CN112782124B - High-resolution continuous terahertz wave lamination imaging method - Google Patents

Abstract

The invention discloses a high-resolution continuous terahertz stack imaging method. The method includes irradiating objects with divergent spherical waves, effectively utilizing all beam energy, and amplifying the object after being collected by an area array detector according to the geometric relationship. Diffraction intensity information, so as to increase sampling, as the recording surface constraints, based on the ePIE algorithm process, the complex amplitude transmittance function and the probe function of the object surface object reconstructed by the ePIE algorithm are used as the initial guess to obtain the phase of the diffraction field information, so enough information can be obtained to uniquely define the outgoing light field of the object. Part of the high diffraction order information not recorded in the periphery of the area array detector is extrapolated by iterative numerical calculation, and the numerical aperture of the system is expanded. This method effectively improves the lateral resolution and imaging quality of the reconstructed object without increasing the complexity of the optical path, and reduces the requirements of the experimental resolution on the target size and pixel size of the area array detector.

Description

A high-resolution continuous terahertz wave stack imaging method

technical field

The present invention relates to a high-resolution continuous terahertz wave stack scanning full-field quantitative phase contrast imaging method, in particular to a high-resolution continuous terahertz wave stack based on a divergent spherical wave probe plus an extrapolation algorithm Scanning full-field quantitative phase-contrast imaging method to improve lateral resolution of imaging.

Background technique

Terahertz wave is an electromagnetic wave with a frequency between 0.1-10THz, corresponding to a wavelength of 0.03 to 3mm. It has many important characteristics such as wide spectrum, high penetration, low energy, and fear of water. It is used in medical imaging, non-destructive testing, and security checks. Many fields such as terahertz, broadband communication, and biosensing have shown their great scientific significance and application value. As one of the main applications of terahertz band, terahertz imaging technology has played an increasingly important role. Terahertz stack imaging technology is a lensless coherent diffraction imaging technology that restores the complex amplitude distribution of the sample by overlapping and collecting redundant diffraction pattern information. On the basis of changing the positions of the illumination light and the sample, the complex amplitude transmittance function and the illumination The complex amplitude distribution of the beam. The phase distribution of the reconstructed object by the stack imaging method is not affected by other factors except the characteristics of the object itself, and the imaging optical path is simple. The resolution of the system is only related to the numerical aperture of the system and the recording distance. The numerical aperture of the system can obtain more diffraction information. At present, the detectors in the terahertz band are limited by the manufacturing process, and it is impossible to expand the target size of the detector and reduce the pixel size of the detector. The synthetic aperture is used as a mechanical movement of the area array detector to collect more high-resolution images. The method of using high-frequency information will increase the data volume and recording time of the terahertz stack imaging method, and the long recording time puts forward higher requirements for the stability of the equipment and the environment. To this end, we propose a high-resolution terahertz stack imaging method. Using divergent spherical waves as a stack probe can effectively utilize all the beam energy, and obtain amplified diffraction information according to the geometric relationship, thereby increasing sampling. Improve the imaging lateral resolution of the system; use the low-resolution object complex amplitude transmittance function and probe function reconstructed by the ePIE algorithm on the object plane as the initial guess, and the low-diffraction order intensity information collected by the area array detector as the record Surface constraint conditions, extrapolation of part of the high diffraction order information that has not been recorded by iterative numerical calculation. This method effectively improves the lateral resolution and imaging quality of the reconstructed object without increasing the complexity of the optical path, and reduces the requirements of the experimental resolution on the target size and pixel size of the area array detector.

Contents of the invention

The purpose of the present invention is to irradiate the object with divergent spherical waves, and the diffraction intensity information of the object is collected by the area array detector, which is used as the constraint condition of the recording surface. diffraction intensity information. It can be seen from the geometric relationship that the diffraction information of the object is amplified on the recording surface, which can actually be regarded as the physical pixel size of the detector becomes smaller, and the extrapolation of more diffraction intensity information effectively increases the numerical aperture of the system. More diffraction intensity information reconstructs the complex amplitude transmittance function of the object, thereby obtaining a reconstructed image with higher resolution. On the basis of not changing the complexity of the optical path, it reduces the requirements for experimental equipment and experimental environment, and effectively improves imaging. Quality and fidelity, the reconstruction obtains a high-resolution complex amplitude transmittance function of the object.

In order to achieve the above purpose, the technical solution adopted by the present invention is a high-resolution continuous terahertz stack imaging method. The optical path of the imaging system for realizing this method includes a carbon dioxide pumped continuous terahertz laser, two gold-plated off-axis parabolic mirrors, Terahertz lens, measured object, two-dimensional electric translation stage, area array pyroelectric detector. The carbon dioxide pumped continuous terahertz laser is used as the radiation source; the off-axis parabolic mirror expands and collimates the continuous terahertz wave beam radiated by the laser into parallel light; the terahertz lens converges the parallel light; the measured object is fixed in a two-dimensional electric translation On the stage, and placed behind the back focal plane of the terahertz lens, the x-axis and y-axis of the two-dimensional electric translation stage are used to move the measured object horizontally and vertically; the area array pyroelectric detector is placed behind the measured object, Used to collect diffraction intensity information. After the terahertz wave is generated by the laser, it is expanded and collimated by two off-axis parabolic mirrors. The terahertz lens converges the parallel light. The measured object is placed behind the back focal plane of the terahertz lens. The detector records the diffraction information of the object. By moving the two-dimensional electric translation stage, the area array pyroelectric detector collects the object diffraction pattern I _j (u) with overlapping areas on the object plane, where j=1,2,3...J, J is the scanning position total.

The method includes irradiating the object with divergent spherical waves, effectively utilizing all the beam energy, and collecting the amplified diffraction intensity information of the object according to the geometric relationship of the surface array detector, thereby increasing the sampling, which is used as the recording surface constraint condition, and the ePIE algorithm Based on the process, the complex amplitude transmittance function and the probe function of the object reconstructed by the ePIE algorithm are used as the initial guess to obtain the phase information of the diffraction field, so sufficient information can be obtained to uniquely define the outgoing light field of the object. The part of the high diffraction order information not recorded in the periphery of the area array detector is extrapolated by iterative numerical calculation, and the numerical aperture of the system is expanded. This method effectively improves the lateral resolution and imaging quality of the reconstructed object without increasing the complexity of the optical path, and reduces the requirements of the experimental resolution on the target size and pixel size of the area array detector.

A high-resolution continuous terahertz stack imaging method, the process of improving the lateral resolution of the reconstructed object and improving the imaging quality is divided into five steps:

The parallel light after S1 beam expansion and collimation is converged by the terahertz lens, and the distance between the terahertz lens with focal length f and the measured object is adjusted to d ₀ , so that the measured object is located at d ₀ -f behind the back focal plane of the terahertz lens, Assuming that the coordinates of the object plane are r=(x, y), and the coordinates of the recording surface are u=(ξ, η), the complex amplitude distribution on the surface of the measured object and the function of the probe (illumination beam) are expressed as O(r) and P(r). The distance between the measured object and the detector is d, remove the measured object, and use the detector to record the intensity map I ₀ (u) of the divergent spherical wave probe, which is used as the initial probe guess of the ePIE algorithm using the divergent spherical wave as the probe.

S2 adds the measured object into the optical path, and sequentially collects the diffraction intensity I _j (u) of the object according to the scanning path of the electric translation stage, and the expression is as follows:

I _j (u)＝|G _d {O(r)P(rR _j )}| ² , (1)

Where R _j =(x _j ,y _j ) represents the jth translation vector of the sample, j=1,2,3...J, J is the total number of diffraction patterns, G{} is the diffraction propagation operator, we use the angle spectrum Diffraction propagation.

S3 uses the ePIE algorithm to reconstruct the low-resolution complex amplitude transmittance function and probe function of the object plane.

S4 uses the extrapolation algorithm to expand the size of the diffraction pattern, and reconstructs the complex amplitude transmittance function of the object plane with high resolution. The specific process of the extrapolation algorithm is divided into the following seven steps:

S4.1 Taking a reconstruction of the jth position as an example, take the jth diffraction intensity map I _j (u), probe size, translation vector, and diffraction distance as known quantities, and the matrix size of the diffraction intensity map is M ₀ ×N ₀ , using the ePIE algorithm to reconstruct the complex amplitude transmittance function of the object plane as the initial guess object function O _jX , where the matrix size of the object function O _jX is K×L, and X=(k,l) represents the pixel of the object Position, fill the periphery of the probe function of the ePIE algorithm to reconstruct the object surface with random numbers to the size of M×N, as the initial guess probe P _jr , where K>M, L>N, indicating that the local illumination light scans the sample area, M> M ₀ , N>N ₀ .

S4.2 Intercept the M ₀ ×N ₀ area centered on the position R _j = (R _xj , R _yj ) in O _jX , and fill the periphery of the M ₀ ×N ₀ area with random numbers to the size of M×N as a partial function obj _jr , then the outgoing wave of the sample at the jth position is:

ψ _jr ＝obj _jr ×P _jr . (2)

S4.3 The outgoing wave of the sample is multiplied by a Hanning window to eliminate the aperture effect, and then use the angular spectrum diffraction to propagate the distance d to the recording surface to obtain the complex amplitude distribution of the diffraction field on the recording surface:

Φ _jr ＝G _d {Hanning(ψ _jr )}. (3)

S4.4 Use the root mean square of the diffraction intensity I _j (u) collected by the detector to replace the amplitude in the M ₀ × N ₀ area at the central position of Φ _jr in the formula (3), and the amplitude and all phases outside the M ₀ × N ₀ area Retaining, the new complex amplitude distribution is obtained:

Wherein, S ₀ is the M ₀ ×N ₀ area.

S4.5 Multiply the complex amplitude distribution of the diffraction field on the new recording surface by the Hanning window, and then diffract back the distance d to the object surface to obtain the updated object exit light field ψ' _jr .

S4.6 Update the initial guessed partial object function and probe function through the update function, the update function is as follows:

Among them, α and β are weight coefficients, and the value is generally between 0.9-1; ε is an adjustment coefficient, which is used to adjust the denominator to be different from 0, and the value is 0.01.

S4.7 Intercept the center M ₀ ×N ₀ area of the updated part of the object function matrix obj' _jr , put it back to the intercepted position in S4.2, and obtain the updated object function distribution O _j ' _X .

S5 adopts the updated physical function as the initial physical function guess, and the updated probe function as the initial probe function guess, and performs the processing of S4.2-S4.7 on the diffraction pattern of the next sample position in sequence until the whole is updated. surface once. The entire object surface is iterated n times, and after the algorithm converges, the high-resolution object transmittance function and probe function distribution of the final reconstructed object surface can be obtained.

The experimental results of a typical embodiment of the present invention show that the divergent spherical wave is used as a laminated probe to irradiate the sample, the area array detector collects the amplified diffraction field intensity information, and the complex amplitude transmittance function of the object reconstructed by the ePIE algorithm and the probe function as the initial guess, the low-diffraction-order intensity information collected by the detector is used as the constraint condition of the recording surface, and the higher-order diffraction information that is not recorded around the detector is extrapolated by iterative numerical calculation. Without increasing the complexity of the optical path, the lateral resolution and imaging quality of the reconstructed object are effectively improved, and the requirements for the experimental resolution on the target size and pixel size of the area array detector are reduced.

Compared with the prior art, the present invention proposes a high-resolution continuous terahertz stack imaging method, using divergent spherical waves as a stack probe to irradiate the sample, and an area array detector to collect the amplified diffraction field intensity information , effectively utilizes all beam energy, and obtains amplified diffraction information according to the geometric relationship, increases the sampling of the diffraction pattern, and improves the imaging resolution of the system; the complex amplitude transmittance function and the probe function of the object reconstructed by the ePIE algorithm As an initial guess, the low-diffraction-order intensity information collected by the detector is used as the constraint condition of the recording surface, and the higher-order diffraction information that is not recorded around the detector is extrapolated by iterative numerical calculation. This method does not increase the complexity of the optical path. Under the premise of stability, the lateral resolution and imaging quality of the reconstructed object are effectively improved, and the requirements of the experimental resolution on the target size and pixel size of the area array detector are reduced.

Description of drawings

Figure 1 is a system optical path of a high-resolution continuous terahertz stack imaging method. In the figure: 1. FIRL295 carbon dioxide pumped continuous terahertz laser, 2. The first off-axis parabolic mirror, 3. The second off-axis parabolic mirror, 4. Terahertz lens, 5. The measured object, 6. Two-dimensional electric motor Translation stage, 7. PY-Ⅳ area array pyroelectric detector.

detailed description

As shown in Figure 1, a high-resolution continuous terahertz stack imaging method is characterized in that: the optical path of the imaging system for realizing the method includes a FIRL295 carbon dioxide pumped continuous terahertz laser 1, and the first laser beam with a focal length of 50.8mm Gold-plated off-axis parabolic mirror 2, second gold-plated off-axis parabolic mirror 3 with a focal length of 152.4 mm, terahertz TPX lens 4, object to be measured 5, two-dimensional electric translation stage 6, PY-IV area array pyroelectric detector 7. FIRL295 carbon dioxide pumped continuous terahertz laser 1 is used as the radiation source, the working gas is methanol, the output frequency is 2.52THz, the corresponding center wavelength is 118.83μm, and the maximum output power is 500mW; the first off-axis parabolic mirror 2, the second off-axis The system composed of parabolic mirror 3 expands the continuous terahertz wave beam radiated by laser 1 three times, and collimates it into parallel light with a diameter of 21 mm; the focal length of terahertz lens 4 is 100 mm, and converges the parallel light; the measured object 5 Fixed on the two-dimensional electric translation stage 6, and placed at a position 16mm behind the focal plane of the terahertz lens 4, the maximum range of the two-dimensional electric translation stage 6 is 25mm, and the x-axis and y-axis are used to move the sample to be tested horizontally and vertically, In the experiment, the step length is set to 0.8mm, and the scanning path is in a serpentine order, ensuring that the overlapping area of adjacent diffraction patterns is about 75%; the PY-IV area array pyroelectric detector 7 is placed behind the measured object, Collect the overlapping diffraction pattern of the sample, the number of pixels is 320×320, and the pixel size is 80×80 μm.

The method includes irradiating the object with divergent spherical waves, effectively utilizing all the beam energy, and collecting the amplified diffraction intensity information of the object according to the geometric relationship of the surface array detector, thereby increasing the sampling, which is used as the recording surface constraint condition, and the ePIE algorithm Based on the process, the complex amplitude transmittance function and the probe function of the object reconstructed by the ePIE algorithm are used as the initial guess to obtain the phase information of the diffraction field, so sufficient information can be obtained to uniquely define the outgoing light field of the object. The part of the high diffraction order information not recorded in the periphery of the area array detector is extrapolated by iterative numerical calculation, and the numerical aperture of the system is expanded. This method effectively improves the lateral resolution and imaging quality of the reconstructed object without increasing the complexity of the optical path, and reduces the requirements of the experimental resolution on the target size and pixel size of the area array detector.

A high-resolution continuous terahertz stack imaging method, the process of improving the lateral resolution of the reconstructed object and improving the imaging quality is divided into five steps:

The parallel light after S1 beam expansion and collimation is converged by the terahertz lens, and the distance between the terahertz lens with focal length f and the measured object is adjusted to d ₀ , so that the measured object is located at d ₀ -f behind the back focal plane of the terahertz lens, Assuming that the coordinates of the object plane are r=(x, y), and the coordinates of the recording surface are u=(ξ, η), the complex amplitude distribution on the surface of the measured object and the function of the probe (illumination beam) are expressed as O(r) and P(r). The distance between the measured object and the detector is d, remove the measured object, and use the detector to record the intensity map I ₀ (u) of the divergent spherical wave probe as the initial probe guess of the ePIE algorithm using the divergent spherical wave as the probe.

S2 adds the measured object into the optical path, and sequentially collects the diffraction intensity I _j (u) of the object according to the scanning path of the electric translation stage, and the expression is as follows:

I _j (u)＝|G _d {O(r)P(rR _j )}| ² , (1)

Where R _j =(x _j ,y _j ) represents the jth translation vector of the sample, j=1,2,3...J, J is the total number of diffraction patterns, G{} is the diffraction propagation operator, we use the angle spectrum Diffraction propagation.

S3 uses the ePIE algorithm to reconstruct the low-resolution complex amplitude transmittance function and probe function of the object plane.

S4 uses the extrapolation algorithm to expand the size of the diffraction pattern, and reconstructs the complex amplitude transmittance function of the object plane with high resolution. The specific process of the extrapolation algorithm is divided into the following seven steps:

S4.1 Taking a reconstruction of the jth position as an example, take the jth diffraction intensity map I _j (u), probe size, translation vector, and diffraction distance as known quantities, and the matrix size of the diffraction intensity map is M ₀ ×N ₀ , using the ePIE algorithm to reconstruct the complex amplitude transmittance function of the object plane as the initial guess object function O _jX , where the matrix size of the object function O _jX is K×L, and X=(k,l) represents the pixel of the object Position, fill the periphery of the probe function of the ePIE algorithm to reconstruct the object surface with random numbers to the size of M×N, as the initial guess probe P _jr , where K>M, L>N, indicating that the local illumination light scans the sample area, M> M ₀ , N>N ₀ .

S4.2 Intercept the M ₀ ×N ₀ area centered on the position R _j = (R _xj , R _yj ) in O _jX , and fill the periphery of the M ₀ ×N ₀ area with random numbers to the size of M×N as a partial function obj _jr , then the outgoing wave of the sample at the jth position is:

ψ _jr ＝obj _jr ×P _jr . (2)

S4.3 The outgoing wave of the sample is multiplied by a Hanning window to eliminate the aperture effect, and then use the angular spectrum diffraction to propagate the distance d to the recording surface to obtain the complex amplitude distribution of the diffraction field on the recording surface:

Φ _jr ＝G _d {Hanning(ψ _jr )}. (3)

S4.4 Use the root mean square of the diffraction intensity I _j (u) collected by the detector to replace the amplitude in the M ₀ × N ₀ area at the central position of Φ _jr in the formula (3), and the amplitude and all phases outside the M ₀ × N ₀ area Retaining, the new complex amplitude distribution is obtained:

Wherein, S ₀ is the M ₀ ×N ₀ area.

S4.5 Multiply the complex amplitude distribution of the diffraction field on the new recording surface by the Hanning window, and then diffract back the distance d to the object surface to obtain the updated object exit light field ψ' _jr .

S4.6 Update the initial guessed partial object function and probe function through the update function, the update function is as follows:

Among them, α and β are weight coefficients, and the value is generally between 0.9-1; ε is an adjustment coefficient, which is used to adjust the denominator to be different from 0, and the value is 0.01.

S4.7 Intercept the center M ₀ ×N ₀ area of the updated part of the object function matrix obj' _jr , put it back to the intercepted position in S4.2, and obtain the updated object function distribution O′ _jX .

S5 adopts the updated physical function as the initial physical function guess, and the updated probe function as the initial probe function guess, and performs the processing of S4.2-S4.7 on the diffraction pattern of the next sample position in sequence until the whole is updated. surface once. The entire object surface is iterated n times, and after the algorithm converges, the high-resolution object transmittance function and probe function distribution of the final reconstructed object surface can be obtained.

The experimental results of a typical embodiment of the present invention show that the divergent spherical wave is used as a laminated probe to irradiate the sample, the area array detector collects the amplified diffraction field intensity information, and the complex amplitude transmittance function of the object reconstructed by the ePIE algorithm and the probe function as the initial guess, the low-diffraction-order intensity information collected by the detector is used as the constraint condition of the recording surface, and the higher-order diffraction information that is not recorded around the detector is extrapolated by iterative numerical calculation. Without increasing the complexity of the optical path, the lateral resolution and imaging quality of the reconstructed object are effectively improved, and the requirements for the experimental resolution on the target size and pixel size of the area array detector are reduced.

Claims (1)

1. A high-resolution continuous terahertz stack imaging system, characterized in that it includes a carbon dioxide pumped continuous terahertz laser, two gold-plated off-axis parabolic mirrors, a terahertz lens, an object to be measured, and a two-dimensional electric translation stage And area array pyroelectric detector; carbon dioxide pumped continuous terahertz laser as radiation source; off-axis parabolic mirror expands and collimates the continuous terahertz wave beam radiated by the laser into parallel light; terahertz lens converges parallel light The object to be measured is fixed on the two-dimensional electric translation stage and placed behind the back focal plane of the terahertz lens. The x-axis and y-axis of the two-dimensional electric translation stage are used to move the object under test horizontally and vertically; The electrical detector is placed behind the measured object to collect diffraction intensity information; after the terahertz wave is generated by the laser, it is expanded and collimated by two off-axis parabolic mirrors, and the terahertz lens converges the parallel light, and the measured object Placed behind the back focal plane of the terahertz lens, the area-array pyroelectric detector records the diffraction information behind the object; by moving the two-dimensional electric translation stage, the area-array pyroelectric detector collects the overlapping area of the object plane. Object diffraction pattern I _j (u), where j=1,2,3...J, where J is the total number of scanning positions; The imaging method of the continuous terahertz stack imaging system includes the following steps. The parallel light beam expanded and collimated by S1 is converged by the terahertz lens, and the distance between the terahertz lens with the focal length f and the measured object is adjusted to be d ₀ , so that the measured object The object is located at d ₀ -f behind the back focal plane of the terahertz lens, assuming that the coordinates of the object plane are r=(x,y), and the coordinates of the recording surface are u=(ξ,η), the complex amplitude distribution and The probe functions are denoted as O(r) and P(r) respectively; the distance between the measured object and the detector is d, the measured object is removed, and the intensity map I ₀ (u) of the divergent spherical wave probe is recorded by the detector as Initial probe guesses for the ePIE algorithm using divergent spherical waves as probes; S2 adds the measured object into the optical path, and sequentially collects the diffraction intensity I _j (u) of the object according to the scanning path of the electric translation stage, and the expression is as follows: I _j (u)＝|G _d {O(r)P(rR _j )}| ² , (1) Where R _j =(x _j ,y _j ) represents the jth translation vector of the sample, j=1,2,3...J, J is the total number of diffraction patterns, G{} is the diffraction propagation operator, we use the angle spectrum Diffraction propagation; S3 uses the ePIE algorithm to reconstruct the low-resolution complex amplitude transmittance function and probe function of the object plane; S4 uses the extrapolation algorithm to expand the size of the diffraction pattern, and reconstructs the complex amplitude transmittance function of the object with high resolution on the object plane; S5 uses the updated object function as the initial object function guess, and the updated probe function as the initial probe function guess, and reconstructs the diffraction pattern of the next sample position in sequence until the entire object surface is updated once; for the entire object Surface iteration n times, after convergence, the high-resolution object transmittance function and probe function distribution of the final reconstructed object surface are obtained; The specific process of the extrapolation algorithm is divided into the following seven steps: S4.1 In the reconstruction of the jth position, the jth diffraction intensity map I _j (u), probe size, translation vector, and diffraction distance are taken as known quantities, and the matrix size of the diffraction intensity map is M ₀ ×N ₀ , using the ePIE algorithm to reconstruct the complex amplitude transmittance function of the object plane as the initial guess object function O _jX , where the matrix size of the object function O _jX is K×L, X=(k,l) represents the pixel position of the object, and The ePIE algorithm reconstructs the probe function of the object surface and fills the periphery with random numbers to the size of M×N, as the initial guess probe P _jr , where K>M, L>N, which means that the local illumination light scans the sample area, M>M ₀ , N>N ₀ ; S4.2 Intercept the M ₀ ×N ₀ area centered on the position R _j = (R _xj , R _yj ) in O _jX , and fill the periphery of the M ₀ ×N ₀ area with random numbers to the size of M×N as a partial function obj _jr , then the outgoing wave of the sample at the jth position is: ψ _jr ＝obj _jr ×P _jr . (2) S4.3 The outgoing wave of the sample is multiplied by a Hanning window to eliminate the aperture effect, and then use the angular spectrum diffraction to propagate the distance d to the recording surface to obtain the complex amplitude distribution of the diffraction field on the recording surface: Φ _jr ＝G _d {Hanning(ψ _jr )}. (3) S4.4 Use the root mean square of the diffraction intensity I _j (u) collected by the detector to replace the amplitude in the M ₀ × N ₀ area at the central position of Φ _jr in the formula (3), and the amplitude and all phases outside the M ₀ × N ₀ area Retaining, the new complex amplitude distribution is obtained:

Among them, S ₀ is the area of M ₀ ×N ₀ ; S4.5 Multiply the complex amplitude distribution of the diffraction field on the new recording surface by the Hanning window, and then diffract back the distance d to the object surface to obtain the updated object exit light field _ψ'jr ; S4.6 Update the initial guessed partial object function and probe function through the update function, the update function is as follows:

Among them, α and β are weight coefficients; ε is an adjustment coefficient, which is used to adjust the denominator not equal to 0, and the value is 0.01; S4.7 Intercept the center M ₀ ×N ₀ area of the updated part of the object function matrix obj' _jr , put it back to the intercepted position in S4.2, and obtain the updated object function distribution O' _jX .

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