Image Edge Detection Method, Device and Complex Amplitude Modulation Module Based on Differential Operation of Complex Amplitude Filtering Topological Space

Abstract

The present invention discloses an image edge detection method, device and complex amplitude modulation module based on complex amplitude filtering topological space differential operation, relating to the technical field of optical information processing. The technical points of the present invention include: placing a two-dimensional image on the object plane of a 4f system composed of two Fourier transform lenses, and performing Fourier transform on the signal light beam carrying the two-dimensional image information through the first Fourier transform lens to generate the Fourier spectrum of the two-dimensional image on the spectrum plane of the 4f system; at the same time, placing a complex amplitude filtering device for realizing complex amplitude filtering topological space differential operation on the spectrum plane of the 4f system to generate a corresponding transfer function, so as to modulate the Fourier spectrum of the two-dimensional image; performing inverse Fourier transform on the Fourier spectrum modulated by the two-dimensional image through the second Fourier transform lens, and obtaining the result of image differential operation on the image plane of the 4f system to realize image edge detection. The present invention can be applied to aspects such as image edge detection and biological microscopic imaging.

Description

Technical Field

[0001] The present invention relates to the technical field of optical information processing, and particularly relates to an image edge detection method, device and complex amplitude modulation module based on complex amplitude filtering topological space differential operation. Background Art

[0002] Image edge enhancement or detection is one of the core contents of image processing. Among them, spatial differentiation is widely used in edge detection. Therefore, differentiation is a key mathematical operation that is particularly important in image processing. Traditionally, image processing is achieved through digital computing, that is, digital image processing. In digital image processing, convolution kernels known as edge detection operators (such as Sobel operators, Laplace operators, etc.) are usually used to perform convolution operations with the original image in the spatial domain to achieve image edge detection. However, due to the analog-to-digital conversion and discretization processes, such a differential operation process is limited by low energy efficiency and low operation speed. On the contrary, optical analog computing provides advantages such as high-speed operation, parallel processing, low power consumption, etc. Moreover, optical spatial differentiation can not only achieve edge detection of amplitude objects, but also achieve the conversion of phase gradient to intensity information. Therefore, in recent years, it has attracted the research interest of many scientific researchers again.

[0003] Related research focuses on achieving optical spatial differentiation through various methods, such as surface plasmons, photonic crystals, spin-orbit interactions, Brewster effects, photonic spin Hall effects, geometric spin Hall effects, etc. In addition, with the development of nanophotonic technologies, due to the advantage of easy integration, metasurfaces and liquid crystal photonic platforms based on Pancharatnam-Berry phase and nanophotonic devices of photonic chips are used to achieve spatial differentiation and optical analog information processing. In particular, an optical analog spatial differentiator based on the concept of topological photonics has also been proposed. The article "Topological optical differentiator" published on page 680 of Volume 12 of "Nature Communications" in 2021 established the connection between two-dimensional optical spatial differentiation and topological charges in the transfer function. Based on this connection, isotropic two-dimensional differentiation with a wide spectral bandwidth was achieved only by reflecting light waves at the dielectric interface. Subsequently, an isotropic topological second-order spatial differentiator operating in the transmission mode under normal incidence was proposed in the article "Isotropic topological second-order spatial differentiator operating in transmission mode" published on page 3247 of Volume 46 of "Optics Letters" in 2021, showing a transfer function with topological charges of ±2. It is undeniable that differentiators based on planar optical elements have the advantage of being easily integrated into a compact imaging system.

[0004] However, existing differential operations usually require input / output orthogonal polarization control or complex manufacturing processes. Therefore, further simplifying the design and implementation of spatial differentiators has become an urgent problem to be solved in practical applications. Summary of the Invention

[0005] To this end, the present invention proposes an image edge detection method, device and complex amplitude modulation module based on complex amplitude filtering topological spatial differential operations, in an attempt to solve or at least alleviate at least one of the problems mentioned above.

[0006] According to one aspect of the present invention, there is provided an image edge detection method based on complex amplitude filtering topological spatial differential operations, the image edge detection method comprising the following steps:

[0007] Place a two-dimensional image on the object plane of a 4f system composed of two Fourier transform lenses. The signal beam carrying the two-dimensional image information is Fourier-transformed by the first Fourier transform lens to generate the Fourier spectrum of the two-dimensional image on the spectrum plane of the 4f system;

[0008] At the same time, place a complex amplitude filtering device for realizing complex amplitude filtering topological spatial differential operations on the spectrum plane of the 4f system to generate a corresponding transfer function, thereby modulating the Fourier spectrum of the two-dimensional image;

[0009] Perform an inverse Fourier transform on the Fourier spectrum modulated by the two-dimensional image through the second Fourier transform lens to obtain the result of the image differential operation on the image plane of the 4f system, and realize image edge detection.

[0010] Furthermore, the transfer function corresponding to the complex amplitude filtering device is generated based on complex amplitude filtering topological spatial differential operations, wherein,

[0011] For one-dimensional anisotropic nth-order differential operations, the corresponding transfer function is expressed as T1(k x , k y ), and the following relationship exists for this function:

[0012]

[0013] For two-dimensional isotropic nth-order differential operations, the corresponding transfer function is expressed as T2(k x , k y ), and the following relationship exists for this function:

[0014] T2(k x , k y ) ∝ (k x ± ik y ) n

[0015] In the formula, kx , k y respectively represent the components of the wave vector of the incident light field on the x and y axes in the frequency-domain Cartesian coordinate system.

[0016] Furthermore, for two-dimensional isotropic nth-order differential operations, the corresponding transfer function is expressed using the frequency-domain polar coordinate system and Euler's formula as:

[0017]

[0018] In the formula, represents the projection of the wave vector of the incident light field along the radial direction on the Fourier plane; represents the polar axis k on the Fourier plane r and the axis k x angle.

[0019] Furthermore, the complex amplitude filtering device is a metasurface optical device, a liquid crystal planar optical device, a spatial light modulator, or a digital micromirror device.

[0020] According to another aspect of the present invention, there is provided an image edge detection device based on complex amplitude filtering topological space differential operation, the device comprising: a 4f system composed of a first Fourier transform lens and a second Fourier transform lens, a complex amplitude filtering device, and a camera; wherein,

[0021] A two-dimensional image is placed on the object plane of the 4f system, and the complex amplitude filtering device is placed on the spectrum plane of the 4f system; the first Fourier transform lens is used to perform Fourier transform on the signal beam carrying two-dimensional image information; the complex amplitude filtering device is used to modulate the Fourier spectrum of the two-dimensional image; the second Fourier transform lens is used to perform inverse Fourier transform on the Fourier spectrum modulated by the two-dimensional image, obtain the image differential operation result on the image plane of the 4f system, and image it onto the camera.

[0022] Furthermore, the transfer function corresponding to the complex amplitude filtering device for modulating the Fourier spectrum of the two-dimensional image is generated based on complex amplitude filtering topological space differential operation, wherein,

[0023] For one-dimensional anisotropic nth-order differential operation, its corresponding transfer function is expressed as T1(k x , k y ), and the following relationship exists for this function:

[0024]

[0025] For two-dimensional isotropic nth-order differential operation, its corresponding transfer function is expressed as T2(k x , k y ), and the following relationship exists for this function:

[0026] T2(k x , k y ) ∝ (k x ±ik y ) n

[0027] In the formula, k x , k y respectively represent the components of the wave vector of the incident light field on the x and y axes in the frequency-domain Cartesian coordinate system.

[0028] Furthermore, for a two-dimensional isotropic nth-order differential operation, the corresponding transfer function is expressed using the frequency-domain polar coordinate system and Euler's formula as:

[0029]

[0030] In the formula, represents the projection of the wave vector of the incident light field along the radial direction on the Fourier plane; represents the polar axis k r and the axis k x angle.

[0031] Furthermore, the complex amplitude filtering device is a metasurface optical device, a liquid crystal planar optical device, a spatial light modulator, or a digital micromirror device.

[0032] According to another aspect of the present invention, a complex amplitude modulation module is also provided. The complex amplitude modulation module is used to modulate the Fourier spectrum of a two-dimensional image, and the corresponding transfer function during its modulation process is generated based on complex amplitude filtering topological space differential operations; wherein,

[0033] For a one-dimensional anisotropic nth-order differential operation, its corresponding transfer function is expressed as T1(k x , k y ), and the following relationship exists for this function:

[0034]

[0035] For a two-dimensional isotropic nth-order differential operation, its corresponding transfer function is expressed as T2(k x , k y ), and the following relationship exists for this function:

[0036] T2(k x , k y ) ∝ (k x ±ik y ) n

[0037] In the formula, k x , k yThey respectively represent the components of the wave vector of the incident light field on the x-axis and y-axis in the frequency-domain Cartesian coordinate system.

[0038] Furthermore, for a two-dimensional isotropic nth-order differential operation, the corresponding transfer function is expressed using the frequency-domain polar coordinate system and Euler's formula as:

[0039]

[0040] In the formula, represents the projection of the wave vector of the incident light field along the radial direction on the Fourier plane; represents the polar axis k on the Fourier plane r and the axis k x angle.

[0041] The beneficial technical effects of the present invention are:

[0042] The present invention provides a method based on Fourier space complex amplitude filtering for isotropic, anisotropic multi-order spatial differential operations and edge detection of images, and provides a Fourier space complex amplitude filtering device with simple operation and strong practicability; compared with traditional digital operations, optical operations have the advantages of fast speed, low power consumption, and large-scale parallel computing; in addition, compared with existing differential operation edge detection methods, since devices such as input and output polarization control are not required, the differential system device is more concise, and it can simplify the design and manufacture of planar optical devices such as metasurfaces and liquid crystals, facilitating the formation of a miniaturized image edge detection system, and has important applications in biological microscopy, computer and robot vision, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] By reading the following detailed description with reference to the accompanying drawings, the above and other objects, features, and advantages of the exemplary embodiments of the present invention will become readily understood. In the drawings, several embodiments of the present invention are shown in an exemplary rather than restrictive manner, wherein:

[0044] Figure 1 is a flowchart of an image edge detection method according to an embodiment of the present invention.

[0045] Figure 2 is a schematic structural diagram of an image edge detection device according to an embodiment of the present invention.

[0046] Figure 3It is an exemplary diagram of isotropic multi - order topological differentiation of an amplitude object in an embodiment of the present invention. Among them, (a0) is the simulation result of the output image without any filter; (a1)-(a4) correspond to the simulation results of the first - order, second - order, third - order, and fourth - order image differential images respectively; (b0) is the experimental result of the output image without any filter; (b1)-(b4) correspond to the experimental results of the first - order, second - order, third - order, and fourth - order image differential images respectively; (c1)-(c4) depict the intensity distributions of the differential images along the white dashed lines in the corresponding simulation and experimental results.

[0047] Figure 4 It is an exemplary diagram of anisotropic differentiation of an amplitude object in an embodiment of the present invention. Among them, (a1)–(a4) and (b1)-(b4) are the simulation and experimental results of anisotropic - direction first - order differentiation along the horizontal, vertical, diagonal, and anti - diagonal directions respectively; (c1)–(c4) and (d1)-(d4) are the simulation and experimental results in the corresponding directions in the case of second - order differentiation respectively.

[0048] Figure 5 It is an exemplary diagram of isotropic topological differentiation of a phase object in an embodiment of the present invention. Among them, (a1) and (b1) correspond to the target phase distribution and bright - field imaging result without any filter respectively; (a2) and (a3) correspond to the simulation results of the first - order and second - order differentiation respectively; (b2) and (b3) correspond to the experimental results of the first - order and second - order differentiation respectively. Detailed implementation manners

[0049] The principles and spirit of the present invention will be described below with reference to several exemplary implementation manners. It should be understood that these implementation manners are provided only to enable those skilled in the art to better understand and then implement the present invention, rather than limiting the scope of the present invention in any way. On the contrary, these implementation manners are provided to make the present disclosure more thorough and complete, and to be able to convey the scope of the present disclosure completely to those skilled in the art.

[0050] By establishing the relationship between multi - order differential operations in real space and transfer functions in Fourier space, the present invention proposes a topological space differentiator based on complex - amplitude filtering in Fourier space. By using a complex - amplitude filter with a topological - charge spiral phase that is consistent with the differential order in the frequency domain, two - dimensional multi - order differential operations are realized, thereby realizing isotropic multi - order differential image edge detection. In addition, through a direction - differential filtering device that includes an azimuth parameter, one - dimensional multi - order differentiation in any direction is realized, and further, anisotropic multi - order differential image edge detection in any direction is realized. The present invention provides a method for spatial differential operations and image edge detection based on complex - amplitude filtering in Fourier space. The present invention will be elaborated in detail below.

[0051] An embodiment of the present invention proposes an image edge detection method based on complex amplitude filtering topological space differential operation, as Figure 1 shown. The method includes the following steps:

[0052] Place the two-dimensional image on the object plane of a 4f system composed of two Fourier transform lenses. The signal beam carrying the two-dimensional image information is Fourier-transformed by the first Fourier transform lens to generate the Fourier spectrum of the two-dimensional image on the spectrum plane of the 4f system;

[0053] At the same time, place a complex amplitude filtering device for realizing complex amplitude filtering topological space differential operation on the spectrum plane of the 4f system to generate a corresponding transfer function, thereby modulating the Fourier spectrum of the two-dimensional image;

[0054] Perform an inverse Fourier transform on the Fourier spectrum of the two-dimensional image modulated by the second Fourier transform lens to obtain the result of the image differential operation on the image plane of the 4f system, realizing image edge detection.

[0055] According to the embodiment of the present invention, compared with traditional digital signal processing, optical analog signal processing realizes various computing functions by manipulating the propagation of light waves. According to Fourier optical theory, the relationship between multi-order differential and the corresponding transfer function is first established. Considering the incident light field U in (x, y) carrying object information in the Cartesian coordinate system and the filtered outgoing light field U out (x, y), the incident and outgoing light fields can be expanded as the sum of a series of spatial frequency components, as follows:

[0056] U in (x, y) = ∫∫A in ((k x , k y )exp(ik x x + ik y y)dk x dk y , (1)

[0057] U out (x, y) = ∫∫A out (k x , k y )exp(ik x x + ik y y)dk x dk y , (2)

[0058] A out (k x , k y ) = T(k x , k y )Ain (k x ,k y ). (3)

[0059] Among them, A in (k x ,k y ) and A out (k x ,k y ) are the spatial Fourier spectra of the scalar fields U in (x, y) and U out (x, y) respectively; k x ,k y represent the x and y axis components of the wave vector of the incident light field in the frequency domain Cartesian coordinate system respectively; T(k x ,k y ) represents the transfer function of the entire differential filtering system.

[0060] To achieve the first-order differential in a certain direction, assuming along the x-axis direction, the output light field should satisfy Therefore, for the Fourier expansion of the incident light field U in (x, y), that is, equation (1), taking the partial derivative with respect to x on both sides gives:

[0061]

[0062] Substitute equation (3) into equation (2) and compare it with equation (4), the transfer function can be expressed as:

[0063] T(k x ,k y ) = ik x ∝ k x . (5)

[0064] In addition, for higher-order one-dimensional anisotropic differentials, the transfer function T(k x ,k y ) is derived in the same way as follows:

[0065]

[0066] Therefore, for the one-dimensional anisotropic nth-order differential, its corresponding transfer function T1(k x ,k y ) can be expressed as:

[0067]

[0068] To achieve the first-order differential of two-dimensional isotropy, the transfer function should be rotation-invariant. In such a case, the transfer function T(k x ,k y) is a complex scalar field carrying topological charge and exhibits topological invariance, i.e., k x and k y The coefficients of should differ by a multiple of ±i. Assuming the transfer function satisfies equation (8), the corresponding output optical field is then derived as formula (9).

[0069] T(k x , k y ) ∝ k x + ik y , (8)

[0070]

[0071] Thus, for a given input optical field U in (x, y), the corresponding output intensity distribution can be written as:

[0072]

[0073] Therefore, for two-dimensional isotropic nth-order differentiation, the corresponding transfer function T2(k x , k y ) can be expressed as:

[0074] T2(k x , k y ) ∝ (k x ± ik y ) n (11)

[0075] Furthermore, to more clearly illustrate the transfer function corresponding to isotropic two-dimensional first-order differentiation, the transfer function is expressed using polar coordinates and Euler's formula as:

[0076]

[0077] where is the projection of the wave vector along the radial direction on the Fourier plane, is the polar axis on the Fourier plane k r and the axis k x The included angle between

[0078] Similarly, to achieve two-dimensional isotropic high-order differentiation, the transfer function in polar coordinates should have the following form:

[0079]

[0080] where n represents the order of differentiation.

[0081] It should be noted that the transfer function has the characteristics of amplitude filtering and phase filtering, that is, complex amplitude filtering, where the phase distribution is a helical phase carrying the topological charge corresponding to the differential order, so it is called topological differentiation.

[0082] Another embodiment of the present invention further proposes an image edge detection device, as Figure 2 shown, the device includes: a 4f system composed of a first Fourier transform lens and a second Fourier transform lens, a complex amplitude filtering device, and a camera; wherein,

[0083] A two-dimensional image is placed on the object plane of the 4f system, and the complex amplitude filtering device is placed on the spectral plane of the 4f system; the first Fourier transform lens is used to perform Fourier transform on the signal beam carrying the two-dimensional image information; the complex amplitude filtering device is used to modulate the Fourier spectrum of the two-dimensional image; the second Fourier transform lens is used to perform inverse Fourier transform on the Fourier spectrum modulated by the two-dimensional image, obtain the result of image differential operation on the image plane of the 4f system, and image it onto the camera.

[0084] The working principle of the device is as follows: First, the target image 1 is placed on the object plane of the 4f system composed of the first Fourier transform lens 2 and the second Fourier transform lens 4. The signal beam carrying the information of the target image 1 performs Fourier transform through the first Fourier transform lens 2, thereby generating the Fourier spectrum of the target image on the spectral plane of the 4f system. Then, the complex amplitude filtering device 3 corresponding to different spatial differential operations (which can be a metasurface optical device, a liquid crystal planar optical device, a spatial light modulator, or a digital micromirror device, etc.) is placed on the spectral plane of the 4f system to modulate the Fourier spectrum of the target image 1. Then, the second Fourier transform lens 4 performs inverse Fourier transform on the modulated Fourier spectrum of the target image 1. Finally, the result of image differential operation is obtained on the image plane of the 4f system, realizing multi-order differential edge detection of the image, and finally imaging it onto the camera 5.

[0085] Another embodiment of the present invention further proposes a complex amplitude modulation module, which is used to modulate the Fourier spectrum of a two-dimensional image, and the corresponding transfer function during its modulation process is generated based on complex amplitude filtering topological space differential operation; wherein,

[0086] For one-dimensional anisotropic n-order differential operation, its corresponding transfer function is expressed as T1(k x , k y ), and the following relationship exists for this function:

[0087]

[0088] For two-dimensional isotropic n-order differential operation, its corresponding transfer function is expressed as T2(k x , ky ), the following relationship exists for this function:

[0089] T2(k x , k y ) ∝ (k x ± ik y ) n

[0090] where k x , k y respectively represent the x and y axis components of the wave vector of the incident light field in the frequency domain Cartesian coordinate system.

[0091] For two-dimensional isotropic nth-order differential operation, using the frequency domain polar coordinate system and Euler's formula, its corresponding transfer function is expressed as:

[0092]

[0093] where represents the projection of the wave vector of the incident light field along the radial direction on the Fourier plane; represents the polar axis k r and the axis k x angle.

[0094] Furthermore, through Matlab numerical simulation and actual experiments, complex amplitude filtering is used to achieve multi-order topological space differential operation, and multi-line edge detection of amplitude objects (with amplitude change and phase unchanged after modulation) and phase objects (with phase change and amplitude unchanged after modulation) is realized to verify the technical effects of the present invention.

[0095] In both the simulation and the experiment, a 532 nm laser light source is used to illuminate the object, and the focal lengths of the two Fourier transform lenses forming the 4f system are both set to 100 mm. In the experiment, based on complex amplitude modulation, a hologram is generated by coding calculation to generate a target differential filter on a liquid crystal spatial light modulator. The number "2" of the negative USAF-1951 resolution test target is selected as the amplitude object, and isotropic first-order, second-order, third-order, and fourth-order topological differential operations are performed, generating corresponding multi-order differential edges. The number of multi-edges corresponds to the order of differentiation, and the local intensity curve of the edge at the position of the white dotted line of the differential image generated by the simulation and the experiment is depicted for comparison, as Figure 3 shown. First-order and second-order differential operations in the anisotropic horizontal, vertical, diagonal, and anti-diagonal directions are also applied to the amplitude object, and the differential images are as Figure 4As shown. For phase objects, a phase target with a phase difference of π between the left half and the right half is selected for numerical simulation. In the experiment, a spiral phase plate with a phase mutation is used as an analogue. Through isotropic differential operations, clear single-line edges and double-line edges corresponding to isotropic first-order and second-order differentials are achieved, as Figure 5 shown.

[0096] Although the operations of the method of the present invention are described in a specific order in the accompanying drawings, this does not require or imply that these operations must be performed in that specific order, or that all the operations shown must be performed to achieve the desired result. Additionally or alternatively, certain steps may be omitted, multiple steps may be combined into one step for execution, and / or one step may be decomposed into multiple steps for execution.

[0097] Although the spirit and principles of the present invention have been described with reference to several specific embodiments, it should be understood that the present invention is not limited to the specific embodiments disclosed, and the division of each aspect does not mean that the features in these aspects cannot be combined for benefit. This division is only for the convenience of expression. The present invention aims to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.

Claims

1. An image edge detection method based on differential operation of complex amplitude filtering topological space, characterized in that: The image edge detection method includes the following steps: Place the two-dimensional image on the object plane of a 4f system composed of two Fourier transform lenses. The signal beam carrying the two-dimensional image information is Fourier-transformed by the first Fourier transform lens to generate the Fourier spectrum of the two-dimensional image on the spectrum plane of the 4f system; Meanwhile, a complex amplitude filtering device for implementing complex amplitude filtering topological space differential operation is placed on the spectrum plane of the 4f system to generate a corresponding transfer function, thereby modulating the Fourier spectrum of the two-dimensional image; the transfer function corresponding to the complex amplitude filtering device is generated based on the complex amplitude filtering topological space differential operation, wherein, for one-dimensional anisotropic n-order differential operation, the corresponding transfer function is expressed as T1(k x ,k y ), and the following relationship exists for this function: For a two-dimensional isotropic nth-order differential operation, its corresponding transfer function is expressed as T2(k x , k y ), and the following relationship exists for this function: T2(k x ,k y )∝(k x ±ik y ) n where k x , k y respectively represent the x- and y-axis components of the wave vector of the incident light field in the frequency-domain Cartesian coordinate system; For two-dimensional isotropic n-order differential operations, the corresponding transfer function is expressed using the frequency-domain polar coordinate system and Euler's formula as: In the formula, represents the projection of the wave vector of the incident light field along the radial direction on the Fourier plane; represents the polar axis k on the Fourier plane r and the axis k x angle; The Fourier spectrum modulated by the two-dimensional image through the second Fourier transform lens is subjected to inverse Fourier transform to obtain the image differential operation result on the image plane of the 4f system, thereby realizing image edge detection.

2. The image edge detection method according to claim 1, wherein: The complex amplitude filtering device is a metasurface optical device, a liquid crystal planar optical device, a spatial light modulator, or a digital micromirror device.

3. An image edge detection device based on differential operation of complex amplitude filtering topological space, characterized in that: It includes: A 4f system composed of a first Fourier transform lens and a second Fourier transform lens, a complex amplitude filtering device, and a camera; where A two-dimensional image is placed on the object plane of the 4f system, and the complex amplitude filtering device is placed on the spectrum plane of the 4f system; the first Fourier transform lens is used to perform Fourier transform on the signal beam carrying two-dimensional image information; the complex amplitude filtering device is used to modulate the Fourier spectrum of the two-dimensional image; the second Fourier transform lens is used to perform inverse Fourier transform on the Fourier spectrum after modulating the two-dimensional image, obtain the image differential operation result on the image plane of the 4f system, and image it onto the camera; the transfer function corresponding to the complex amplitude filtering device for modulating the Fourier spectrum of the two-dimensional image is generated based on the complex amplitude filtering topological space differential operation, wherein, for one-dimensional anisotropic nth-order differential operation, its corresponding transfer function is expressed as T1(k x ,k y ), and the following relationship exists for this function: For a two-dimensional isotropic nth-order differential operation, its corresponding transfer function is expressed as T2(k x , k y ), and the following relationship exists for this function: T2(k x ,k y )∝(k x ±ik y ) n where k x , k y respectively represent the x- and y-axis components of the wave vector of the incident optical field in the frequency-domain Cartesian coordinate system; For two-dimensional isotropic n-order differential operations, the corresponding transfer function is expressed using the frequency-domain polar coordinate system and Euler's formula as: In the formula, represents the projection of the wave vector of the incident light field along the radial direction on the Fourier plane; represents the polar axis k on the Fourier plane r and the axis k x angle.

4. The image edge detection device according to claim 3, wherein: The complex amplitude filtering device is a metasurface optical device, a liquid crystal planar optical device, a spatial light modulator, or a digital micromirror device.

5. A complex amplitude modulation system, characterized in that: The complex amplitude modulation system is used to modulate the Fourier spectrum of a two-dimensional image, and the corresponding transfer function during the modulation process is generated based on the differential operation of the complex amplitude filtering topological space; among them, for the one-dimensional anisotropic nth-order differential operation, the corresponding transfer function is expressed as T1(k x , k y ), and the following relationship exists for this function: For a two-dimensional isotropic nth-order differential operation, its corresponding transfer function is expressed as T2(k x , k y ), and the following relationship exists for this function: T2(k x ,k y )∝(k x ±ik y ) n where k x , k y respectively represent the x- and y-axis components of the wave vector of the incident light field in the frequency-domain Cartesian coordinate system; For two-dimensional isotropic n-order differential operations, the corresponding transfer function is expressed using the frequency-domain polar coordinate system and Euler's formula as: In the formula, represents the projection of the wave vector of the incident light field along the radial direction on the Fourier plane; represents the polar axis k on the Fourier plane γ and the axis k x angle.

Images