An ultra-high-speed digital micromirror spatial light modulator detection device and method

The detection device, composed of a laser, a computer, a Fourier lens, and a photodetector, solves the problem that existing technologies cannot detect ultra-high-speed digital micromirror spatial light modulators, and realizes a low-cost and efficient detection method.

CN116007905BActive Publication Date: 2025-12-19JINHUA FEMTOSECOND LIGHT TECH CO LTD
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Patent Information

Application Number
CN202310072452.X
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2023-02-07
Publication Date
2025-12-19
Estimated Expiration
2043-02-07

AI Technical Summary

Technical Problem

Existing detection methods cannot effectively detect the technical parameters of ultra-high-speed digital micromirror spatial light modulators, especially due to insufficient camera acquisition speed, which makes it impossible to complete the acquisition of high frame rate images.

Method used

The detection device consists of a laser, a computer, a Fourier lens, and a photodetector. The computer processes the image and loads it onto an ultra-high-speed digital micromirror spatial light modulator. The collimated laser beam and Fourier lens are used to perform image transformation, and the photodetector detects the position of relevant peaks to determine whether the device is qualified.

Benefits of technology

It enables effective detection of ultra-high-speed digital micromirror spatial light modulators, with low cost and wide applicability. It can detect whether the bright spot frequency at the relevant peak position is equal to half of the set frequency, and determine whether the device is qualified.

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Abstract

The application discloses a kind of ultra-high-speed digital micro-mirror spatial light modulator detection device and method, it is related to spatial light modulator test field, including: laser, computer, ultra-high-speed digital micro-mirror spatial light modulator, fourier lens and photoelectric detector;The frequency spectrum image of two identical images is directly calculated by computer, the obtained frequency spectrum image and full black image are loaded to ultra-high-speed digital micro-mirror spatial light modulator, the irradiation of collimated laser beam is passed through, makes ultra-high-speed digital micro-mirror spatial light modulator with set frequency output frequency spectrum image and full black image, after the processing of fourier lens, the presence or absence of bright spot in two image correlation peak positions is detected using photoelectric detector, if the frequency of bright spot is equal to half of set frequency, then it indicates that ultra-high-speed digital micro-mirror spatial light modulator is qualified.The scheme of the present application is low in cost, wide in scope of application, and can detect ultra-high-speed digital micro-mirror spatial light modulator.
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Description

TECHNICAL FIELD

[0001] The present application relates to the field of spatial light modulator testing, in particular to a high-speed digital micro-mirror spatial light modulator detection device and method. BACKGROUND

[0002] Spatial light modulators (SLM) can modulate spatial light, and according to their structures, they can be divided into liquid crystal spatial light modulators, digital micro-mirror spatial light modulators, magnetic spatial light modulators, multi-quantum well spatial light modulators, electro-optic spatial light modulators, and acousto-optic spatial light modulators. Spatial light modulators can be divided into amplitude type, phase type, and hybrid type according to their modulation characteristics. Amplitude type spatial light modulators only modulate the amplitude of the incident light field, and the typical device is a digital micro-mirror spatial light modulator; phase type spatial light modulators only modulate the phase of the incident light field. Spatial light modulators are widely used in projectors, displays, holographic data storage, laser processing, information coding, beam shaping, compressed sensing, maskless lithography, and other fields.

[0003] Spatial light modulators need to be tested for technical parameters such as frequency, resolution, and number of bad pixels when they are shipped. These parameters are tested by professional measuring instruments. For example, the resolution, speed, and pixel condition of a traditional amplitude type spatial light modulator can be measured by an industrial camera, but for a high-speed digital micro-mirror spatial light modulator, the binary image modulation speed can reach 20Khz or more at a resolution of 1920*1080. Due to the insufficient speed of the camera, it is unable to complete the task of collecting high frame rate images modulated by the digital micro-mirror spatial light modulator, and thus it is unable to detect the high-speed digital micro-mirror spatial light modulator.

[0004] Therefore, there is an urgent need for a method capable of detecting a high-speed digital micro-mirror spatial light modulator. SUMMARY

[0005] The purpose of the present application is to provide a high-speed digital micro-mirror spatial light modulator detection device and method, so as to provide a method capable of detecting a high-speed digital micro-mirror spatial light modulator.

[0006] To achieve the above-mentioned purpose, the present application provides the following solutions:

[0007] A high-speed digital micro-mirror spatial light modulator detection device, comprising: a laser, a computer, a high-speed digital micro-mirror spatial light modulator, a Fourier lens, and a photodetector.

[0008] The computer is connected with the ultra-high-speed digital micromirror spatial light modulator; the computer is used for acquiring two same gray scale images and one all-black image and processing the two gray scale images to obtain a spectrum image, and uploading the spectrum image and the all-black image to the ultra-high-speed digital micromirror spatial light modulator;

[0009] The laser is used for generating a collimated laser beam;

[0010] The ultra-high-speed digital micromirror spatial light modulator is arranged on an outgoing light path of the laser; the collimated laser beam is incident on the ultra-high-speed digital micromirror spatial light modulator and reflects the spectrum image or the all-black image;

[0011] The Fourier lens is arranged on a reflected light path of the ultra-high-speed digital micromirror spatial light modulator; the Fourier lens is used for performing inverse Fourier transform on the spectrum image or the all-black image and outputting a transformed spectrum image or a transformed all-black image;

[0012] The photoelectric detector is arranged on an outgoing light path of the Fourier lens and connected with the computer; the photoelectric detector is used for detecting a correlation peak position of the transformed spectrum image or a correlation peak position of the transformed all-black image and uploading a detection result to the computer; the detection result is a frequency of the correlation peak position having a bright spot;

[0013] The computer is further used for determining whether the ultra-high-speed digital micromirror spatial light modulator is qualified according to the detection result.

[0014] Optionally, the laser is a solid laser, a fiber laser, a gas laser or a semiconductor laser.

[0015] Optionally, the photoelectric detector is a photosensitive diode or a photosensitive triode.

[0016] Optionally, the Fourier lens is a cemented lens or a diffractive optical element.

[0017] An ultra-high-speed digital micromirror spatial light modulator detection method, applied to the ultra-high-speed digital micromirror spatial light modulator detection device, comprises:

[0018] Acquiring two same gray scale images and one all-black image and processing the two gray scale images to obtain a spectrum image;

[0019] upload the spectrum image and the all-black image to a high-speed digital micro-mirror spatial light modulator; the high-speed digital micro-mirror spatial light modulator outputs the spectrum image and the all-black image at a set frequency;

[0020] The collimated laser beam is incident on the high-speed digital micro-mirror spatial light modulator and reflects the spectrum image or the all-black image;

[0021] The Fourier lens inversely transforms the spectrum image or the all-black image to obtain a transformed spectrum image or a transformed all-black image;

[0022] The photoelectric detector detects a correlation peak position of the transformed spectrum image or a correlation peak position of the transformed all-black image to obtain a detection result; the detection result is a frequency at which the correlation peak position has a bright spot;

[0023] When the frequency at which the correlation peak position has the bright spot is equal to half of the set frequency, it is determined that the high-speed digital micro-mirror spatial light modulator is qualified.

[0024] Optionally, the two gray-scale images are processed to obtain a spectrum image, and the processing specifically includes:

[0025] The two gray-scale images are combined into one gray-scale image to obtain a merged gray-scale image;

[0026] The joint power spectrum of the merged gray-scale image is calculated;

[0027] The useful power spectrum is extracted from the joint power spectrum;

[0028] The useful power spectrum is binarized to obtain the spectrum image.

[0029] Optionally, the joint power spectrum of the two gray-scale images is calculated, and the calculation specifically includes:

[0030] The joint power spectrum of the two gray-scale images is calculated by using the formula ; wherein, F(u,v) represents the Fourier transform of one gray-scale image; G(u,v) represents the Fourier transform of another gray-scale image; u represents the spatial frequency in the x direction; v represents the spatial frequency in the y direction; a represents half of the center distance in the x direction of the two gray-scale images; f represents the spatial frequency; G*(u,v) represents the complex conjugate function of G(u,v); and F*(u,v) represents the complex conjugate function of F(u,v).

[0031] According to the specific embodiments of the present application, the following technical effects are provided:

[0032] The super high-speed digital micro-mirror spatial light modulator detection device and method of the present application directly calculates the spectrum images of two same images by a computer, loads the obtained spectrum images and full black images to the super high-speed digital micro-mirror spatial light modulator, makes the super high-speed digital micro-mirror spatial light modulator output the spectrum images and full black images at a set frequency through the irradiation of a collimated laser beam, processes through a Fourier lens, detects the presence or absence of a bright spot in the related peak position of the two images by a photoelectric detector, and if the frequency of the bright spot is equal to half of the set frequency, it indicates that the super high-speed digital micro-mirror spatial light modulator is qualified. The scheme of the present application has low cost, wide application range, and can detect the super high-speed digital micro-mirror spatial light modulator. BRIEF DESCRIPTION OF DRAWINGS

[0033] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed in the embodiments will be briefly introduced as follows. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0034] Figure 1 The super high-speed digital micro-mirror spatial light modulator detection device provided by the present application is shown in the schematic diagram.

[0035] Figure 2 The joint transform correlator schematic diagram is provided.

[0036] Symbol explanation:

[0037] 1, computer; 2, photoelectric detector; 3, Fourier lens; 4, super high-speed digital micro-mirror spatial light modulator; 5, collimated laser beam. DETAILED DESCRIPTION

[0038] The technical solutions in the embodiments of the present application will be described clearly and completely in the following with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only some embodiments of the present application, not all. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the scope of protection of the present application.

[0039] The purpose of the present application is to provide a super high-speed digital micro-mirror spatial light modulator detection device and method, to provide a method capable of detecting a super high-speed digital micro-mirror spatial light modulator.

[0040] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the present application will be further described in detail below with reference to the drawings and specific embodiments.

[0041] Optical correlator has the characteristics of parallel processing and high speed, so the application detects the digital micromirror spatial light modulator by the principle of optical correlator. Joint transform correlator (JTC) is a typical optical correlator, which was first proposed by C.S. Weaver and J.W. Goodman in the early 1960s. JTC requires the target image and the reference image to be arranged on both sides of the input plane, and the joint transform power spectrum (JPS) is obtained by Fourier lens to complete Fourier transform. The power spectrum image is then inversely Fourier transformed to obtain the correlation operation result. When the target image and the reference image are completely consistent, a pair of conjugate correlation peaks can be obtained on the output plane. The principle diagram of JTC is shown in the following figure. Figure 2

[0042] In JTC, the input reference image and target image are arranged on both sides of the input plane. As shown in the following figure, Figure 2 the reference image f(x+a, y) is arranged on the right side, and the target image g(x-a, y) is arranged on the left side, with a center distance of 2a. Thus, the entire input plane can be denoted as s(x, y)=f(x+a, y)+g(x-a, y), which is transformed by Fourier transform lens L1, and the joint spectrum on the frequency spectrum plane P2 is:

[0043]

[0044] In the formula, G(u, v), F(u, v), and S(u, v) are the Fourier transforms of g(x, y), f(x, y), and s(x, y), respectively. On the P2 plane, a square rate detector such as a CCD is used to record, which only responds to light intensity, so the joint power spectrum is:

[0045]

[0046] After the CCD collects the joint power spectrum, it is input to the spatial light modulator (SLM), and the same unit amplitude plane wave is read out, so that the output function is obtained on the output plane P3 after inverse Fourier transform by Fourier lens L2.

[0047]

[0048] In the formula, “*” is convolution operation, ​are the autocorrelations of f(ξ,η) and g(ξ,η) respectively, which form the 0th order diffraction, centered at the center of the output plane, belonging to the interference term, the last two terms are the cross-correlations of f(ξ,η) and g(ξ,η), which form the 1st order diffraction, centered at (ξ=±2a,η=0), which are the correlation terms needed for identification, appearing as two correlation peaks on the image. From the above analysis, the spectrum needed for identification is:

[0049]

[0050] where "Real" represents the real part, and "Image" represents the imaginary part. Further analysis shows that the presence or absence of the cross-correlation peak is determined by the above formula, and its position is determined by the shift factor .

[0051] Example 1

[0052] As shown in Figure 1 , the super-speed digital micro-mirror spatial light modulator detection device provided by the application comprises a laser, a computer 1, a super-speed digital micro-mirror spatial light modulator 4, a Fourier lens 3 and a photoelectric detector 2. In actual application, the computer 1 is provided with super-speed digital micro-mirror spatial light modulator control software for controlling the super-speed digital micro-mirror spatial light modulator; the super-speed digital micro-mirror spatial light modulator 4 is located before the Fourier lens 3, and the Fourier lens 3 is located before the photoelectric detector 2. The laser is not shown in the figure.

[0053] The computer 1 is connected with the super-speed digital micro-mirror spatial light modulator 4; the computer 1 is used for acquiring two identical gray-scale images and one all-black image and processing the two gray-scale images to obtain a spectrum image, and uploading the spectrum image and the all-black image to the super-speed digital micro-mirror spatial light modulator 4.

[0054] The laser is used for generating a collimated laser beam 5; the laser is one of a solid-state laser, a fiber laser, a gas laser and a semiconductor laser, which is obtained through a beam expansion collimation system and is used for reading out the spectrum image from the super-speed digital micro-mirror spatial light modulator 4.

[0055] The super-speed digital micro-mirror spatial light modulator 4 is arranged on the light path of the laser; the collimated laser beam 5 is incident on the super-speed digital micro-mirror spatial light modulator 4 and reflects the spectrum image or the all-black image.

[0056] The Fourier lens 3 is arranged on the reflection light path of the super high speed digital micromirror spatial light modulator 4; the Fourier lens 3 is used for inverse Fourier transform of the frequency spectrum image or the all-black image, and outputs the transformed frequency spectrum image or the transformed all-black image. The Fourier lens 3 is a cemented lens or a diffractive optical element, and is used for inverse Fourier transform of spatial light information.

[0057] The photoelectric detector 2 is arranged on the exit light path of the Fourier lens 3 and is connected with the computer 1; the photoelectric detector 2 is used for detecting the correlation peak position of the transformed frequency spectrum image or the correlation peak position of the transformed all-black image, and uploading the detection result to the computer 1; the detection result is the frequency of the correlation peak position having a bright spot. In actual application, the photoelectric detector 2 is a photosensitive diode or a photosensitive triode, converts the light signal into an electric signal, and is used for detecting whether the bright spot of the correlation peak position exists; if the bright spot exists, the super high speed digital micromirror spatial light modulator 4 modulates the binary frequency spectrum image; if the bright spot does not exist, the super high speed digital micromirror spatial light modulator 4 modulates the all-black image.

[0058] The computer 1 is further used for determining whether the super high speed digital micromirror spatial light modulator 4 is qualified according to the detection result.

[0059] In practical application, when testing the super high speed digital micro-mirror spatial light modulator 4, two same images (gray scale images) are taken, the joint power spectrum of the two images is directly calculated by the computer 1, the useful power spectrum is extracted according to the joint power spectrum, the useful power spectrum is binarized, the binarized frequency spectrum image and a full black image are uploaded to the super high speed digital micro-mirror spatial light modulator 4 through the super high speed digital micro-mirror spatial light modulator control software, the super high speed digital micro-mirror spatial light modulator 4 is set to cyclically display the uploaded two images at a certain frequency through the super high speed digital micro-mirror spatial light modulator control software, and finally a collimated laser beam 5 is used for reading. The super high speed digital micro-mirror spatial light modulator 4 modulates light in a reflective manner and does not emit light itself, so it needs to be matched with an illumination light source. Therefore, the two images are modulated or read out in time division. After inverse Fourier transform by the Fourier lens 3, the correlation peak position on the output plane is detected by the photodetector 2. If the photodetector 2 detects that the frequency of the bright spot is half of the set frequency of the super high speed digital micro-mirror spatial light modulator 4, the super high speed digital micro-mirror spatial light modulator 4 passes the technical index test. The frequency spectrum image and the full black image have been loaded into the super high speed digital micro-mirror spatial light modulator 4. When the super high speed digital micro-mirror control spatial light modulator 4 displays the binarized frequency spectrum image, the super high speed digital micro-mirror spatial light modulator 4 modulates the frequency spectrum image. When the super high speed digital micro-mirror control spatial light modulator 4 displays the full black image, the super high speed digital micro-mirror spatial light modulator 4 modulates the full black image. If the super high speed digital micro-mirror spatial light modulator 4 currently modulates the binarized frequency spectrum image, the correlation peak position will have a bright spot. If the super high speed digital micro-mirror spatial light modulator 4 currently modulates the full black image, the correlation peak position will be a dark spot. The super high speed digital micro-mirror spatial light modulator 4 cyclically modulates the two loaded images at a set frequency, so that the correlation peak position will periodically appear bright spots at half of the set frequency.

[0060] Embodiment two

[0061] In order to realize the corresponding functions and technical effects of embodiment one, a super high speed digital micro-mirror spatial light modulator detection method is provided below, which is applied to the super high speed digital micro-mirror spatial light modulator detection device of embodiment one. The super high speed digital micro-mirror spatial light modulator detection method comprises:

[0062] Step 301: acquiring two same gray scale images and a full black image, and processing the two gray scale images to obtain a frequency spectrum image.

[0063] As an optional implementation, the two gray scale images are processed to obtain a frequency spectrum image, specifically comprising:

[0064] The two gray images are combined into one gray image to obtain a combined gray image. In practical applications, the two gray images are combined into one gray image in left-right direction.

[0065] The joint power spectrum of the combined gray image is calculated.

[0066] The useful power spectrum is extracted from the joint power spectrum.

[0067] The useful power spectrum is binarized to obtain a spectrum image.

[0068] As an optional embodiment, the joint power spectrum of the two gray images is calculated, specifically including:

[0069] Step S1: taking two same images f(x+a, y) and g(x-a, y), and combining them into one image s(x, y) = f(x+a, y) + g(x-a, y) by using a computer.

[0070] Step S2: calculating the joint power spectrum by using a computer, and the calculation formula is as follows:

[0071]

[0072] The useful power spectrum is extracted from the joint power spectrum by using a computer.

[0073]

[0074] The useful power spectrum is binarized, and the binarized image (spectrum image) and a full 0 image (full black image) are loaded into a super high-speed digital micromirror spatial light modulator by using super high-speed spatial light modulator control software. The spatial light modulator is set to display the uploaded two images in frequency cycle.

[0075] Step 302: uploading the spectrum image and the full black image to a super high-speed digital micromirror spatial light modulator; the high-speed digital micromirror spatial light modulator outputs the spectrum image and the full black image in set frequency cycle.

[0076] Step 303: collimating a laser beam to be incident on the high-speed digital micromirror spatial light modulator and reflecting the spectrum image or the full black image.

[0077] Step 304: performing inverse Fourier transform on the spectrum image or the full black image by using a Fourier lens to obtain a transformed spectrum image or a transformed full black image.

[0078] Step 305: the photoelectric detector detects the correlation peak position of the transformed spectrum image or the correlation peak position of the transformed all-black image, and obtains a detection result; the detection result is the frequency of the correlation peak position having a bright spot.

[0079] Step 306: when the frequency of the correlation peak position having a bright spot is equal to half of the set frequency, it is determined that the super-high-speed digital micro-mirror spatial light modulator is qualified.

[0080] Step S3: the photoelectric detector detects at the correlation peak position, and the frequency of the photoelectric detector detecting light is half of the set frequency of the super-high-speed digital micro-mirror spatial light modulator; therefore, the technical index of the super-high-speed digital micro-mirror spatial light modulator meets the requirements, and the product test is qualified.

[0081] For the test or detection of the spatial light modulator, the traditional method is to complete the test through an industrial camera, but for the test of the super-high-speed digital micro-mirror spatial light modulator, a super-high-speed camera must be used to complete the test, and the test method is limited due to the high price of the high-resolution super-high-speed camera. The present application directly calculates the spectrum image through a computer, modulates the spectrum light field on the Fourier plane through the spatial light modulator, then performs inverse Fourier transform through an optical Fourier lens, and finally detects the presence or absence of the bright spot of the correlation peak position through the photoelectric detector. Compared with the prior art, the present application has low cost and is easy to use.

[0082] The embodiments in the specification are described in a progressive manner, and each embodiment focuses on the difference from other embodiments. The same or similar parts of each embodiment can be referred to each other.

[0083] The principles and implementation manners of the present application are described by using specific examples, and the above description of the embodiments is only used to help understand the method of the present application and its core idea; meanwhile, for those skilled in the art, the specific implementation manners and application ranges can be changed according to the idea of the present application. In conclusion, the content of the specification should not be understood as a limitation of the present application.

Claims

1. A device for detecting an ultra-high-speed digital micromirror spatial light modulator, characterized in that, The application relates to a high-speed digital micro-mirror spatial light modulator detection method and device. The application comprises a laser, a computer, a high-speed digital micro-mirror spatial light modulator, a Fourier lens and a photoelectric detector. The computer is connected with the high-speed digital micro-mirror spatial light modulator; the computer is used for acquiring two same gray images and one all-black image, processing the two gray images to obtain a spectrum image, and uploading the spectrum image and the all-black image to the high-speed digital micro-mirror spatial light modulator. The laser is used for generating a collimated laser beam. The high-speed digital micro-mirror spatial light modulator is arranged on an outgoing light path of the laser; the collimated laser beam is incident on the high-speed digital micro-mirror spatial light modulator and reflects the spectrum image or the all-black image. The Fourier lens is arranged on a reflected light path of the high-speed digital micro-mirror spatial light modulator; the Fourier lens is used for performing inverse Fourier transform on the spectrum image or the all-black image and outputting a transformed spectrum image or a transformed all-black image. The photoelectric detector is arranged on an outgoing light path of the Fourier lens and is connected with the computer; the photoelectric detector is used for detecting a correlation peak position of the transformed spectrum image or a correlation peak position of the transformed all-black image and uploading a detection result to the computer; the detection result is a frequency of the correlation peak position having a bright spot. The computer is further used for determining whether the high-speed digital micro-mirror spatial light modulator is qualified according to the detection result.

2. The detection device of the ultra-high-speed digital micromirror spatial light modulator according to claim 1, wherein, The laser is a solid laser, a fiber laser, a gas laser or a semiconductor laser.

3. The detection device of the ultra-high speed digital micromirror spatial light modulator according to claim 1, wherein, The photoelectric detector is a photosensitive diode or a photosensitive triode.

4. The detection device of the ultra-high speed digital micromirror spatial light modulator according to claim 1, wherein, The Fourier lens is a cemented lens or a diffractive optical element.

5. A method for detecting a spatial light modulator with ultra-high speed digital micromirror, characterized in that, The high-speed digital micro-mirror spatial light modulator detection method is applied to the high-speed digital micro-mirror spatial light modulator detection device, and the high-speed digital micro-mirror spatial light modulator detection method comprises the following steps: acquiring two same gray images and one all-black image, processing the two gray images to obtain a spectrum image; uploading the spectrum image and the all-black image to the high-speed digital micro-mirror spatial light modulator; the high-speed digital micro-mirror spatial light modulator cyclically outputs the spectrum image and the all-black image at a set frequency; a collimated laser beam is incident on the high-speed digital micro-mirror spatial light modulator and reflects the spectrum image or the all-black image; a Fourier lens performs inverse Fourier transform on the spectrum image or the all-black image to obtain a transformed spectrum image or a transformed all-black image; a photoelectric detector detects a correlation peak position of the transformed spectrum image or a correlation peak position of the transformed all-black image to obtain a detection result; the detection result is a frequency of the correlation peak position having a bright spot; when the frequency of the correlation peak position having the bright spot is equal to half of the set frequency, it is determined that the high-speed digital micro-mirror spatial light modulator is qualified.

6. The method of claim 5, wherein the method further comprises: processing the two gray images to obtain a spectrum image, specifically comprising the following steps: combining the two gray images to obtain a combined gray image; calculating a joint power spectrum of the merged gray-scale images; extracting a useful power spectrum from the joint power spectrum; performing a binaryzation process on the useful power spectrum to obtain a frequency spectrum image.

7. The method of claim 5, wherein the method further comprises: calculating a joint power spectrum of the two gray-scale images, specifically including: calculating a joint power spectrum of the two said gray scale images using the formula F(u, v) * G(u, v) = a2* f2+ (u - a)2+ (v - a)2 where F(u, v) represents the Fourier transform of one gray scale image; G(u, v) represents the Fourier transform of another gray scale image; u represents the spatial frequency in the x direction; v represents the spatial frequency in the y direction; a represents half of the center distance in the x direction of the two gray scale images; and f represents the spatial frequency. G*(u,v) represents a complex conjugate function of G(u,v); F*(u,v) represents a complex conjugate function of F(u,v).

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