension of the input is set to [batch_size, 52430, 1, 1]. Here, batch_size represents the number of samples input into the network at one time, 52429 indicates that there are 52429 channels, and 1×1 is the spatial dimension of the image. These preprocessed features are used as initial data to enter the phase-shifting network. Specifically, in step 6, the upsampling part of the encoder processes the input features. The upsampling part consists of a series of transposed convolutional blocks. Each transposed convolutional block is composed of a ReLU activation function and a 4×4 transposed convolutional kernel with a kernel stride of 2. The transposed convolutional block upsamples the original one-dimensional signal 52430×1×1 to a three-dimensional feature space of 128×512×512. Subsequently, the network introduces a convolution-attention fusion module, which achieves synergistic enhancement of local features and global dependencies through the fusion of convolution operations and self-attention mechanisms.

5. The single-pixel compressed sensing phase-shift holographic reconstruction method based on KAN-CAFM network according to claim 1, characterized in that: Specifically, step 7 involves downsampling in the encoder. In downsampling, a 4×4 convolutional kernel with a stride of 2 is used. The convolutional layer is preceded and followed by an activation function layer and a normalization layer. The activation function used is LeakyReLU. α is set to 0.2; based on this, the encoder's downsampling stage includes three standard convolutional blocks and three KA-C blocks, forming a hierarchical feature extraction network; in the standard convolutional blocks, the feature map is progressively extracted to 512×64×64; The features are then further processed by three KA-C blocks. The features are projected onto the shape adapted by the KAN linear layer through PatchEmbed. After KAN processing, the features are passed through a normalization layer. Then, the output of the KAN linear layer is residually connected to the input to ensure the stability of the deep network. Finally, the features are further downsampled by a convolutional layer. After processing by three KA-C blocks, the encoder finally extracts the features into a compact representation of 512×8×8. Specifically, in step 8, the decoder process employs a U-shaped architecture with dense skip connections. This architecture concatenates the high-resolution features from the encoder with the low-resolution features from the decoder, enhancing the model's ability to preserve high-resolution details in the input data. Simultaneously, a multi-branch structure is used, replicating the 512×8×8 feature map output by the encoder into four independent branches H. k k∈{1,2,3,4}, each branch corresponds to a specific phase shift control, and the encoder features are shared and utilized through a parameter copying mechanism; each decoder branch is applied first; The CAFM attention module enhances the features, and then gradually restores the feature map size through a six-layer upsampling process. Similar to the encoder's downsampling stage, the decoder's upsampling process also employs a two-stage upsampling module: first, it passes through three KA-DC blocks, then through three transposed convolutional layers, finally expanding to 1×512×512, resulting in four interferogram outputs H′ with specific phase shifts. k ,k∈{1,2,3,4}.

6. The single-pixel compressed sensing phase-shift holographic reconstruction method based on KAN-CAFM network according to claim 1, characterized in that: Specifically, step 9 involves the CAFM block, which employs a dual-branch parallel design. This module contains two important branches: local and global. The local branch extracts local features using convolution and channel rearrangement operations, while the global branch captures long-distance dependencies through a self-attention mechanism. The two branches complement each other in the feature space. In the local branch, a 1×1 kernel convolution is first used to adjust the channel dimensions to enhance cross-channel information integration, followed by channel shuffling. Finally, a 3×3 kernel is used to extract features. The output F of the local branch is... conv It can be represented as: F conv =W 3×3×3 (CS(W 1×1 (X)))(6) In the formula, W 1×1 and W 3×3×3 represents 1×1 and 3×3 convolution kernels, while CS represents the channel shuffling operation; In the global branch, the query (Q), key (K), and value (V) are first generated using depthwise convolutions with kernels of 1×1 and 3×3, resulting in a shape of [formula missing]. Three tensors, then through and Calculate the attention weight matrix: In the formula, α is a learnable parameter used to control... The magnitude of the product; the output of the global branch can be represented as: The outputs of the local and global branches are merged; the final output is: F CAFM =F conv +F att (9); Specifically, step 10 involves the KAN block. The design of the KAN block is based on the Kolmogorov-Arnold representation theorem, which states that any multivariate continuous function can be represented as a combination of a series of single-variable functions, i.e., for an input vector x... p KAN is represented as: Where Φ q and φ q,p All are univariate functions, representing the q-th layer of the entire KAN network; each KAN layer has n variables. in dimensional input and n out The output of dimension, therefore Φ q Contains n in ×n out A learnable activation function φ: F q ={φ q,p },p=1,2,…,n in ,q=1,2…,n out (11) In the KAFE-SPN network, the linear weight matrix in the traditional MLP is replaced with a learnable univariate function, and the univariate function combination property of KAN is used to improve the nonlinear expressive power: φ(x)=w b ·b(x)+w s ·spline(x) (12) Here, b(x) is the SigmoidLinearUnit (SiLU) basis function, while spline(x) is the parameterized B-spline function; therefore, in our designed KAFE-SPN network, the KAN linear layer linearly combines the outputs of the basis function branch and the B-spline branch: KANLinear(x)=Linear(SiLU(x)·w b )+Linear(bases(x)·w s ) (13) Here, bases(x) is the combination result of B-spline basis functions, w b and w s These are the learnable weights for the basis function branch and the spline branch, respectively; to ensure the training stability of deep networks, the KAN block uses a residual connection mechanism, and the final output is: F KAN =Norm(KANLinear(x))+shortcut(x) (14) Where shortcut(x) is the projection layer of the input features.

7. The single-pixel compressed sensing phase-shift holographic reconstruction method based on KAN-CAFM network according to claim 1, characterized in that: Specifically, in step 11, the loss function definition uses the L1 loss function as the loss function for network training in the KAFE-SPN network; the mathematical expression of the L1 loss function is: Among them, y i f(x) represents the true interference intensity value of the i-th pixel. i The value represents the predicted interference intensity at the corresponding location, and n is the total number of pixels. In the KAFE-SPN network, N = 4 × 512 × 512, corresponding to the total number of pixels in the four-frame phase-shifted interferogram. The L1 loss function is directly applied to the output of the encoder-decoder structure, guiding network parameter optimization by comparing the pixel-level differences between the predicted and actual phase-shifted interferograms. For the four-channel phase-shifted output of the KAFE-SPN network, the L1 loss function can be extended to... Among them, y i (c,h,w) f(x) represents the true interference intensity value of the c-th phase shift channel, the h-th row, and the w-th column. i ) (c,h,w) This represents the interference intensity value predicted by the network at the corresponding location.

8. The single-pixel compressed sensing phase-shift holographic reconstruction method based on KAN-CAFM network according to claim 1, characterized in that: Specifically, in step 13, after obtaining the four-frame phase-shifted holograms output by the network, the phase-shifting technique can be used to record and reconstruct the complex-valued hologram of the same object using the four-frame phase-shifted hologram; the network generates a hologram with a phase shift θ. k The form of a single-frame hologram (k = 1, 2, 3, 4) is as follows: I k =|A(x,y)| 2 +|R(x,y)| 2 +2A(x,y)R(x,y)cos(φ+θ k ) (17) Equation (5) is used to obtain a four-frame phase-shifted composite hologram: I F (x,y)=(I1-jI2-I3+jI4) / 4 (18) Specifically, step 14 involves phase-shift synthesis of hologram I. F The portions corresponding to the conjugate image and the zeroth-order image of the reconstructed light field were eliminated in (x,y), thus the light wave at the reconstructed distance Z was simulated using the Fresnel diffraction formula. r The image is reconstructed by propagation at a certain point; the specific formula is as follows:

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