A resolution enhancement method for spaceborne one-dimensional synthetic aperture microwave radiometer

Abstract

This invention belongs to the field of microwave remote sensing observation technology and relates to a resolution enhancement method for a spaceborne one-dimensional synthetic aperture microwave radiometer. The steps include: calculating the harmonic basis functions of the spaceborne one-dimensional synthetic aperture microwave radiometer corresponding to different baseline intervals; obtaining the number of overlaps in the ground field of view of the one-dimensional synthetic aperture microwave radiometer array; updating the visibility samples collected by the one-dimensional synthetic aperture microwave radiometer array with visibility samples carrying high-precision information along the orbital direction; and solving for the high-precision brightness temperature value using an inversion algorithm. This invention utilizes a BG-inspired resolution enhancement method for spaceborne one-dimensional synthetic aperture microwave radiometers to enhance resolution along the orbital direction without requiring additional high-resolution data. By leveraging the concept of real aperture microwave radiometer resolution enhancement and fully utilizing the information redundancy of its own data along the orbital direction, it significantly improves the resolution of the spaceborne one-dimensional synthetic aperture microwave radiometer along the orbital direction.

Description

Technical Field

[0001] The invention belongs to the technical field of microwave remote sensing observation, relates to a resolution enhancement technology for a satellite-borne one-dimensional synthetic aperture microwave radiometer, and specifically relates to a resolution enhancement method for a satellite-borne one-dimensional synthetic aperture microwave radiometer. Background Art

[0002] The introduction of synthetic aperture technology enables synthetic aperture microwave radiometers to replace real aperture microwave radiometers, thereby achieving high spatial resolution Earth observation in the low-frequency microwave band. This approach solves the technical challenges of deploying large scanning antennas and provides a cost-effective solution for high-resolution observations. In addition, synthetic aperture microwave radiometers use electronically controlled antenna beams to avoid the complexity and limitations of mechanical control.

[0003] Aperture synthesis technology has developed rapidly in the field of earth remote sensing applications. The one-dimensional (1D) aperture synthesis microwave radiometer (SAIR) is the first spaceborne 1D SAIR in the world to use aperture synthesis technology. However, the resolution of spaceborne 1D SAIR in the along-track direction determined by the real aperture is often relatively coarse, especially in the L band. This limitation poses a major challenge to obtaining high-resolution data required for detailed oceanographic studies. Moreover, there is no research on the enhancement of the resolution of spaceborne 1D SAIR in the along-track direction. Therefore, designing a resolution enhancement method suitable for spaceborne 1D SAIR is a technical problem that needs to be solved urgently.

[0004] The information disclosed in this background technology section is only intended to enhance the understanding of the overall background of the invention and should not be regarded as an acknowledgment or any form of suggestion that the information constitutes the prior art already known to a person skilled in the art. Summary of the invention

[0005] The purpose of the present invention is to solve the problem of relatively coarse resolution in the along-track direction of a one-dimensional synthetic aperture microwave radiometer. A resolution enhancement method for a satellite-borne one-dimensional synthetic aperture microwave radiometer is proposed. By using the idea of ​​improving the resolution of a real aperture microwave radiometer and making full use of the information redundancy of its own data in the along-track direction, the resolution of the satellite-borne one-dimensional synthetic aperture microwave radiometer in the along-track direction is significantly improved.

[0006] The technical solution of the present invention is:

[0007] A resolution enhancement method for a spaceborne one-dimensional synthetic aperture microwave radiometer comprises the following steps:

[0008] The harmonic basis functions of different interval baselines are calculated according to the antenna arrangement and antenna pattern matrix of the satellite-borne one-dimensional synthetic aperture microwave radiometer;

[0009] Using the configuration parameters, flight altitude, and assembly angle of the satellite-borne one-dimensional synthetic aperture microwave radiometer, an appropriate cutoff threshold is selected to obtain the overlap number of the satellite-borne one-dimensional synthetic aperture microwave radiometer array in the ground field of view.

[0010] Combining the visibility samples collected by the satellite-borne one-dimensional synthetic aperture microwave radiometer array, formula (1) is used to obtain visibility samples carrying high-precision information in the along-track direction:

[0011] (1)

[0012] in, Represents the number of overlapping ground fields of view, m represents the ordinal number of overlapping fields of view, m≤n, Represents the coefficient corresponding to No. Weight coefficients; Represents the coefficient corresponding to The visibility samples at the location where the field of view overlaps samples; Represents the coefficient corresponding to Visibility samples carrying high-precision information in the along-track direction; represents the direction cosine coordinates, Corresponding to the coordinates The brightness temperature information of the target scene in Zhang Cheng space, Represents the corresponding The abscissa in the cosine coordinates of the basis functions, Represents the corresponding The ordinate in the cosine coordinates of the basis functions, Represents the coefficient corresponding to Harmonic orthogonal basis functions at ;

[0013] Weight coefficient Satisfy the constraints:

[0014] ;

[0015] Weight coefficient By minimizing the overall error get:

[0016] ;

[0017] in, represents the fitting error,

[0018] ;

[0019] represents the variance, ;

[0020] Represents the dimension parameter, represents the noise tuning parameter; represents the normalized harmonic basis function, given by the formula Normalize to ensure satisfy ; represents the target harmonic basis function, satisfying represents the normalized target harmonic basis function, given by the formula Normalize to ensure ; Upper right corner mark represents transpose, Represents the covariance matrix of measurement errors;

[0021] The high-precision visibility samples obtained are combined with the inversion algorithm to reconstruct the brightness temperature and output high-precision brightness temperature values. It is calculated by the following formula (2):

[0022] in, (2)

[0023] Represented by The two-dimensional matrix composed of -1 represents the inverse of the matrix. Represented by The one-dimensional matrix composed of Represents the transpose of a matrix; Represented by A one-dimensional matrix composed of; represents the random noise variance; Represents the dimension parameter, represents the noise tuning parameter; represent Take the cosine, represent Take the sine.

[0024] Furthermore, the harmonic basis functions of the different interval baselines The following formula (3) is used for calculation:

[0025] (3)

[0026] in, Represents the antenna array located at The antenna pattern of the antenna at the position of the signal; Represents direction cosine coordinates, with a superscript represents conjugation, Represents the antenna pair number, Indicates the antenna pair The coefficient of the relationship between the spacing between the antennas and the minimum antenna spacing.

[0027] The harmonic basis function is a complex number. Baselines with different intervals correspond to different harmonic basis functions. The harmonic basis function corresponding to the zero interval is a real number, that is, it only contains the real part and the imaginary part is zero.

[0028] Furthermore, the truncation threshold is used to adjust the overlap of the ground field of view of the satellite-borne one-dimensional synthetic aperture microwave radiometer. The recommended value of the truncation threshold is -11dB to ensure a sufficient number of overlapping fields of view. The truncation threshold can be adjusted according to actual needs. A lower truncation threshold will produce more overlapping fields of view. Using more overlapping fields of view in the inversion process can increase the accuracy of the inversion result, but it will also increase the computational burden and increase the time consumption.

[0029] Furthermore, the configuration parameters include effective integration time and sampling interval.

[0030] Furthermore, the inversion algorithm is a G matrix inversion algorithm, and the least squares solution method is selected to obtain a high-precision brightness temperature value using the following formula (4):

[0031] (4)

[0032] in, Represents the high-precision brightness temperature value obtained by inversion, with the upper right corner marked represents the pseudo-inverse of the matrix, represent The method of the present invention can be combined with any G matrix inversion method. In the implementation process, selecting a better G matrix inversion method can optimize the resolution in the cross-track direction to a certain extent.

[0033] Beneficial effects of the present invention:

[0034] The present invention utilizes the Backus-Gilbert (BG) inspired spaceborne one-dimensional synthetic aperture microwave radiometer resolution enhancement method to enhance the resolution in the along-track direction. It does not require the use of other high-resolution data. By making full use of the redundant information of visibility samples in the overlapping field of view, the visibility samples collected by the original spaceborne one-dimensional synthetic aperture microwave radiometer array are combined to calculate the visibility samples containing higher precision information. Finally, the G matrix inversion method is used to convert the visibility into a high-precision brightness temperature value, which significantly improves the resolution of the spaceborne one-dimensional synthetic aperture microwave radiometer in the along-track direction. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 The figure is a flowchart of a resolution enhancement method for a spaceborne one-dimensional synthetic aperture microwave radiometer;

[0036] Figure 2 A comparison diagram of the inversion results of the synthetic brightness temperature image by the one-dimensional synthetic aperture microwave radiometer resolution enhancement method provided by the present invention and the traditional one-dimensional synthetic aperture microwave radiometer inversion algorithm;

[0037] Figure 2 (a) is a synthetic brightness temperature image; (b) is a brightness temperature image inverted by the traditional one-dimensional synthetic aperture microwave radiometer inversion algorithm; (c) is the resolution enhancement method of the one-dimensional synthetic aperture microwave radiometer inspired by the BG of the present invention. Take the inversion result at 0.001;

[0038] in, Figure 2 (b) and (c) are blurred and out of focus to varying degrees during the reconstruction process relative to the original synthetic test image (a);

[0039] Figure 3 A comparison chart of the inversion results of the one-dimensional synthetic aperture microwave radiometer resolution enhancement method provided by the present invention and the traditional one-dimensional synthetic aperture microwave radiometer inversion algorithm for the observation samples of the Amazon Delta;

[0040] Figure 3 (a) is the brightness temperature image of the Amazon Delta observation sample in the L band of the simulated one-dimensional synthetic aperture microwave radiometer; (b) is the brightness temperature image of the Amazon inverted by the traditional one-dimensional synthetic aperture microwave radiometer inversion algorithm; (c) is the inversion result of the BG-inspired one-dimensional synthetic aperture microwave radiometer resolution enhancement method for the Amazon Delta observation sample; (d) is the difference image between (b) and (a); (e) is the difference image between (c) and (a). DETAILED DESCRIPTION

[0041] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0042] In order to further understand the present invention, the present invention will be further described in conjunction with the accompanying drawings and embodiments.

[0043] The present invention uses a one-dimensional (1D) synthetic aperture microwave radiometer (SAIR). The 1D SAIR adopts a real aperture mode in the along-track direction and a synthetic aperture mode in the cross-track direction. The payload assembly angle is 30°, the swath width is 1200 km, and the sampling density is 6 km.

[0044] The L band is selected as the research sample. The along-track resolution of the data in this band is 100 km, and the target resolution of the inverted along-track direction is set to 6 km.

[0045] like Figure 1 As shown, the present invention provides a resolution enhancement method for a satellite-borne one-dimensional synthetic aperture microwave radiometer, which calculates the harmonic basis functions of different interval baselines according to the antenna arrangement and the antenna pattern matrix of the satellite-borne one-dimensional synthetic aperture microwave radiometer, and calculates the projection of the harmonic basis functions on the ground;

[0046] Harmonic basis functions at different intervals of baseline The calculation formula is:

[0047] ;

[0048] in, Represents the antenna array located at The antenna pattern of the antenna at the position of the signal; Represents direction cosine coordinates, with a superscript represents conjugation, Represents the antenna pair number, Indicates the antenna pair The coefficient of the multiple relationship between the interval between the two antennas and the minimum antenna interval. The harmonic basis function is a complex number. Baselines with different intervals correspond to different harmonic basis functions. The harmonic basis function corresponding to the zero interval is a real number, that is, it only contains the real part and the imaginary part is zero.

[0049] The configuration parameters (effective integration time, sampling interval), flight altitude, and assembly angle of the satellite-borne one-dimensional synthetic aperture microwave radiometer are used to select a suitable truncation threshold to obtain the overlap number of the satellite-borne one-dimensional synthetic aperture microwave radiometer array in the ground field of view. In this embodiment, the truncation threshold selected is -11dB. If the increase in running time is acceptable to the user, more overlap numbers can be obtained by reducing the truncation threshold. The increase in the overlap number will increase the accuracy of the inversion result.

[0050] Combined with the visibility samples collected by the onboard one-dimensional synthetic aperture microwave radiometer array, the visibility samples are updated to visibility samples carrying higher precision information in the along-track direction;

[0051] The visibility samples that carry higher precision information in the along-track direction use the following formula:

[0052] in, Represents the number of overlapping ground fields of view, m represents the ordinal number of overlapping fields of view, m≤n, Represents the coefficient corresponding to No. Weight coefficients; Represents the coefficient corresponding to The visibility samples at the location where the field of view overlaps samples; Represents the coefficient corresponding to Visibility samples carrying high-precision information in the along-track direction; represents the direction cosine coordinates, Corresponding to the coordinates The brightness temperature information of the target scene in Zhang Cheng space, Represents the corresponding The abscissa in the cosine coordinates of the basis functions, Represents the corresponding The ordinate in the cosine coordinates of the basis functions, Represents the coefficient corresponding to Harmonic orthogonal basis functions at

[0053] Weight coefficient Satisfy the constraints:

[0054] ;

[0055] Weight coefficient By minimizing the overall error The overall error is obtained:

[0056]

[0057] in, represents the overall error, represents the fitting error, represents the variance, Represents the dimension parameter, The recommended value of is 0.05, and represents the noise tuning parameter, The adjustment range is represents the normalized harmonic basis function, given by the formula Normalize to ensure satisfy represents the target harmonic basis function, satisfying represents the normalized target harmonic basis function, given by the formula Normalize to ensure ; The upper right corner subscript T represents the transpose, and E represents the covariance matrix of the measurement error.

[0058] Weight coefficient Determined by the following formula:

[0059] in,

[0060] Represented by The two-dimensional matrix composed of -1 represents the inverse of the matrix. Represented by The one-dimensional matrix composed of Represents the transpose of a matrix; Represented by A one-dimensional matrix composed of; represents the random noise variance; Represents the dimension parameter, represents the noise tuning parameter; represent Take the cosine, represent Take the sine. Use the inversion algorithm to solve the high-precision brightness temperature value. The inversion algorithm used is the traditional G matrix inversion algorithm, and the least squares solution method is selected, using the formula: .

[0061] in, Represents the high-precision brightness temperature value obtained by inversion, with the upper right corner marked represents the pseudo-inverse of the matrix, represent The method of the present invention can be combined with any G matrix inversion method. In the implementation process, selecting a better G matrix inversion method can optimize the resolution in the cross-track direction to a certain extent.

[0062] The advantages of the present invention are further illustrated below in conjunction with two simulation embodiments.

[0063] Example 1

[0064] This embodiment uses the one-dimensional synthetic aperture microwave radiometer resolution enhancement method provided by the present invention and the traditional one-dimensional synthetic aperture microwave radiometer inversion algorithm to perform brightness temperature inversion on the synthetic brightness temperature image. The synthetic image contains the brightness temperature in the ξ direction and the brightness temperature in the ξ direction. Lines of different widths and lower brightness temperature (in blue) extending in the ξ direction are used to test the different methods in the ξ direction and At the same time, two rectangular blocks with different sizes and different brightness temperatures are set, which are higher than the background brightness temperature (dark red and orange respectively). The small rectangular block with higher brightness temperature (dark red) is placed directly below the large rectangular block with slightly lower brightness temperature (orange). The two rectangular blocks are used to simulate targets with different sizes and different brightness temperatures in actual observations (such as land). The distance between the two rectangular blocks is set very close, which is very challenging for remote sensing inversion.

[0065] For the synthetic brightness temperature image, the adjustment parameters selected in this area are .

[0066] like Figure 2 As shown, the original synthetic test image is Figure 2 In (a), the brightness temperature image reconstructed by the traditional 1D SAIR inversion method is Figure 2 In (b), the reconstructed brightness temperature image of the BG-inspired one-dimensional synthetic aperture microwave radiometer resolution enhancement method provided by the present invention is Figure 2 (c). It should be noted that the inversion image Figure 2 In (b) and (c), stripes appear in the brightness temperature background. This is because 1D SAIR is equipped with 10 antennas arranged in the same straight line. Antenna pairs with different spacings form a baseline. The inversion of 1D SAIR is generated with the help of visibility samples collected from different baselines. The visibility samples and brightness temperature are similar to Fourier transforms. A complete Fourier transform basis will not produce the stripes shown in the figure, but due to the limitation of the number of antennas, the samples collected by the visibility function here are not complete. Affected by the edge set in the synthetic image The influence of the lower brightness temperature distributed in the direction, using the incomplete basis here, causes the Directional stripes, see Figure 2 Middle (b).

[0067] Figure 2 The reason for the stripes in the along-track direction in (c) is similar to that in the conventional 1D SAIR inversion method. The stripes produced are consistent. The stripes in the direction are due to the fact that the proposed method enhances When the resolution of the direction is improved, a compromise is made in enhancing the resolution of the main beam, and the oscillation outside the main beam cannot be avoided. Targets with a brightness temperature higher or lower than the background in the directional distribution are affected by The oscillation outside the main beam in the direction of Direction stripes.

[0068] The BG-inspired method of the present invention improves image quality beyond the quality degradation caused by the oscillation outside the main beam. Figure 2 Middle (c), Figure 2In (b), for the original synthetic test image, the lines with a certain width and low brightness temperature distributed laterally (along ξ) are blurred and blurred to a wider range during the reconstruction process, and the corresponding low brightness temperature values ​​are raised to varying degrees. Two rectangular blocks of different sizes that are close to each other and have a higher brightness temperature than the background are blurred to varying degrees, and the degree to which the brightness temperature of the original rectangular blocks is underestimated is very obvious. In particular, for two rectangular blocks of different sizes and different brightness temperatures that are close to each other, the traditional 1D SAIR inversion method reconstructs images that can hardly distinguish that these are two independent parts, and the brightness temperature difference between the two cannot be distinguished. Therefore, the brightness temperature image inverted by the BG-inspired one-dimensional synthetic aperture microwave radiometer resolution enhancement method is more reliable.

[0069] Example 2

[0070] For the inversion of the observed samples in the Amazon Delta, the adjustment parameters selected in this region are .

[0071] like Figure 3 The simulated true brightness temperature image of the Amazon Delta is shown in Figure 3 (a) shows the inversion result compared with the traditional 1D SAIR inversion algorithm (see Figure 3 Compared with (b), the image obtained by the method proposed in this embodiment shows a clearer temperature gradient and contains more detailed information. It can be seen that the Amazon River basin in the image inverted by the traditional 1D SAIR inversion method is basically unrecognizable, and the changes in the Guyana Plateau due to the changes in the spatial range of vegetation are not reflected in the inverted image. However, these two details can be clearly identified in the inverted image of the BG-inspired one-dimensional synthetic aperture microwave radiometer resolution enhancement method (see Figure 3 (c)).

[0072] By comparing the difference image with a simulated ground truth image of the Amazon Delta ( Figure 3 In (d) and (e) , it can be seen that the image differences inverted by the BG-inspired one-dimensional synthetic aperture microwave radiometer resolution enhancement method are significantly smaller.

[0073] It should be emphasized here that the inversion process of the BG-inspired one-dimensional synthetic aperture microwave radiometer resolution enhancement method for the space-borne one-dimensional synthetic aperture microwave radiometer includes a visibility sample update process and a visibility to brightness temperature conversion process. The update of the visibility sample is obtained by solving an optimization problem, while the conversion of visibility to brightness temperature follows the traditional method.

[0074] The updated solution of visibility samples makes full use of the redundant information carried in the original visibility samples. By solving the optimization problem, updated visibility samples with higher resolution along-track information are obtained. The process of converting visibility to brightness temperature follows the traditional G matrix inversion method. Here, the least squares method is selected. The proposed method can be combined with any traditional G matrix method, which is highly flexible. Choosing a G matrix inversion method that is better than the least squares method can obtain higher accuracy in the cross-track direction. Compared with the traditional 1D SAIR inversion algorithm, the BG-inspired one-dimensional synthetic aperture microwave radiometer resolution enhancement method can significantly enhance the resolution in the along-track direction.

[0075] The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art can still modify the technical solutions described in the above embodiments, or replace some of the technical features therein by equivalents. Any modification, equivalent replacement, modification, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A resolution enhancement method for a satellite-borne one-dimensional synthetic aperture microwave radiometer, characterized in that: The following steps are involved: The harmonic basis functions of different interval baselines are calculated according to the antenna arrangement and antenna pattern matrix of the satellite-borne one-dimensional synthetic aperture microwave radiometer; The harmonic basis functions G of the different interval baselines i The following formula (3) is used for calculation: Among them, F p (ξ,η) represents the antenna pattern at position p in the antenna array; (ξ,η) represents the direction cosine coordinates, the superscript * represents conjugation, p, q represent the antenna pair numbers, and i represents the coefficient of the multiple relationship between the spacing between antenna pairs p, q and the minimum antenna spacing; The configuration parameters, flight altitude and assembly angle of the satellite-borne one-dimensional synthetic aperture microwave radiometer are used to select the cutoff threshold and obtain the overlap number of the satellite-borne one-dimensional synthetic aperture microwave radiometer array in the ground field of view. Combining the visibility samples collected by the satellite-borne one-dimensional synthetic aperture microwave radiometer array, formula (1) is used to obtain visibility samples carrying high-precision information in the along-track direction: Where n represents the number of overlapping ground fields of view, m represents the ordinal number of overlapping fields of view, m≤n, a i (m) represents the mth weight coefficient corresponding to coefficient i; v i (m) represents the mth sample corresponding to the visibility sample at coefficient i with field overlap; represents the visibility sample with high-precision information along the track direction corresponding to coefficient i; (ξ,η) represents the direction cosine coordinates, T B (ξ,η) corresponds to the brightness temperature information of the target scene in the space spanned by the coordinates (ξ,η), ξ(m) represents the horizontal coordinate in the cosine coordinate corresponding to the mth basis function, η(m) represents the vertical coordinate in the cosine coordinate corresponding to the mth basis function, G i represents the harmonic orthogonal basis function corresponding to coefficient i; Weight coefficient a i Satisfy the constraints: Weight coefficient a i By minimizing the overall error Q, we obtain: Q=Q0cosγ+ωQ N sinγ; Where Q0 represents the fitting error, Q N represents the variance, Q N =α T Eα; ω represents the dimension parameter, γ represents the noise tuning parameter; G i ' represents the normalized harmonic basis function, which is given by the formula Normalize to ensure G i 'Satisfaction ∫ ξ ∫ η G i 'dξdη=1;F i represents the target harmonic basis function, satisfying F i (ξ,η=0)=∫ η G i (ξ,η)dη;F i ' represents the normalized target harmonic basis function, which is given by the formula Normalize to ensure that ∫ ξ ∫ η F i 'dξdη=1; the upper right corner subscript T represents the transpose, and E represents the covariance matrix of the measurement error; The acquired high-precision visibility samples are used in combination with the inversion algorithm to reconstruct the brightness temperature and output high-precision brightness temperature values.

2. The resolution enhancement method for a satellite-borne one-dimensional synthetic aperture microwave radiometer according to claim 1, characterized in that: Weight coefficient a i Determined by formula (2): among them, u i =∫ ξ ∫ η G i 'dξdη;b i =∫ ξ ∫ η G i 'F i 'dξdη; Z ij =cosγ∫ ξ ∫ η G i 'G j 'dξdη+(ΔT rms ) 2 ωsinγ; Z stands for Z ij The two-dimensional matrix is ​​composed of u. The superscript -1 represents the inverse of the matrix; u represents the matrix composed of u i The one-dimensional matrix composed of β, the superscript T represents the transpose of the matrix; β represents the i The one-dimensional matrix composed of (ΔT rms ) 2 represents the variance of random noise; ω represents the dimensionality parameter, γ represents the noise tuning parameter; cosγ represents the cosine of γ, and sinγ represents the sine of γ.

3. The resolution enhancement method for a satellite-borne one-dimensional synthetic aperture microwave radiometer according to claim 1, characterized in that: The cutoff threshold is -11dB.

4. The resolution enhancement method for a satellite-borne one-dimensional synthetic aperture microwave radiometer according to claim 1, characterized in that: Configuration parameters include effective integration time and sampling interval.

5. The resolution enhancement method for a satellite-borne one-dimensional synthetic aperture microwave radiometer according to claim 1, characterized in that: The inversion algorithm is a G matrix inversion algorithm, which selects the least squares solution method and uses the following formula (4) to obtain a high-precision brightness temperature value: in, represents the high-precision brightness temperature value obtained by inversion, and the upper right corner subscript + represents the pseudo-inverse of the matrix. represent A visibility sample matrix that carries high-precision information.

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