Detector rectification and stitching method based on filter cross target projection matrix

Abstract

The present application relates to a kind of detector calibration splicing method based on filter cross target projection matrix, belong to the high-precision splicing technical field of space remote sensing camera detector.Solve the technical problem of image quality decline caused by different detector not collinear in the prior art detector splicing method.The method of the present application uses filter target projection image position, and obtains the coordinate matrix of detector splicing by centroid extraction algorithm, linear fitting algorithm, coordinate matrix normalization processing and other methods, solves the problem of detector not collinear caused by filter tilt splicing.The method does not need to refer to the preset mark point of detector itself, and can still realize the linear splicing of detector under the condition of only filter mark point, and can ensure that the maximum tilt of the same row of pixels is less than 0.5 pixel size, and the misregistration accuracy between rows is less than 0.3 pixel size, avoids the random error caused by filter tilt, and further solves the problem of detector collinearity, improves the splicing accuracy.

Description

Technical Field

[0001] This invention relates to the field of high-precision stitching technology for aerospace remote sensing camera detectors, specifically to a detector correction and stitching method based on a filter cross-shaped target projection matrix. Background Technology

[0002] In the field of aerospace remote sensing, multispectral and hyperspectral cameras are core equipment for acquiring detailed spectral information of ground objects. Their imaging swath width determines the coverage efficiency of Earth observation, while imaging quality is a key prerequisite for the accuracy of data interpretation in subsequent applications such as ground object classification, resource surveys, and disaster monitoring. With the large-scale popularization of high-resolution multispectral and hyperspectral remote sensing missions, traditional single-chip linear array detectors, limited by hardware size, can no longer simultaneously meet the observation requirements of wide swath coverage and sub-meter resolution. Therefore, multi-detector stitching technology has become the core solution to overcome this bottleneck. However, this technology places stringent requirements on system accuracy, requiring micrometer-level linearity and flatness of the linear array direction of multiple detectors. At the same time, the assembly overlap accuracy of the detector and filter needs to reach the sub-pixel level, and the stitching method needs to meet long-term on-orbit stability and reliability. If the accuracy is not up to standard, the collinearity error of the linear array direction of the multi-detector after stitching and the perpendicularity error of the camera along the orbital flight direction will directly cause problems such as imaging strip misalignment and spectral registration deviation, ultimately leading to a serious deterioration in camera image quality and even rendering the camera unusable.

[0003] Currently, the traditional detector stitching methods in the industry are based on detector marker points and based on filter marker points, but both methods have significant limitations:

[0004] 1. Stitching method based on detector marker points: This method uses pre-set physical marker points (such as metal lines or photolithographic marks) on the detector surface for positioning. Its advantage is that it can directly rely on the detector's own reference and ensure collinearity in the linear array direction. However, in multispectral cameras, the marker points will be covered after the detector surface is covered with filters to achieve spectral dispersion, making it difficult to capture effectively and limiting its practicality.

[0005] 2. Stitching method based on filter marker points: This method directly uses the marker points on the filter surface as a reference, which solves the problem of marker point capture. However, during the filter pasting process, it is affected by factors such as assembly process deviation and thermal expansion and contraction of materials, which inevitably produce tilting errors. This error will be transmitted to the detector linear array direction, resulting in differences in the parallelism of the linear arrays of multiple detectors. Ultimately, this leads to pixel misalignment and blurring of different detectors during imaging, reducing the quality of image data. Summary of the Invention

[0006] This invention addresses the technical problem of image quality degradation caused by the non-collinearity of different detectors in existing detector stitching methods. It provides a detector correction stitching method based on a filter cross-shaped target projection matrix. The method primarily improves stitching accuracy by incorporating target projection into the filter marker point stitching method. Through centroid extraction, linear fitting, and coordinate matrix normalization algorithms, a corrected stitching matrix is ​​obtained from the projection matrix, thereby enhancing stitching accuracy.

[0007] To solve the above-mentioned technical problems, the technical solution of the present invention is as follows:

[0008] A detector trimming and stitching method based on a filter cross-shaped target projection matrix includes the following steps:

[0009] Step 1: Pre-set multiple rows and columns of high-precision cross-shaped target patterns on the surface of each multispectral filter. Paste each multispectral filter containing the cross-shaped target pattern onto the detector. Then, use a uniformly integrating sphere on the detector surface to image the cross-shaped target pattern, which will be projected onto the detector to obtain an image containing the cross-shaped target.

[0010] Step 2: The image containing the cross-shaped target obtained in Step 1 is processed by grayscale value flipping and grayscale weighted centroid extraction algorithm to obtain the actual distance coordinates. Then, the single target error is eliminated by least squares fitting to obtain the global distribution law of the target. Finally, the target coordinates of all detectors are unified to the detector reference coordinate system through coordinate matrix normalization to generate a high-precision detector stitching coordinate matrix, so as to realize the micron-level accurate correction and stitching of multiple detectors.

[0011] Step 3: After stitching together the detector imaging images, the detector windowing technology is used to select filters that can still completely cover the same pixel areas of each detector after tilting.

[0012] In the above technical solution, in step 1, a high-precision cross-shaped target pattern with 2 rows and M columns is preset on the surface of the multispectral filter, where M≥5.

[0013] In the above technical solution, in step 2, the actual distance coordinates are obtained from the image containing the crosshair target obtained in step 1 through grayscale value flipping and grayscale weighted centroid extraction algorithms. The specific steps are as follows:

[0014] Step 21: Perform grayscale flipping on the image data containing the crosshair target obtained in Step 1, flipping the grayscale value of the target from low to high;

[0015] Step 22: Based on the target position in the flipped image, match the size and position of the extraction box to enclose the target;

[0016] Assume the first row and first column of the target are the origin. The coordinates of the reference point of the extraction box are The grayscale value corresponding to this reference point is Take the reference point of the extraction box in both horizontal and vertical directions. [Number] pixels, meaning the extraction box size is [Number] pixels. The squares;

[0017] Step 23: Combine pixel edge detection and use the gray-scale weighted centroid extraction algorithm to calculate the centroid pixel coordinates of each target on each multispectral filter;

[0018] Assume the centroid pixel coordinates of one of the targets are The extraction method is as follows:

[0019]

[0020]

[0021] in, Represents pixel coordinates grayscale value, This represents the x-coordinate of the pixel. This represents the ordinate of the pixel; This represents the x-coordinate of the reference point of the extraction box. This represents the ordinate of the reference point of the extraction box. This indicates the number of pixels taken in both the horizontal and vertical directions from the reference point of the extraction box;

[0022] Step 24: Based on the detector pixel size calibration value, convert the centroid pixel coordinates into actual distance coordinates. ;

[0023] Assuming the cell size is micrometer;

[0024]

[0025]

[0026] in, Indicates the size of the pixel.

[0027] In the above technical solution, step 2 involves eliminating single-target errors using least-squares fitting to obtain the global distribution pattern of the target. The specific steps are as follows:

[0028] Step 25: For all actual distance coordinates obtained through grayscale value flipping and grayscale weighted centroid extraction algorithms, outlier removal is performed, followed by least-squares linear fitting to obtain two fitted curves, respectively. and Where a1, b1, a2, and b2 are all obtained using the least squares method, a1 and a2 represent coefficient values, and b1 and b2 represent constant values; the formula is as follows:

[0029]

[0030]

[0031] in, For the sample size, , For sample number, Indicates the first The coordinates of each sample and These represent the first-order coefficients and constant values ​​of the least squares fitting, respectively;

[0032] Step 26: Calculate the final horizontal slope by averaging and weighting the slopes of the two fitted curves from Step 25. By calculating the angle between the filter and the ideal horizontal line, the horizontal tilt angle of the multispectral filter can be accurately determined. ;

[0033] Step 27: Perform least squares fitting with outlier removal on the actual distance coordinates of each column to obtain multiple fitting curves, using the same method as in step 25.

[0034] Step 28: The slopes of the multiple fitted curves from Step 27 are averaged and weighted to obtain the final vertical slope. The vertical tilt angle of the multispectral filter is then accurately calculated by averaging the slopes with respect to the ideal vertical line. The method is the same as step 26;

[0035] Step 29: Using the target coordinates of the first row and first column as a reference, calculate the target coordinates of the remaining points using trigonometric functions according to the actual distance coordinates and tilt angle.

[0036] Step 210: Following the above method, calculate the coordinates of all targets for all detectors in sequence.

[0037] In the above technical solution, in step 2, the target coordinates of all detectors are unified to the detector reference coordinate system through coordinate matrix normalization processing to generate a high-precision detector stitching coordinate matrix. The specific steps are as follows:

[0038] Step 211: Using the first row and first column of the first detector as the origin, normalize the detector target coordinates. That is, subtract the first row and first column coordinates from all target coordinates to obtain the coordinate information of all detectors relative to the reference origin. This is the generated high-precision detector stitching coordinate matrix, as shown below:

[0039]

[0040]

[0041]

[0042] in, , , , , These represent the second, third, and fourth columns of the first row of the first detector. , No. The coordinates of the crosshair target in the column; , , , , , These represent the first, second, third, and fourth columns of the second row of the first detector. , No. The coordinates of the crosshair target in the column; , , , , , These represent the first, second, third, and fourth columns of the first row of the Nth detector, respectively. , No. The coordinates of the crosshair target in the column; , , , , , These represent the first, second, third, and fourth columns of the second row of the Nth detector, respectively. , No. The coordinates of the crosshair target in the column.

[0043] In the above technical solution, step 2, which achieves precise micron-level correction and splicing of multiple detectors, specifically involves:

[0044] Step 212: Assemble the detectors according to the priority of ensuring the coordinate positions of the first row and first column and the last row and last column of each detector.

[0045] The beneficial effects of this invention are:

[0046] This invention presents a detector correction and stitching method based on a filter cross-shaped target projection matrix. Utilizing the position of the filter target projection image, the coordinate matrix for detector stitching is obtained through centroid extraction, linear fitting, and coordinate matrix normalization algorithms, thus resolving the issue of detector non-collinearity caused by filter tilt stitching. This method does not require reference to pre-set marker points on the detector itself; it can still achieve straight-line stitching of the detector even with only filter marker points. Furthermore, it ensures that the maximum tilt of pixels in the same row is less than 0.5 pixel size, and the misalignment accuracy between rows is less than 0.3 pixel size, avoiding random errors caused by filter tilt, thereby solving the detector collinearity problem and improving stitching accuracy. Attached Figure Description

[0047] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0048] Figure 1 This is a schematic diagram of a cross-shaped target multispectral filter. In the diagram, L represents the actual distance on the horizontal axis, and H represents the actual distance on the vertical axis.

[0049] Figure 2 This is a schematic diagram of the projection of the cross-shaped target onto the detector after it is tilted. This represents the horizontal tilt angle of the multispectral filter. This represents the vertical tilt angle of the multispectral filter.

[0050] Figure 3 This is a schematic diagram showing the effect after stitching together the normalized coordinates of multiple detectors.

[0051] Figure 1 and 2 The spectral bands 1-3 in the text represent three different multispectral filters. Detailed Implementation

[0052] The core technical requirement of this invention is that when a remote sensing camera uses multiple detectors for on-orbit imaging, the linear array direction of all detectors must be strictly perpendicular to the satellite's flight direction. This is a crucial prerequisite for avoiding image shift mismatch and preventing image quality degradation. Furthermore, to ensure that the ground features output by each detector are not misaligned, the initial rows of each detector must also be on the same straight line. Therefore, the camera places extremely high demands on the collinearity of the detectors. The basis for achieving precise collinear stitching of detectors is that each detector needs a clearly defined marker point as a stitching reference. However, in practical applications, some remote sensing cameras, to meet the imaging requirements of multispectral and hyperspectral imaging, require custom filters to be pasted on the detector surface. Some of these filters completely cover the detector's own preset marker points, making it impossible to observe the traditional stitching reference. Simultaneously, theoretically, the marker points on the filter surface should be strictly aligned with the detector pixels. However, due to factors such as assembly process deviations and material thermal expansion and contraction, the actual assembled filters may have a certain tilt error. Stitching using the filter reference point will transmit this error to the detector's linear array direction, thereby reducing the collinearity between detectors and severely affecting image quality.

[0053] To address the above problems, this invention innovatively proposes a detector correction and stitching method based on the projection matrix of a filter cross-shaped target. This method ensures that the collinearity of the detectors is maintained even when the filter marker points are tilted, thus solving the image quality degradation problem caused by detector non-collinearity. This invention uses the filter cross-shaped target projection matrix as a reference to correct detector stitching, ensuring that the collinearity of the detectors is maintained even when the filter marker points are tilted. The core technology revolves around collinearity assurance for multi-detector on-orbit imaging, covering key aspects such as detector collinearity calibration, cross-shaped target design, linear array detector imaging control, centroid extraction preprocessing, centroid extraction algorithm, linear fitting algorithm, coordinate matrix normalization processing, and detector windowing.

[0054] The present invention will now be described in detail with reference to the accompanying drawings.

[0055] This invention takes a high-precision cross-shaped target pattern with two rows and five columns preset on the surface of a multispectral filter as an example. The number of multispectral filters is 3. The invention specifically describes a detector correction and splicing method based on the cross-shaped target projection matrix of the filter, including the following steps:

[0056] Step 1: Pre-set a high-precision cross-shaped target pattern of "two rows and five columns" on the surface of the multispectral filter. The error in row spacing and column spacing is within the micrometer level. Figure 1 As shown, Figure 1Holes 1-10 in the diagram represent 10 cross-shaped targets. Each multispectral filter containing a cross-shaped target pattern is attached to the detector. Then, the detector is aligned with a uniform light source integrating sphere and imaged in area array mode. The cross-shaped target pattern is projected onto the detector, resulting in an image of the target projected onto the detector containing the cross-shaped target.

[0057] Step 2: The image containing the cross-shaped target obtained in Step 1 is processed by grayscale value flipping and grayscale weighted centroid extraction algorithm to obtain the actual distance coordinates. The single target error is eliminated by least squares fitting to obtain the global distribution law of the target. Finally, the target coordinates of all detectors are unified to the detector reference coordinate system through coordinate matrix normalization to generate a high-precision detector stitching coordinate matrix, so as to realize the micron-level accurate correction and stitching of multiple detectors.

[0058] Step 2 specifically includes the following steps:

[0059] Step 21: Perform grayscale flipping on the image data containing the crosshair target obtained in Step 1, flipping the grayscale value of the target from low to high to enhance the contrast with the background;

[0060] Step 22: Based on the target position in the flipped image, match the size and position of the extraction box to accurately surround the target;

[0061] Assume the first row and first column of the target are the origin. The coordinates of the reference point of the extraction box are The grayscale value corresponding to this reference point is Take the reference point of the extraction box in both horizontal and vertical directions. [Number] pixels, meaning the extraction box size is [Number] pixels. The squares;

[0062] Step 23: Combine pixel edge detection with grayscale weighted centroid extraction algorithm to calculate the coordinates of 10 target centroid pixels;

[0063] Assume that the extracted coordinates (i.e., centroid pixel coordinates) of one of the targets are The extraction method is as follows:

[0064]

[0065]

[0066] in, Represents pixel coordinates grayscale value, This represents the x-coordinate of the pixel. This represents the ordinate of the pixel; This represents the x-coordinate of the reference point of the extraction box. This represents the ordinate of the reference point of the extraction box. This indicates the number of pixels taken in both the horizontal and vertical directions from the reference point of the extraction box;

[0067] Step 24: Based on the detector pixel size calibration value, convert the centroid pixel coordinates into actual distance coordinates. ;

[0068] Assuming the cell size is micrometer;

[0069]

[0070]

[0071] in, Indicates the size of the pixel;

[0072] Step 25: The two rows obtained through the grayscale value flipping and grayscale weighted centroid extraction algorithm are... The actual distance coordinates of each target were used to remove outliers, and then least-squares linear fitting was performed to obtain two fitted curves, which are respectively and a1, b1, a2, and b2 are all obtained using the least squares method. a1 and a2 represent coefficient values, and b1 and b2 represent constant values. The formulas are as follows:

[0073]

[0074]

[0075] in, For the sample size, , For sample number, Indicates the first The coordinates of each sample and These represent the first-order coefficients and constant values ​​of the least squares fitting, respectively;

[0076] Step 26: Calculate the final horizontal slope by averaging and weighting the slopes of the two fitted curves from Step 25. By calculating the angle between the filter and the ideal horizontal line, the horizontal tilt angle of the multispectral filter can be accurately determined. ;

[0077] Step 27: Perform least squares fitting with outlier removal on the actual distance coordinates of the five columns on the left and right to obtain five fitting curves, using the same method as in step 25.

[0078] Step 28: The slopes of the five fitted curves from Step 27 are averaged and weighted to obtain the final vertical slope. The vertical tilt angle of the multispectral filter is then accurately calculated by averaging the slopes with respect to the ideal vertical line. The method is the same as step 26;

[0079] Step 29: Using the target coordinates of the first row and first column as a reference, according to the target spacing (i.e., the actual distance coordinates) ) and tilt angle, and use trigonometric functions to obtain the target coordinates of the remaining points respectively;

[0080] Step 210: Following the method described above, calculate the coordinates of all targets for all detectors in sequence;

[0081] Step 211: Using the first row and first column of the first detector as the origin, normalize the target coordinates of the detectors, that is, subtract the first row and first column coordinates from all target coordinates to obtain the coordinate information of all detectors relative to the reference origin, which is the generated high-precision detector splicing coordinate matrix.

[0082]

[0083]

[0084]

[0085] in, , , , These represent the coordinates of the crosshair targets in the second, third, fourth, and fifth columns of the first row of the first detector; , , , , These represent the coordinates of the crosshair targets in the first, second, third, fourth, and fifth columns of the second row of the first detector. , , , , These represent the coordinates of the crosshair targets in the first, second, third, fourth, and fifth columns of the first row of the second detector. , , , , These represent the coordinates of the crosshair targets in the first, second, third, fourth, and fifth columns of the second row of the second detector. , , , , These represent the coordinates of the crosshair targets in the first, second, third, fourth, and fifth columns of the first row of the third detector. , , , , These represent the coordinates of the crosshair targets in the first, second, third, fourth, and fifth columns of the second row of the third detector.

[0086] Step 212: The detectors are stitched together according to the principle of prioritizing the first row and first column, and the last row and last column coordinates of each detector piece. This ensures the filters are tilted and the detectors are horizontal. See the image for the stitched result. Figure 3 .

[0087] Step 3: After stitching, combine the detector's imaging image and adjust the detector's windowing position parameters appropriately. Select areas where the same pixel region is still covered by each filter after tilting, such as... Figure 2 As shown.

[0088] In this step, since the number of rows of the detector linear array covered by the filter is greater than the actual number of rows used, the detector windowing technology is used to specifically select pixels in the same area of ​​the sensor that can still be completely covered by the filter after it is tilted for imaging. This ensures that the maximum tilt of pixels in the same row of the detector is less than 0.5 pixel size, the misalignment accuracy between rows is less than 0.3 pixel size, and all effective pixels are covered by the filter and are collinear. This comprehensively solves the problem of image quality degradation caused by the non-collinearity of detector stitching due to filter tilt.

[0089] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A detector rectification stitching method based on a filter cross target projection matrix, characterized in that, Includes the following steps: Step 1: Pre-set multiple rows and columns of high-precision cross-shaped target patterns on the surface of each multispectral filter. Paste each multispectral filter containing the cross-shaped target pattern onto the detector. Then, use a uniformly integrating sphere on the detector surface to image the cross-shaped target pattern, which will be projected onto the detector to obtain an image containing the cross-shaped target. Step 2: The image containing the cross-shaped target obtained in Step 1 is processed by grayscale value flipping and grayscale weighted centroid extraction algorithm to obtain the actual distance coordinates. Then, the single target error is eliminated by least squares fitting to obtain the global distribution law of the target. Finally, the target coordinates of all detectors are unified to the detector reference coordinate system through coordinate matrix normalization to generate a high-precision detector stitching coordinate matrix, so as to realize the micron-level accurate correction and stitching of multiple detectors. Step 3: After stitching together the detector imaging images, the detector windowing technology is used to select filters that can still completely cover the same pixel areas of each detector after tilting.

2. The filter cross target projection matrix based detector rectification stitching method of claim 1, wherein, In step 1, a high-precision cross-shaped target pattern with 2 rows and M columns is preset on the surface of the multispectral filter, where M≥5.

3. The filter cross target projection matrix based detector calibration stitching method of claim 2, wherein, In step 2, the actual distance coordinates are obtained from the image containing the crosshair target obtained in step 1 through grayscale value flipping and grayscale weighted centroid extraction algorithms. The specific steps are as follows: Step 21: Perform grayscale flipping on the image data containing the crosshair target obtained in Step 1, flipping the grayscale value of the target from low to high; Step 22: Based on the target position in the flipped image, match the size and position of the extraction box to enclose the target; Assume the first row and first column of the target are the origin. The coordinates of the reference point of the extraction box are The grayscale value corresponding to this reference point is Take the reference point of the extraction box in both horizontal and vertical directions. [Number] pixels, meaning the extraction box size is [Number] pixels. The squares; Step 23: Combine pixel edge detection and use the gray-scale weighted centroid extraction algorithm to calculate the centroid pixel coordinates of each target on each multispectral filter; Assuming the centroid pixel coordinates of one of the targets is The extraction method is as follows: in, Represents pixel coordinates grayscale value, This represents the x-coordinate of the pixel. This represents the ordinate of the pixel; This represents the x-coordinate of the reference point of the extraction box. This represents the ordinate of the reference point of the extraction box. This indicates the number of pixels taken in both the horizontal and vertical directions from the reference point of the extraction box; Step 24, convert the centroid pixel coordinates to actual distance coordinates according to the detector pixel size calibration value ; Assuming a pixel size of microns; wherein, represents the size of the pixel.

4. The filter cross target projection matrix based detector rectification stitching method of claim 3, wherein, In step 2, the single-target error is eliminated by combining least squares fitting to obtain the global distribution law of the target. The specific steps are as follows: Step 25: For all actual distance coordinates obtained through grayscale value flipping and grayscale weighted centroid extraction algorithms, outlier removal is performed, followed by least-squares linear fitting to obtain two fitted curves, respectively. and Where a1, b1, a2, and b2 are all obtained using the least squares method, a1 and a2 represent coefficient values, and b1 and b2 represent constant values; the formula is as follows: in, For the sample size, , For sample number, Indicates the first The coordinates of each sample and These represent the first-order coefficients and constant values ​​of the least squares fitting, respectively; Step 26, average the slope of the two fitted curves in step 25 to get the final lateral slope , calculate the horizontal tilt angle of the multispectral filter by the included angle calculation with the ideal horizontal line ; Step 27: Perform least squares fitting with outlier removal on the actual distance coordinates of each column to obtain multiple fitting curves, using the same method as in step 25. Step 28, average the slope of the fitting curve in step 27 to get the final vertical slope, through the angle calculation with the ideal vertical line, accurately calculate the vertical tilt angle of the multi-spectral filter , method as in step 26; Step 29: Using the target coordinates of the first row and first column as a reference, calculate the target coordinates of the remaining points using trigonometric functions according to the actual distance coordinates and tilt angle. Step 210: Following the above method, calculate the coordinates of all targets for all detectors in sequence.

5. The filter cross target projection matrix based detector rectification stitching method of claim 4, wherein, In step 2, the target coordinates of all detectors are unified to the detector reference coordinate system through coordinate matrix normalization, generating a high-precision detector stitching coordinate matrix. The specific steps are as follows: Step 211: Using the first row and first column of the first detector as the origin, normalize the detector target coordinates. That is, subtract the first row and first column coordinates from all target coordinates to obtain the coordinate information of all detectors relative to the reference origin. This is the generated high-precision detector stitching coordinate matrix, as shown below: in, , , , , These represent the second, third, and fourth columns of the first row of the first detector. , No. The coordinates of the crosshair target in the column; , , , , , These represent the first, second, third, and fourth columns of the second row of the first detector. , No. The coordinates of the crosshair target in the column; , , , , , These represent the first, second, third, and fourth columns of the first row of the Nth detector, respectively. , No. The coordinates of the crosshair target in the column; , , , , , These represent the first, second, third, and fourth columns of the second row of the Nth detector, respectively. , No. The coordinates of the crosshair target in the column.

6. The filter cross target projection matrix based detector calibration stitching method of claim 5, wherein, In step 2, precise micron-level correction and splicing of multiple detectors is achieved, specifically as follows: Step 212: Assemble the detectors according to the priority of ensuring the coordinate positions of the first row and first column and the last row and last column of each detector.

Images