A phase recovery method based on single fractional fourier transform amplitude observation

Abstract

The application discloses a phase recovery method based on single fractional Fourier transform amplitude observation, and comprises the following steps: S1, a near-field observation model based on fractional Fourier transform is established according to optical propagation theory; S2, corresponding single fractional Fourier transform amplitude observation information is obtained by simulation according to the established near-field observation model; S3, a phase recovery algorithm based on generalized alternating projection method is established, the single fractional Fourier transform amplitude observation information and signal prior information are taken as inputs, iterative calculation is carried out by using the phase recovery algorithm, and signal reconstruction is realized. The application has the advantages that the requirement of excessive dependence on oversampling or multiple observations is broken, the near-field propagation problem is described by combining a signal processing tool fractional Fourier transform, more accurate and efficient signal reconstruction is realized only from single amplitude observation, and the reconstruction quality is high.

Description

Technical Field

[0001] This invention relates to phase retrieval technology in the field of optical imaging technology, and particularly to a phase retrieval method based on single fractional Fourier transform amplitude observation. Background Technology

[0002] Currently, existing phase retrieval methods mainly focus on amplitude spectrum observations of Fourier transforms. To address the ambiguity caused by phase loss, researchers often focus on optical observation systems, introducing more additional information to achieve signal reconstruction. For example, coherent diffraction imaging technology... [1] The introduction of a signal support domain to obtain oversampled Fourier transform amplitude spectrum observations achieves information redundancy, and the improved coded diffraction imaging technique... [2] The introduction of optical modulation devices such as masks to obtain diffraction patterns after multiple modulations achieves information redundancy, and is compatible with current mainstream superposition or scanning diffraction imaging techniques. [3] Overlapping illumination patterns were introduced to acquire a series of observations, achieving information redundancy. The existing imaging systems mentioned above all introduce additional physical components and require the acquisition of a large number of redundant observations to establish the phase retrieval observation model, hindering the development of practical applications such as real-time and live-body imaging.

[0003] Other prior art related to this invention includes: a phase recovery method based on multiple fractional Fourier transform amplitudes. [4] However, it remains at the level of theoretical research, without considering the actual application background, and still requires a large number of observations, which cannot meet the application needs such as real-time imaging.

[0004] In summary, the existing technologies mentioned above mainly rely on a large number of redundant observations to achieve phase recovery, which cannot meet the requirements of rapid acquisition and real-time imaging. Currently, there is no technology that can achieve phase recovery from a single diffraction observation.

[0005] References

[0006] [1]Fienup J R.Reconstruction of an object from the modulus of itsFourier transform[J].Optics letters,1978,3(1):27-29;

[0007] [2]Candes E J,Li X,Soltanolkotabi M.Phase retrieval from codeddiffraction patterns[J].Applied and Computational Harmonic Analysis,2015,39(2):277-299;

[0008] [3]Pfeiffer F.X-ray ptychography[J].Nature Photonics,2018,12(1):9-17;

[0009] [4]Su X,Tao R,Li Y.Phase retrieval from multiple FRFT measurementsbased on nonconvex low-rank minimization[J].Signal Processing,2022,198:108601;

[0010] [5]Candes E J,Li X,Soltanolkotabi M.Phase retrieval via Wirtingerflow:Theory and algorithms[J].IEEE Transactions on Information Theory,2015,61(4):1985-2007;

[0011] [6]Yuan X.Generalized alternating projection based total variationminimization for compressive sensing[C] / / 2016IEEE International conference onimage processing(ICIP).IEEE,2016:2539-2543;

[0012] [7]Gu S, Zhang L, Zuo W, et al. Weighted nuclear norm minimization with application to image denoising[C] / / Proceedings of the IEEE conference oncomputer vision and pattern recognition.2014:2862-2869;

[0013] [8]Dabov K, Foi A, Katkovnik V, et al.Image denoising with block-matching and3D filtering[C] / / Image processing: algorithms and systems, neural networks, and machine learning. SPIE, 2006, 6064:354-365;

[0014] [9] Zhang K, Zuo W, Chen Y, et al. Beyond a gaussian denoiser: Residuallearning of deep cnn for image denoising[J]. IEEE transactions on imageprocessing, 2017, 26(7): 3142-3155. Summary of the Invention

[0015] This invention aims to overcome the bottleneck of existing phase retrieval methods that rely excessively on oversampling or multiple observations of the Fourier transform amplitude, thus failing to recover the signal from a single observation. It provides a phase retrieval method based on single-observation fractional Fourier transform amplitude observation. This method is widely used in optical imaging fields such as X-ray diffraction imaging, microscopic imaging, and holographic imaging.

[0016] To achieve the above-mentioned objectives, the technical solution adopted by the present invention is as follows:

[0017] A phase retrieval method based on single fractional Fourier transform amplitude observation includes:

[0018] S1: A near-field observation model based on fractional Fourier transform was established from the optical propagation theory;

[0019] S2: Based on the established near-field observation model, the corresponding single fractional Fourier transform amplitude observation information is obtained through simulation;

[0020] S3: Establish a phase retrieval algorithm based on the generalized alternating projection method;

[0021] By taking the amplitude observation information of the single fractional Fourier transform and the prior information of the signal as input, the phase retrieval algorithm is used for iterative calculation to achieve signal reconstruction.

[0022] Furthermore, S1 established a near-field observation model based on fractional Fourier transform, as follows:

[0023] According to optical diffraction theory, the near-field propagation model can be expressed using Fresnel diffraction integrals:

[0024]

[0025] Where λ and z are two physical parameters, representing the wavelength and propagation distance of light, respectively, and U0 and U z Let x, y, u, and v represent the target light field and the probe light field, respectively, and x, y, u, and v be the spatial coordinates of the two planes.

[0026] Based on signal processing theory, a two-dimensional fractional Fourier transform F is performed on the light field distribution U0(x,y). α Defined as:

[0027]

[0028] in It is the kernel function of the fractional Fourier transform. p = 2α / π is denoted as the fractional order, and its value range is 0 < |p| < 2. In particular, when the fractional order p = ±1, the fractional Fourier transform degenerates into the (forward and inverse) Fourier transform.

[0029] To establish the connection between Fresnel diffraction integrals and two-dimensional fractional Fourier transforms, a scaling field is introduced:

[0030]

[0031] Among them, U z The light field distributions at different locations are represented by s1 and s2, which are scaling factors. This represents a scaled light field, where x, y, u, and v represent the spatial coordinates of the two planes, respectively.

[0032] Equation (1) is obtained:

[0033]

[0034] Where e is the natural constant, i is the imaginary number, and F α This represents the fractional Fourier transform with parameter α.

[0035] Considering that the optical detector only collects the amplitude observation of light, the spherical phase factor in equation (1) is ignored, resulting in equation (2):

[0036]

[0037] Furthermore, before performing numerical calculations of the fast fractional Fourier transform in the near-field observation model, dimensional normalization must be performed. This is done by normalizing the fractional order and scaling factor:

[0038]

[0039] Where N is the number of sampling points and L is the length of the light field sampling area.

[0040] Furthermore, the phase retrieval algorithm calculation process is as follows:

[0041] The phase recovery problem of single fractional Fourier transform amplitude observations is mathematically described as follows:

[0042] Given the observation y = |F α x|, solve for signal x.

[0043] Assume M and S are two complex spaces C n×n The constraint set ensures signal observation consistency and signal prior, respectively. Given a feasible initial point x0, the generalized alternating projection framework alternately maps the signal onto sets M and S, as shown in equation (3):

[0044] v k =P M (x k-1 ),x k =P S (v k ),k=1,2,3,..., (3)

[0045] Among them, P M and P S This indicates mapping to M and S, v k and x k These are the reconstruction results after the k-th iteration.

[0046] P M The operation is to impose observation fidelity constraints, as shown in equation (4):

[0047]

[0048] Where ° represents the Hadama product, F -α It is F α The inverse transform of .

[0049] For P SThe implementation is achieved by implicitly applying an inherent image prior, as shown in equation (5):

[0050]

[0051] Where R(x) represents a prior regularization term, and σ is a parameter related to the noise intensity.

[0052] Equation (5) is solved using a noise reduction device or a noise reduction algorithm, i.e., Equation 6:

[0053] P S (v k )=H σ (v k ), (6)

[0054] Among them, H σ It is a noise reducer.

[0055] Compared with the prior art, the advantages of the present invention are as follows:

[0056] For the first time, phase recovery was achieved from a single fractional Fourier transform amplitude observation, which overturned the previous phase recovery technology based on Fourier transform and broke through the requirement of over-sampling or multiple observations in previous phase recovery methods, providing a new approach for efficient and real-time phase recovery technology.

[0057] By innovatively combining the signal processing tool fractional Fourier transform to describe the near-field propagation problem, a more accurate, efficient, and unified optical propagation model was obtained.

[0058] Signal reconstruction can be achieved from a single amplitude observation, and the reconstruction quality (measured by the peak signal-to-noise ratio of the image) is far superior to that reconstructed from Fourier transform amplitude observations. Attached Figure Description

[0059] Figure 1 This is a schematic diagram comparing the accuracy (peak signal-to-noise ratio) of different propagation observation models at different propagation distances according to embodiments of the present invention;

[0060] Figure 2 This is a flowchart of the detection and identification process according to an embodiment of the present invention;

[0061] Figure 3 These are test diagrams from embodiments of the present invention;

[0062] Figure 4 This is the single amplitude observation diagram obtained from the simulation of an embodiment of the present invention;

[0063] Figure 5 This is a diagram showing the signal reconstruction effect according to an embodiment of the present invention. Detailed Implementation

[0064] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and examples.

[0065] A phase recovery method based on single fractional Fourier transform amplitude observation mainly consists of two parts: a near-field observation model based on fractional Fourier transform and a phase recovery algorithm based on generalized alternating projection method.

[0066] First, a physical forward model based on fractional Fourier transform was established from optical theory, thereby defining the fractional Fourier phase retrieval problem. Then, an effective phase retrieval algorithm was proposed, which can combine the amplitude observation information of a single fractional Fourier transform with the prior information of the signal to achieve signal reconstruction.

[0067] I. Near-field observation model based on fractional Fourier transform

[0068] The most widely used optical diffraction propagation model can be represented by Fresnel diffraction integrals:

[0069]

[0070] Where λ and z are two physical parameters, representing the wavelength and propagation distance of light, respectively, and U represents the light field distribution at different locations.

[0071] The two-dimensional fractional Fourier transform of the light field distribution U0(x,y) can be defined as:

[0072]

[0073] in It is the kernel function of the fractional Fourier transform. p = 2α / π is denoted as the fractional order, and its value range is 0 < |p| < 2. In particular, when the fractional order p = ±1, the fractional Fourier transform degenerates into the (forward and inverse) Fourier transform.

[0074] To establish the connection between Fresnel integral (1) and fractional Fourier transform (2), this embodiment introduces a scaling field:

[0075]

[0076] in This is the scaling factor.

[0077] Therefore, this embodiment yields the following:

[0078]

[0079] Considering that the optical detector only collects the amplitude observation of light, the spherical phase factor in (3) can be ignored. This embodiment proposes:

[0080]

[0081] Thus, this embodiment establishes a near-field observation model based on fractional Fourier transform, meaning that the amplitude distribution of light scaling on the intermediate plane of diffraction propagation can be interpreted as the amplitude spectrum of the fractional Fourier transform. In practical applications, given the scaling factor s1 and the propagation distance z, this embodiment can accurately determine another scaling factor. And the corresponding fractional order p = 2 / π*arctan(λz / s1) 2 ).

[0082] Therefore, the order of the fractional Fourier transform is not fixed; it is influenced by the wavelength of the light wave, the diffraction distance, and the scaling factor. This choice of degree of freedom is crucial in practical applications because propagation models based on Fresnel integrals lack a unified sampling method for numerical calculation of the integral. The discrete fast algorithm for the fractional Fourier transform allows for free selection of the sampling interval and number of sampling points, provided that the Nyquist sampling theorem is satisfied. Furthermore, the fast algorithm for the fractional Fourier transform can leverage the FFT algorithm, achieving a computational complexity comparable to the FFT at O(NlogN), significantly reducing the computational burden of direct calculation. It is important to note that dimensional normalization must be performed before performing numerical calculations of the fast fractional Fourier transform. In this embodiment, dimensional normalization is applied to the fractional order and the scaling factor:

[0083]

[0084] Where N is the number of sampling points and L is the length of the light field sampling area.

[0085] II. Phase Recovery Algorithm Based on Generalized Alternating Projection Method

[0086] Based on the established fractional Fourier transform near-field observation model, the phase recovery problem of single fractional Fourier transform amplitude observation can be mathematically described as follows:

[0087] Given the observation y = |F α x|, solve for signal x.

[0088] Assume M and S are two complex spaces C n×n The constraint set ensures both signal observation consistency and signal prior. Given a feasible initial point x0, the generalized alternating projection framework can alternately map the signal onto sets M and S:

[0089] v k =P M (x k-1 ),x k =P S (v k),k=1,2,3,..., (6)

[0090] Where P M and P S This indicates mapping to M and S.

[0091] Typically P M The operation is to impose observation fidelity constraints, that is:

[0092]

[0093] Where ° represents the Hadamard product (calculated element-wise), F -α It is F α The inverse transform of .

[0094] For P S The most commonly used priors for implementation are real-valued priors, smooth priors, and sparse priors. Inspired by recent deep plug-and-play techniques, inherent image priors can be implicitly applied by solving the regularization denoising problem:

[0095]

[0096] Where R(x) represents a prior regularization term, and σ is a parameter related to the noise intensity.

[0097] This problem (8) can be solved using any off-the-shelf noise reduction device H. σ Alternatively, a denoising algorithm can be used to solve this problem, namely:

[0098] P S (v k )=H σ (v k ), (9)

[0099] In this way, the phase recovery algorithm based on generalized alternating projection can be iterated according to (6), (7), and (9).

[0100] The most significant advantage of this invention is that it achieves phase recovery from a single fractional Fourier transform amplitude observation for the first time, overturning previous phase recovery techniques that relied primarily on Fourier transforms. It also breaks through the previous phase recovery methods' over-reliance on oversampling or multiple observations, providing a new approach for efficient and real-time phase recovery technology.

[0101] One achieved effect is that existing optical propagation models are mainly based on two methods: angular spectrum method and Fresnel integral method. However, each has certain limitations, mainly in terms of numerical calculation accuracy, computational complexity, and applicability. The optical diffraction propagation model based on fractional Fourier transform proposed in this invention innovatively combines the signal processing tool fractional Fourier transform to describe the near-field propagation problem, resulting in a more accurate, efficient, and unified optical propagation model.

[0102] Experimental Verification: This embodiment mainly compares the angular spectrum method (ASM) and the Fresnel transform (ST-Fresnel) implemented by a single fast Fourier transform. This embodiment uses numerical integration to obtain the true value as a reference standard and uses the peak signal-to-noise ratio of the image as the metric. Table 1 provides the simulation parameter settings. Figure 1 The accuracy of the three propagation observation models described above increases with different propagation distances. It can be observed that the observation model based on fractional Fourier transform (FrFT) proposed in this invention maintains high accuracy with increasing propagation distance, while the observation model based on the angular spectrum method gradually decreases in accuracy with increasing propagation distance. The Fresnel integral observation model based on a single fast Fourier transform becomes completely distorted at short propagation distances. Therefore, the observation model based on fractional Fourier transform proposed in this invention has greater potential and a unified ability to describe the light diffraction propagation process.

[0103] Table 1 Parameter settings for optical propagation simulation

[0104]

[0105] The second effect achieved is that existing phase recovery algorithms based on Fourier transform heavily rely on a large amount of redundant observation information and cannot reconstruct signals from single Fourier transform amplitude observations. However, the phase recovery method proposed in this invention based on single fractional Fourier transform amplitude observations can reconstruct signals from only a single amplitude observation, and the reconstruction quality (measured by the image peak signal-to-noise ratio) is far superior to the results reconstructed from Fourier transform amplitude observations.

[0106] Experimental verification: This embodiment uses six phase retrieval methods, including the classic and universal Wirtinger Flow (WF). [5] In addition to the real-valued image prior, the proposed Generalized Alternating Projection (GAP) method incorporates five different image priors, such as the real-valued image prior and the Total Variation Denoising Prior (tv). [6] Least weighted nuclear norm denoising prior (wnnm) [7] Block matching and 3D filtering denoising priors (bm3d) [8] Convolutional Neural Network Denoising Prior (DNCNN) [9]This embodiment uses the publicly available dataset Set12 to quantitatively evaluate the algorithm's performance. This dataset includes twelve widely used test images, all of which are uniformly sized to 128×128. The number of iterations is empirically set to 200, and this embodiment uses a fully initialized dataset for performance comparison.

[0107] Here, this embodiment uses six PR algorithms to compare the reconstruction performance of fractional Fourier transform amplitude observations (including Fourier transform amplitude observations) of different fractional orders. Table 2 reports the average recovery accuracy on the test dataset in terms of average peak signal-to-noise ratio (PSNR). It can be observed that, except for p=1 representing Fourier transform measurements, all phase recovery methods perform well on fractional Fourier transform measurements of different fractional orders. Furthermore, advanced denoising can significantly improve reconstruction performance when using fractional Fourier transform measurements. Conversely, all phase recovery algorithms suffer from severe stagnation and fail to recover the image from Fourier transform measurements.

[0108] Table 2 shows the reconstruction results of single fractional Fourier transform amplitude observations at various fractional orders using different phase retrieval methods on the dataset (average peak signal-to-noise ratio: dB).

[0109]

[0110]

[0111] Based on the phase retrieval method described above, which is based on single fractional Fourier transform amplitude observation, the following is performed: Figure 2 The detection and identification process shown is mainly divided into optical propagation and observation acquisition, and numerical calculation and signal reconstruction. After the target signal is illuminated by a coherent light source, the diffracted light field propagates through free space and is observed by a near-field detector. This process can be modeled by the near-field observation model based on fractional Fourier transform proposed in this invention. The acquired near-field observations are input into a computer for numerical calculation and reconstruction. Using the phase retrieval algorithm based on the generalized alternating projection method proposed in this invention, the original signal can be perfectly recovered from a single observation.

[0112] Specifically, it includes the following steps:

[0113] I. Simulation yields near-field observations based on fractional Fourier transform

[0114] This embodiment uses the most commonly used test image "cameraman" as an example, with a pixel size of 256×256. Figure 3 ;

[0115] Based on the established fractional Fourier transform near-field observation model, this embodiment can simulate the corresponding single amplitude observation. Here, the fractional order used in this embodiment is 0.5, and the results are as follows: Figure 4 As shown:

[0116] II. Signal Reconstruction Using a Phase Recovery Algorithm Based on Generalized Alternating Projection

[0117] Next, using the near-field observations as input, the proposed phase retrieval algorithm based on the generalized alternating projection method is used for iterative calculation. In this embodiment, a total variational denoising prior is employed. The reconstructed signal is then obtained, as shown in the following figure. Figure 5 As shown;

[0118] The methods described above according to the invention can be implemented in hardware, firmware, or as software or computer code that can be stored in a recording medium (such as a CD-ROM, RAM, floppy disk, hard disk, or magneto-optical disk), or as computer code originally stored on a remote recording medium or a non-transitory machine-readable medium and subsequently stored on a local recording medium, downloaded via a network. Thus, the methods described herein can be stored as software processing on a recording medium using a general-purpose computer, a dedicated processor, or programmable or dedicated hardware (such as an ASIC or FPGA). It is understood that the computer, processor, microprocessor controller, or programmable hardware includes storage components (e.g., RAM, ROM, flash memory, etc.) capable of storing or receiving software or computer code, which, when accessed and executed by the computer, processor, or hardware, implements a phase retrieval method based on single fractional Fourier transform amplitude observations as described herein. Furthermore, when a general-purpose computer accesses the code used to implement the processing shown herein, the execution of the code transforms the general-purpose computer into a dedicated computer for performing the processing shown herein.

[0119] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the implementation methods of the present invention, and should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the present invention.

Claims

1. A phase recovery method based on single fractional Fourier transform amplitude observation, characterized in that, include: S1: A near-field observation model based on fractional Fourier transform was established from the optical propagation theory; S1 establishes a near-field observation model based on fractional Fourier transform, as follows: To establish the connection between Fresnel diffraction integrals and two-dimensional fractional Fourier transforms, a scaling field is introduced: , ， in, and These are two physical parameters, representing the wavelength of light and the distance it travels, respectively. Represents the light field distribution at different locations. and Scaling factor , Represents the scaled light field, where These represent the spatial coordinates of the two planes respectively; Equation (1) is obtained: , (1) Where e is the natural constant and i is the imaginary number. This represents the fractional Fourier transform with parameter a; Considering that the optical detector only collects the amplitude observation of light, the spherical phase factor in equation (1) is ignored, resulting in equation (2): , (2) S2: Based on the established near-field observation model, the corresponding single fractional Fourier transform amplitude observation information is obtained through simulation; S3: Establish a phase retrieval algorithm based on the generalized alternating projection method; The signal is reconstructed by taking the amplitude observation information of the single fractional Fourier transform and the prior information of the signal as input and using the phase retrieval algorithm for iterative calculation. The iterative calculation process of the phase retrieval algorithm is as follows: The phase recovery problem of single fractional Fourier transform amplitude observations is mathematically described as follows: ， assumed and It is two complex spaces The constraint set ensures both signal observation consistency and signal prior; a feasible initial point is given. The generalized alternating projection framework alternately maps signals to sets in sequence. and Above, as in equation (3): (3) in, and Indicates mapping to and , and These are the reconstruction results after the k-th iteration; The operation is to impose observation fidelity constraints, as shown in equation (4): (4) in Represents the Hadama product. yes inverse transform; for The implementation is achieved by implicitly applying an inherent image prior, as shown in equation (5): , (5) in Denotes a prior regularization term, It is a parameter related to noise intensity; Equation (5) is solved using a noise reduction device or a noise reduction algorithm, i.e., equation (6): (6) in, Noise reduction device.

2. The phase recovery method based on single fractional Fourier transform amplitude observation according to claim 1, characterized in that: Before performing numerical calculations of the fast fractional Fourier transform in the near-field observation model, dimensional normalization must be performed. This is done by normalizing the fractional order and scaling factor: ， in It is the number of sampling points. It is the length of the light field sampling area.

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