A Low-Noise Hologram Generation Method Based on Fractional Fourier Transform Algorithm

Abstract

This invention proposes a low-noise holographic 3D display method based on a fractional Fourier transform algorithm. The method includes the following steps: First, setting the signal and noise intervals of the recorded object to obtain a preprocessed image RD(X,Y); Second, superimposing the image RD(X,Y) with a quadratic phase generator to produce a corresponding complex amplitude distribution D0(X,Y) on the object plane; then, performing a fractional Fourier transform based on the complex amplitude distribution D0(X,Y) on the object plane to obtain a complex amplitude distribution D1(X,Y) on the holographic plane; extracting the phase information of D1(X,Y) and... The complex amplitude distribution D2(X,Y) of the object plane is obtained through fractional-order inverse Fourier transform. A double constraint of amplitude and second-order phase is applied to the signal interval of D2(X,Y) to obtain the complex amplitude distribution D3(X,Y) of the object plane, completing the first iteration. The complex amplitude distribution D3(X,Y) is then used to replace D0(X,Y) to start the next loop, continuing until the phase converges to the optimal solution. In the third step, the phase distribution of the optimal solution is extracted to generate a pure phase hologram. By superimposing pure phase holograms of different depths, the final pure phase hologram of the 3D object is obtained. Low-noise holographic 3D display is achieved by illuminating the pure phase hologram of the 3D object with coherent light.

Description

I. Technical Field

[0001] This invention relates to holographic 3D display technology, and more specifically, to a method for generating low-noise holograms based on a fractional Fourier transform algorithm. II. Background Technology

[0002] Holographic 3D display technology is an advanced display technology that uses the principles of interference and diffraction to record and reconstruct the complete wavefront information of 3D objects. To date, despite significant progress in holographic 3D display technology, several challenges remain. For example, due to the coherence of the light source and calculation errors in the hologram phase, randomly distributed speckle noise exists in the holographic reconstruction, reducing the quality of the holographic 3D display and thus limiting the application of holographic technology. To address this, researchers have proposed many different algorithms to suppress speckle noise, mainly categorized as iterative and non-iterative algorithms. Among iterative algorithms, the classic Gerchberg-Saxton algorithm iteratively calculates the complex amplitude light field on the object plane and the hologram plane until the hologram phase reaches its optimal value, at which point a pure phase hologram is derived for holographic reconstruction. However, the initial random phase distribution leads to random speckle noise. To further reduce speckle noise, researchers have proposed various optimization algorithms based on the Gerchberg-Saxton algorithm, adding amplitude and phase constraints. In addition, many non-iterative algorithms can suppress speckle noise to some extent, such as layered pixel scanning algorithms, random phase-free algorithms, and pixel separation algorithms. However, achieving low-noise holographic 3D displays remains a challenge. III. Summary of the Invention

[0003] To suppress speckle noise in holographic 3D displays, this invention proposes a low-noise hologram generation method based on a fractional Fourier transform algorithm. (See attached diagram) Figure 1 As shown, the method includes the following steps: First, the signal range and noise range of the recorded object are set to obtain the preprocessed image RD(X,Y); Second, a quadratic phase is superimposed on the image RD(X,Y). The complex amplitude distribution D0(X,Y) on the object plane is generated. Then, a fractional Fourier transform is performed on the complex amplitude distribution D0(X,Y) on the object plane to obtain the complex amplitude distribution D1(X,Y) on the hologram plane. The phase information of D1(X,Y) is then extracted. The complex amplitude distribution D2(X,Y) of the object plane is obtained through fractional-order inverse Fourier transform. A dual constraint of amplitude and quadratic phase is applied to the signal interval of D2(X,Y) to obtain the complex amplitude distribution D3(X,Y) of the object plane, completing the first iteration. The complex amplitude distribution D3(X,Y) is then used to replace D0(X,Y) to start the next loop, continuing until the phase converges to the optimal solution. In the third step, the phase distribution of the optimal solution is extracted to generate a pure phase hologram. By superimposing pure phase holograms of different depths, the final pure phase hologram of the 3D object is obtained. Low-noise holographic 3D display is achieved by illuminating the pure phase hologram of the 3D object with coherent light. The method proposed in this invention effectively reduces speckle noise caused by random phase distribution and improves the quality of holographic 3D display by introducing dual constraints of amplitude and quadratic phase.

[0004] In step one, the preprocessed image RD(X,Y) consists of a signal interval and a noise interval. The signal interval is the region where the pixels of the recorded object are located, and the noise interval is the region where zero elements are added around the recorded object. If the pixel size of the object plane obtained after fractional Fourier transform and inverse fractional Fourier transform is larger than the pixel size of the signal interval, it will lead to severe crosstalk and speckle noise. Therefore, this invention expands the signal interval of the image RD(X,Y), as shown in the attached figure. Figure 2 As shown, xy is the object plane, uv is the holographic plane, and both planes are at a distance z from the lens. a is the transform order. In a fractional Fourier transform, the relationship between z and a is expressed as:

[0005]

[0006] Where f is the focal length of the lens, a∈(0,2), the value of a affects the imaging position of the image. When a=1, the reconstructed image position is at the focal length of the imaging lens. The larger a is, the farther the imaging position is from the focal length of the lens; the smaller a is, the closer the imaging position is to the focal length of the lens. For a 3D object, the transform order corresponding to the imaging depth z1 is a1, the transform order corresponding to the imaging depth z2 is a2, and the signal range of the image RD(X,Y) should be magnified by a factor of g. The magnification factor g is expressed as:

[0007]

[0008] In step two, the complex amplitude distribution D1(X,Y) of the hologram plane obtained by the fractional Fourier transform is expressed as:

[0009]

[0010] Where m and n are the number of pixels in the horizontal and vertical directions of the complex amplitude distribution D0(X,Y) on the object plane, respectively, and X and Y represent the horizontal and vertical coordinates, respectively. λ is the wavelength of the incident light, and f e dx is the focal length during the Fourier transform, dx and du are the sampling intervals of the object plane and the holographic plane, respectively, and FFT represents the Fast Fourier Transform. The complex amplitude distribution D2(X,Y) of the object plane is expressed as:

[0011]

[0012] in, This is the phase distribution of the complex amplitude distribution D1(X,Y) on the holographic plane. The complex amplitude distribution D3(X,Y) obtained after applying double constraints is expressed as:

[0013]

[0014] Where Sig and Noi represent the signal interval and noise interval, respectively. Let A1 and A2 represent the phase and amplitude of the complex amplitude distribution D2(X,Y), respectively, and c be a constant between 0 and 1. By adjusting the size of the noise interval and iterating repeatedly, the phase converges to the optimal value, thereby effectively suppressing speckle noise. IV. Description of the attached drawings

[0015] Appendix Figure 1 This is a flowchart illustrating a low-noise hologram generation method based on a fractional Fourier transform algorithm according to the present invention.

[0016] Appendix Figure 2 This is a schematic diagram illustrating the relationship between imaging distance and its transformation order in the fractional Fourier transform of this invention.

[0017] Appendix Figure 3 This image shows a comparison of the reconstruction results of a low-noise hologram generation method based on fractional Fourier transform algorithm of the present invention and the traditional Gerchberg-Saxton algorithm. Figure 3 (a)-(b) show the reconstruction results of the traditional Gerchberg-Saxton algorithm. Figure 3 (c)-(d) represent the reconstruction effects of the method proposed in this invention.

[0018] It should be understood that the above figures are only schematic and are not drawn to scale. V. Detailed Implementation Methods

[0019] The following detailed embodiments of the low-noise hologram generation method based on the fractional Fourier transform algorithm proposed in this invention further illustrate the invention. It is important to note that the following embodiments are for illustrative purposes only and should not be construed as limiting the scope of protection of this invention. Any non-essential improvements and adjustments made to this invention by those skilled in the art based on the above description are still within the scope of protection of this invention.

[0020] One embodiment of the present invention is as follows: To verify the feasibility of the proposed method, a 532nm wavelength laser is used as the reconstruction light source, and a reflective pure phase spatial light modulator is used to load the hologram. The pixel pitch and resolution of the spatial light modulator are 3.6μm and 3840×2160, respectively, the phase modulation capability is 2π, and the refresh rate is 120Hz. Two letters, "B" and "H," are selected as the 3D objects to be recorded, each with a resolution of 638×638. The reconstruction distance and transform order of the letter "B" are set to 20cm and 1.1, respectively, and the reconstruction distance and transform order of the letter "H" are set to 23.5cm and 1, respectively. By setting signal and noise regions for "B" and "H," preprocessed images are obtained. Then, the images are subjected to dual constraints of amplitude and secondary phase to finally generate pure phase holograms of the 3D objects. To verify the effectiveness of the method proposed in this invention, holograms of 3D objects were calculated and holographic reconstructions were performed using both the traditional Gerchberg-Saxton algorithm and the method proposed in this invention. The corresponding holographic reconstruction results are attached. Figure 3 As shown. Among them, the appendix Figure 3 (a)-(b) show the focusing effects of "B" and "H" respectively when using the traditional Gerchberg-Saxton algorithm. Figure 3 (c)-(d) show the focusing effects of "B" and "H" respectively when using the method proposed in this invention. The results show that the method proposed in this invention effectively suppresses speckle noise and improves the quality of the holographic reconstruction image.

Claims

1. A method for generating a low-noise hologram based on a fractional Fourier transform algorithm, characterized by, The method comprises the following steps: first, setting the signal interval and noise interval of the recorded object to obtain a preprocessed image RD(X, Y); second, superimposing the image RD(X, Y) with a quadratic phase The corresponding complex amplitude distribution D0(X, Y) on the object plane is generated, then, based on the complex amplitude distribution D0(X, Y) on the object plane, fractional Fourier transform is performed to obtain the complex amplitude distribution D1(X, Y) on the hologram plane, and the phase information of D1(X, Y) is extracted And the complex amplitude distribution D2(X, Y) on the object plane is obtained through inverse fractional Fourier transform, the signal interval of D2(X, Y) is subjected to double constraints of amplitude and quadratic phase to obtain the complex amplitude distribution D3(X, Y) on the object plane, the first iteration is completed, the complex amplitude distribution D3(X, Y) on the object plane is used to replace D0(X, Y) to start the next cycle, and the iteration cycle is stopped until the phase converges to the optimal solution; third, the phase distribution of the optimal solution is extracted to generate a pure phase hologram, different depth pure phase holograms are superimposed to obtain a final pure phase hologram of a 3D object, and the pure phase hologram of the 3D object is irradiated by coherent light to realize low-noise holographic 3D display; In step one, the pre-processed image RD(X, Y) is composed of signal interval and noise interval, the signal interval is the area where the pixel points of the recorded object itself are located, and the noise interval is the area where the elements around the recorded object are added to zero. Once the pixel size of the object plane obtained by using fractional Fourier transform and inverse fractional Fourier transform is greater than the pixel size of the signal interval, it will lead to serious crosstalk and speckle noise. The signal interval of the image RD(X, Y) is expanded, the distance of the two planes from the lens is both, a is the transform order, and the relationship between z and a in the fractional Fourier transform is expressed as: Where f is the focal length of the lens, a ∈ (0, 2), the value of a affects the imaging position of the image, when a = 1, the reproduction image position is at the focal length of the imaging lens, the larger a is, the farther the imaging position is from the focal length of the lens, and the smaller a is, the closer the imaging position is to the focal length of the lens. For a 3D object, when the imaging depth is z1, the corresponding transform order is a1, and when the imaging depth is z2, the corresponding transform order is a2. The signal interval of the image RD(X, Y) should be enlarged by g times, and the enlargement factor g is expressed as: Where the enlargement factor g represents the expansion range of the signal interval; In step two, the complex amplitude distribution D1(X, Y) of the hologram plane obtained by fractional Fourier transform is expressed as: where m and n are the number of pixels in the lateral and longitudinal directions of the object plane complex amplitude distribution D0(X, Y), X and Y represent the lateral and longitudinal coordinates, respectively, λ is the wavelength of the incident light, f e is the focal length in the Fourier transform, dx and du are the sampling intervals of the object plane and the hologram plane, respectively, FFT represents the fast Fourier transform, and the object plane complex amplitude distribution D2(X, Y) is represented as: wherein is the phase distribution of the complex amplitude distribution D1(X,Y) of the hologram plane, the complex amplitude distribution D3(X,Y) after imposing the double constraint is expressed as: where Sig and Noi represent the signal interval and the noise interval, respectively, and A2 represent the phase and the amplitude of the complex amplitude distribution D2(X, Y), respectively, and c is a constant between 0 and 1 that adjusts the size of the noise interval. The multiple loop iterations cause the phase to converge to an optimal value, thereby effectively suppressing speckle noise.

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