Denoising methods for Fourier layered imaging

Abstract

This invention relates to the field of Fourier layered imaging technology, and provides a denoising method for Fourier layered imaging. The method primarily optimizes the objective function of Fourier layered imaging by introducing an L2 paradigm to reduce the impact of noise. By differentiating the optimized objective function, a high-resolution spectrum of the target is obtained. Then, the high-resolution spectrum is inverse Fourier transformed to the spatial domain to obtain a large field-of-view, high-resolution complex image of the target. Compared with traditional Fourier layered imaging methods, the method of this invention can acquire higher-quality reconstructed images under the same noise conditions and effectively suppresses noise.

Description

Technical Field

[0001] This invention relates to the field of Fourier layered imaging technology, and specifically provides a denoising method for Fourier layered imaging based on the L2 paradigm. Background Technology

[0002] In optics, the resolving power of an optical system is positively correlated with its aperture size; the larger the aperture, the stronger the resolving power. However, a larger aperture is more susceptible to aberrations, leading to a sharp decrease in the imaging field of view. Furthermore, a larger aperture means larger volume, higher cost, and an exponential increase in mass. To resolve the conflict between resolution and field of view, Zheng et al. proposed Fourier stacked microscopy (FPM) in 2013, which can achieve a resolution several times that of imaging objectives with low-magnification microscope objectives. This technique acquires high-frequency information of the target beyond the diffraction limit of the objective lens through multi-angle illumination. Then, a phase retrieval algorithm is used to stitch and fuse the acquired images in the frequency domain to obtain a high resolution several times that of the imaging objective lens's diffraction limit. Finally, the high-resolution spectrum is inverse Fourier transformed to the spatial domain to obtain a large-field-of-view, high-resolution complex image of the target. The forward transfer model of Fourier stacked imaging can be expressed as:

[0003] I i =|FPQ i ψ| 2 ;

[0004] Where I represents the intensity image captured by the CCD camera, F represents the Fourier transform, and Q... i ψ represents the sampling matrix, P represents the pupil function, and ψ represents the high-resolution spectrum. By repeating this formula, multiple low-resolution intensity images can be obtained, and phase recovery can be performed to obtain the high-resolution spectrum of the target.

[0005] However, Fourier layer imaging still suffers from significant noise. Most of the images acquired in Fourier layer imaging are dark-field images. Compared to bright-field images, dark-field images have weaker high-frequency information signals and lower signal-to-noise ratios, making them highly susceptible to noise.

[0006] Noise remains a critical problem to be solved in Fourier layer imaging. Summary of the Invention

[0007] To address the aforementioned problems, this invention provides a denoising method for Fourier layered imaging. First, an objective function is established. Second, an L2 norm is introduced into the objective function. The derivative of the objective function is then used to obtain the solution of the high-resolution spectrum of the target. Finally, the high-resolution spectrum is inversely Fourier transformed to the spatial domain to obtain a large field-of-view high-resolution complex image of the target.

[0008] The denoising method for Fourier layered imaging provided by this invention includes the following steps:

[0009] S1. The objective function for Fourier layered imaging is established as follows:

[0010]

[0011] Where F represents the Fourier transform, P represents the pupil function of the optical system, ψ represents the high-resolution spectrum of the target with (m, m) pixels, and Q i Represents the sampling matrix, This represents the spatial complex image after the spectrum is captured by the optical system, i.e., the low-resolution image estimate, with (n, n) pixels. i This represents the intensity of the i-th captured image;

[0012] S2. Introduce the L2 paradigm to denoise the objective function, obtaining the optimized objective function as follows:

[0013]

[0014]

[0015] Where λ represents the penalty coefficient;

[0016] S3. Taking the derivative of the target optimization formula, the high-resolution spectrum ψ of the target is obtained as follows:

[0017]

[0018] Where Diag denotes the diagonalized matrix;

[0019] S4. Perform an inverse Fourier transform of the high-resolution spectrum ψ to the spatial domain to obtain a large field-of-view high-resolution complex image of the target.

[0020] Preferably, the sampling matrix Q i The size is n 2 ×m 2 It is used to convert m 2 The image of size ×1 is downsampled to n 2 An image of size ×1.

[0021] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0022] In the process of Fourier layered imaging, this invention optimizes the objective function by introducing the L2 paradigm into the objective function, which increases the proportion of high-resolution spectrum, reduces the impact of noise, and effectively improves the image reconstruction quality.

[0023] Compared with the sequential iterative update of single images in traditional FPM, this invention optimizes the image update method to batch overall update, which simplifies the solution process of high-resolution spectrum ψ. This invention processes all low-resolution images synchronously, which simplifies the update process, and each frequency domain sub-aperture can play a role simultaneously during the update. Attached Figure Description

[0024] Figure 1 This is a flowchart of Fourier layered imaging using the method of the present invention, provided according to an embodiment of the present invention;

[0025] Figure 2 This is a comparison chart of the amplitude and phase recovery of the target image using the present invention and the traditional GS algorithm under simulated Gaussian noise conditions with a standard deviation of 0.02.

[0026] Figure 3 This is a comparison chart of the reconstruction results of the present invention and the traditional GS algorithm under actual conditions. Detailed Implementation

[0027] In the following description, embodiments of the invention will be described with reference to the accompanying drawings. In the description below, the same modules are denoted by the same reference numerals. Where the same reference numerals are used, their names and functions are also the same. Therefore, their detailed description will not be repeated.

[0028] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not constitute a limitation thereof.

[0029] Figure 1 The flowchart for performing Fourier layered imaging using the method of the present invention is shown according to an embodiment of the present invention.

[0030] like Figure 1 As shown, this embodiment of the invention is based on a Fourier layered imaging system, and mainly aims to solve the noise problem in imaging. The proposed method for denoising Fourier layered imaging includes the following steps:

[0031] S1. Similar to traditional Fourier layered imaging algorithms, the objective function of the Fourier layered imaging system first needs to be established, expressed as:

[0032]

[0033] Where F represents the Fourier transform, P represents the pupil function of the optical system, ψ represents the high-resolution spectrum of the target, and ψ contains (m, m) pixels. This represents the complex spatial image after the spectrum has been captured and imaged by the optical system. The number of pixels contained is (n, n), I i Q represents the intensity of the i-th acquired image, which is the initial image of the target captured by the optical system. i This represents the sampling matrix, specifically the downsampling matrix in the formula, with a size of n. 2 ×m 2 It is used to convert m 2 The image of size ×1 is downsampled to n 2 An image of size ×1.

[0034] The imaging process in a Fourier stacked imaging system is a current technology and can be simply described as follows: the high-resolution spectrum is downsampled to become a low-resolution spectrum, then filtered by the pupil function, and finally inverse Fourier transformed to the spatial domain, where it is captured by the CCD camera as a low-resolution image.

[0035] S2. After the objective function is established, in order to increase the weight of the high-resolution spectrum in the reconstruction process and minimize the influence of noise in the acquired images, this invention introduces the L2 paradigm into the objective function to remove imaging noise in the Fourier layered imaging system. After introducing the L2 paradigm, the optimized objective function is obtained, expressed as:

[0036]

[0037]

[0038] Where λ represents the penalty coefficient.

[0039] The introduction of the L2 paradigm greatly increases the proportion of high-resolution spectrum, reduces the impact of noise, and effectively improves the quality of subsequent reconstruction.

[0040] S3. After establishing the target optimization formula, directly taking the derivative of the target optimization formula yields the high-resolution spectrum ψ of the target, expressed as:

[0041]

[0042] Here, Diag represents the diagonalized matrix.

[0043] Traditional techniques employ sequential iteration of single-image FPM. However, in this invention, the high-resolution spectrum ψ can be solved using only the above formula. In this formula, all low-resolution image estimates are simultaneously Fourier transformed to the frequency domain and upsampled, employing a batch-based overall update. After superposition, modulation is performed using a sampling matrix, ensuring that each frequency domain sub-aperture functions simultaneously during the update. This avoids the aperture information overlay problem in traditional techniques and simplifies the update process. In the actual algorithm, this invention only requires repeatedly updating ψ using the formula for calculating the high-resolution spectrum ψ until the set number of iterations is reached.

[0044] Taking the GS algorithm, commonly used in traditional Fourier layered imaging, as an example, the update process of the GS algorithm mainly involves first initializing the high-resolution spectrum, extracting the sub-aperture region corresponding to the i-th acquired image in the spectrum, and then performing a Fourier transform on it to the spatial domain to obtain a low-resolution image estimate for a single image. The amplitude of the acquired image is then used to replace the amplitude of the low-resolution image estimate, and the replaced low-resolution image estimate is Fourier transformed to the frequency domain. The frequency domain data is then used to update the corresponding region of the spectrum. This process is repeated for each acquired low-resolution image until all low-resolution images have been updated. This process is then repeated until the iteration is complete.

[0045] The GS algorithm updates sequentially from the center outwards. This gradual update from the center sub-aperture outwards can cause the contribution of the previous sub-aperture to be covered by the adjacent unupdated sub-apertures, resulting in aperture information overlay and reducing the clarity of the final reconstructed image.

[0046] S4. The high-resolution spectrum ψ obtained after iteration is then inversely Fourier transformed to the spatial domain to obtain a high-resolution complex image of the target with a large field of view.

[0047] To verify the effectiveness of this invention, a series of verification experiments were conducted:

[0048] Figure 2 The paper presents a comparison of the amplitude and phase recovery of the target image using the present invention and the traditional GS algorithm under simulated Gaussian noise conditions with a standard deviation of 0.02.

[0049] Analog noise: such as Figure 2 As shown, using the same simulated noise environment, i.e., simulated Gaussian noise with a standard deviation of 0.02, the amplitude and phase of the target are recovered using the present invention and the traditional GS algorithm. Figure 2 As can be seen, under the same simulated noise conditions, whether it is an amplitude image or a phase image, the reconstruction results of this invention are far less affected by noise than the traditional GS algorithm.

[0050] Figure 3 The paper presents a comparison of the reconstruction results using the present invention and the traditional GS algorithm under actual conditions.

[0051] Actual data collection: such as Figure 3 As shown, an open-source dataset is selected as the target. Under practical conditions, the reconstructed image of the target is obtained using this invention and the traditional GS algorithm. Figure 3 As can be seen, the reconstruction effect of the present invention is significantly better than that of the traditional GS algorithm.

[0052] The above experiments verified that the reconstruction effect of the present invention on the target is good in both simulated and real environments, proving that the method of the present invention is effective and superior.

[0053] Although embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

[0054] The specific embodiments of the present invention described above do not constitute a limitation on the scope of protection of the present invention. Any other corresponding changes and modifications made in accordance with the technical concept of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A denoising method for Fourier layered imaging, characterized in that, Includes the following steps: S1. The objective function for Fourier layered imaging is established as follows: Where F represents the Fourier transform, P represents the pupil function of the optical system, ψ represents the high-resolution spectrum of the target with (m, m) pixels, and Q i Represents the sampling matrix, This represents the spatial complex image after the spectrum is captured by the optical system, i.e., the low-resolution image estimate, with (n, n) pixels. i This represents the intensity of the i-th captured image; S2. Introduce the L2 paradigm to denoise the objective function, obtaining the optimized objective function as follows: Where λ represents the penalty coefficient; S3. Taking the derivative of the target optimization formula, the high-resolution spectrum ψ of the target is obtained as follows: Where Diag denotes the diagonalized matrix; S4. Perform an inverse Fourier transform of the high-resolution spectrum ψ to the spatial domain to obtain a large field-of-view high-resolution complex image of the target.

2. The denoising method for Fourier layered imaging as described in claim 1, characterized in that, Sampling matrix Q i The size is n 2 ×m 2 It is used to convert m 2 The image of size ×1 is downsampled to n 2 An image of size ×1.

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