A BEV conversion method based on lane line constraints without requiring internal and external parameters

Abstract

This invention belongs to the field of computer vision technology and proposes a lane-constrained BEV conversion method that does not require internal and external parameters. This method proposes a lane-constrained homography matrix calculation model under appropriate internal parameter settings, and uses one or two sets of lane-line parallel constraints to calculate the analytical solution of the homography matrix. An optimized solution method based on multiple lane line inputs is proposed to improve the robustness of the conversion in autonomous driving scenarios. Finally, a processing pipeline is proposed to detect lane lines from an input image and output a BEV image. This invention can solve the problem of BEV perspective image generation relying on internal and external parameters, providing a reliable and robust method for calculating the homography matrix for perspective conversion.

Description

Technical Field

[0001] The present invention relates to the field of computer vision, and in particular to a BEV conversion method based on lane line constraints without requiring internal and external parameters, and is a homography matrix calculation method for BEV perspective conversion. Background Art

[0002] Bird's Eye (BEV), as a scene data representation method, can provide a 360-degree field of view and characterize the height and three-dimensional spatial information of objects in the scene. Compared with the traditional 2D perspective, it can more completely present the multi-dimensional characteristics of the scene, and can significantly improve processing accuracy in tasks such as road reconstruction, target detection and tracking, scene understanding and reasoning, autonomous decision-making and planning.

[0003] Existing BEV perspective conversion schemes rely heavily on the accuracy of the calibration of camera extrinsic and extrinsic parameters using a calibration plate and then calculating the homography matrix using an inverse perspective transform (IPM). This method is highly dependent on the accuracy of the calibration of these parameters, making the camera calibration process prone to drift. Deviations in the calibration parameters can significantly reduce the quality of the generated BEV. While this issue can be circumvented by detecting extrinsic parameters using sensors, this approach is difficult to implement in most application scenarios. A deep learning-based approach (A Geometric Approach to Obtain a Bird's Eye View from an Image) generates BEV images by estimating the vanishing point and detecting the horizon. However, this approach suffers from high computational overhead and only achieves good estimation results in scenes with distinct horizon features, resulting in insufficient robustness and generalization of the BEV conversion. Furthermore, in scenarios where the camera extrinsic and extrinsic parameters are unknown, these methods are unable to calculate the homography matrix required for the conversion.

[0004] Therefore, a reliable, robust and highly generalizable calculation method suitable for BEV conversion is urgently needed to solve the problem of existing technology's dependence on internal and external parameters of the camera and the defect of insufficient generalization ability of deep learning solutions, and to achieve reliable real-time calculation in dynamic environments. Summary of the Invention

[0005] To address the aforementioned issues, the present invention provides a lane-constrained BEV conversion method that requires no extrinsic or extrinsic parameters. This method primarily involves generating end-to-end BEV images without extrinsic or extrinsic parameters, constructing a homography matrix, and solving and optimizing the rotation matrix R. This method automatically detects lane lines and uses their characteristics to establish constraints, ultimately constructing the homography matrix. By establishing equations based on the parallel relationship between lane lines before and after conversion, the homography matrix can be directly solved using only one or two sets of lane lines. Furthermore, optimization using multiple lane line sets enables reliable calculations in dynamic environments, effectively eliminating reliance on camera extrinsic or extrinsic parameters and improving the adaptability and robustness of BEV conversion in complex scenarios.

[0006] In order to achieve the above object, the technical solution of the present invention is as follows:

[0007] A BEV conversion method based on lane line constraints without requiring internal and external parameters includes the following steps:

[0008] Step S1, image preprocessing: perform cropping on the input image to remove irrelevant background at the edge of the image and retain the area containing the road surface and lane lines to reduce invalid calculations and obtain the BEV target area to be converted.

[0009] Step S2: After preprocessing all images, the lane line detection model is used to perform lane line pixel-level detection. The detected lane line pixels are fitted into straight lines using the least squares method. Two lane lines are extracted from each image. These two lane lines are parallel in the BEV perspective. Picture No. The equation of the straight line in the pixel coordinate system is .in, , , is the horizontal coordinate in the pixel plane coordinate system, is the vertical coordinate in the pixel plane coordinate system, is the parameter result of the straight line fitting.

[0010] Step S3: According to the width and height parameters of the input image Set the camera's intrinsic matrix , this matrix does not need to be precisely calibrated, it only serves as an intermediate variable for the transformation between pixel coordinates and normalized coordinates, and its form satisfies the affine transformation characteristics of camera imaging.

[0011] Step S4: Solve the rotation matrix based on the parameter results fitted in S2 and the orthogonality and rotation characteristics of the camera extrinsic rotation matrix From the perspective of BEV projection geometry in 3D, the lane lines that were originally parallel in the perspective view appear to converge due to the perspective effect. Axis rotation angle This perspective effect can be eliminated, so that the lane lines can be restored to parallel in the BEV perspective; if there are two sets of lane lines that are not parallel to each other in spatial direction (such as horizontal and vertical lane lines), the lane lines can be restored to parallel in the BEV perspective. Axis rotation angle With around Axis rotation angle The synergistic effect eliminates the perspective effect. In the case of multiple pictures input, the solution is obtained through optimization method. The solution is as follows:

[0012] Step S4.1: Input a single image and construct a vector map of the direction of the parallel line projection using the direction vector relationship of the parallel line projection in perspective projection. The geometric equation of Axis rotation angle .

[0013] Step S4.2: Input two images. When the lane lines in these two images are not parallel to each other from the BEV perspective, set the Euler angle rotation order to (First go around Axis rotation and rewind Axis rotation), combining the direction vector constraints of the two sets of lines to construct the equation group, and solving them synchronously to obtain Axis rotation angle , Axis rotation angle .

[0014] Step S4.3: In the data sampling process, multiple frames of data are a common situation. Therefore, the present invention considers the scenario of multiple sets of parallel lines. For the scenario of multiple sets of parallel lines formed by multiple frames of data, the sum of the squares of the angle differences of the direction vectors between each set of straight lines in the BEV space is used as the optimization target. The lane lines are fitted on each image to obtain Set straight line parameters and gradually optimize them using gradient descent method and , until the sum of the angle differences converges to a preset threshold.

[0015] Step S5: The internal parameter matrix obtained in step S3 and the rotation matrix solved in step S4 , perform rotation transformation on the transformed image to correct the orientation deviation, and adjust the image position by translation operation so that the effective area of ​​the BEV perspective falls completely within the image coordinate system, and obtain the transformation matrix for boundary adjustment .

[0016] Step S6: Constructing the homography matrix ; Homography matrix By transformation matrix , the rotation matrix and internal inverse moment Cascade structure, the calculation formula is , this matrix can realize the direct mapping from the pixel coordinate system to the BEV coordinate system.

[0017] Step S7: The pixel coordinates of the image are transformed through the homography matrix Map it to the target view coordinate system and use bilinear interpolation to generate the BEV image.

[0018] The beneficial effects of the present invention include: This method effectively addresses the dependency of BEV image generation on camera intrinsic parameters, enabling conversion without the need for precise calibration of intrinsic parameters, thus resolving the issue of intrinsic parameters impacting conversion quality in traditional methods. In lane-rich scenarios, such as autonomous driving, this method, compared to IPM (inverse perspective transformation) methods, eliminates the need for reliance on camera extrinsic parameters, significantly improving practical convenience and practicality, while reducing hardware calibration requirements. Furthermore, compared to deep learning-based BEV generation methods, this method leverages the geometric principles of lane line constraints to achieve conversion, avoiding reliance on large-scale annotated data. This method offers advantages in computational efficiency, along with enhanced generalization and robustness. Particularly in the presence of lane lines, it can stably produce high-quality BEV conversion results in complex and changing environments, enhancing the adaptability of BEV perspective conversion in a variety of practical scenarios. Given camera intrinsic parameters, this method can be further applied to the coarse calibration of camera extrinsic parameters, expanding the method's applicability and providing a simple and efficient supplementary solution for camera parameter calibration. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 is a pipeline input and output diagram of the method provided by an embodiment of the present invention;

[0020] Figure 2 is a schematic diagram of a camera model and rotation angle provided by an embodiment of the present invention;

[0021] Figure 3 This is a flowchart of a BEV conversion method based on lane line constraints without requiring internal and external parameters, provided by an embodiment of the present invention;

[0022] Figure 4 Schematic diagram of a rotation matrix solution model provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0023] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0024] This embodiment provides a BEV conversion method based on lane line constraints without the need for internal and external parameters, such as Figure 3 As shown, the method includes steps S1 to S7. For example, five pictures of the same autonomous driving scene (such as Figure 1 As shown), the pictures with the same camera external and internal parameters are processed:

[0025] Step S1, image preprocessing: crop the input image to retain the image area that needs to be converted to BEV;

[0026] Step S2: Input After preprocessing the images (in this example ), use lane line detection models (such as UltraFast Lane Detection, LaneATT, CLRNet, etc.) to extract lane lines, perform straight line detection on the detected lane lines, select two similar straight lines as input and fit them into a straight line. Two lane lines are extracted from each image, of which the first Picture No. The equation of the straight line in the pixel coordinate system is .in, , , is the horizontal coordinate, is the vertical coordinate, is the parameter result of the straight line fitting.

[0027] Step S3: According to the input picture parameters, the width and height of the picture are , determine the camera intrinsic parameter matrix Internal parameter matrix By camera focal length ,center constitute, Internal parameter matrix for:

[0028]

[0029] From the perspective of camera imaging principle, the core function of the internal parameter is to convert the coordinates in the pixel coordinate system to the normalized coordinate system, where the focal length With center coordinates They bear the effects of stretching and translation respectively. In the normalized coordinate system After directly performing a rotation transformation on the pixel coordinates, reprojecting them to this coordinate system completes the generation of the BEV, at which point the lane lines appear parallel. Since the subsequent transformation matrix can reproject the imaging results to the pixel imaging interval, the camera intrinsic parameters do not need to be accurately measured and set. Essentially, it can be considered a stretching transformation matrix. After projecting to a normalized coordinate system via an inverse matrix operation, a rotation transformation is performed on the pixel coordinates in this coordinate system, and then the projection operation returns to the normalized coordinate system. Combined with the subsequent transformation matrix, the reprojection process within the pixel imaging interval can be achieved.

[0030] Step S4: Calculate the rotation matrix based on the extracted line group parameters and the orthogonality and rotation characteristics of the camera extrinsic rotation matrix. The schematic diagram of the calculation method is as follows Figure 4 .

[0031] Rotation Matrix The calculation process of the camera extrinsic parameters is first specified as the Euler angle rotation order: (First go around Rotate around the axis, then around Axis rotation, where Shaft rotation has no significant effect on BEV conversion, so it can be fixed to 0 in subsequent calculations). Set The clockwise rotation angle of the axis is , The clockwise rotation angle of the axis is The spatial relationship between the camera coordinate system and the ground is as follows: the origin of the camera coordinate system is located at the optical center of the lens, The axis points horizontally to the right side of the vehicle. The axis points vertically upwards towards the vehicle. The axis points to the front of the camera along the optical axis, and the ground is in the camera coordinate system. Below the plane ( area), by bypassing Axis rotation angle , around Axis rotation angle Eliminates perspective effects.

[0032] The relationship between the camera coordinate system and the ground is as follows Figure 2 .

[0033]

[0034]

[0035] in, Matrix represents the three-dimensional coordinate axis around Axis rotation clockwise , Matrix represents the three-dimensional coordinate axis around Axis rotation clockwise .

[0036] Based on the lane line parameters extracted in step S2 and the constraint that the lane lines must remain parallel from the BEV perspective, a rotation matrix is ​​constructed. Geometric relationship with the straight line direction vector: When the lane line is parallel from the BEV perspective, it is rotated in the normalized coordinate system by the matrix The transformed direction vector must satisfy the parallelism constraint (the vector cross product is zero). This constraint can be used to establish and equation, and then solve the rotation matrix Specific parameters.

[0037] The processing steps of step S4 are divided into three categories according to the input lane line conditions.

[0038] Step S4.1: Input a set of lane lines. That is, input an image and a set of lane lines:

[0039] Step S4.1.1: Input a set of lines , ,in and are the homogeneous coordinates of the two lane lines in the image. This set of parallel lane lines appears non-parallel in the pixel coordinate system due to the perspective effect. According to the geometric relationship, the rotation matrix only contains the rotation around The rotation parameters of the axis are set according to the geometric relationship, and the rotation matrix is ​​set .

[0040]

[0041] Step S4.1.2: Define the matrix , The matrix simulates the camera reprojection process to calculate the rotation matrix of the external parameters The input set of straight lines is transformed by the matrix After transformation:

[0042]

[0043]

[0044] in, yes Through the matrix Then the homogeneous coordinates of the lane lines from the BEV perspective are obtained, The direction vector is:

[0045]

[0046]

[0047] Step S4.1.3: Based on the constraint that lane lines are parallel from the BEV perspective, we can get The analytical solution is:

[0048]

[0049] Step S4.2: Input two sets of lane lines. That is, input two images. If the lane lines in these two images are not parallel to each other from the BEV's perspective:

[0050] Step S4.2.1: Input two sets of parallel lines that are not parallel in the BEV perspective, that is, input ,in are the two lane lines in the first picture, These are the two lane lines in the second picture, which can constrain and . Set the rotation matrix according to the geometric relationship for:

[0051]

[0052] The rotation matrix Indicates first winding Axis rotation , then go around Axis rotation .

[0053] Step S4.2.2: This step is similar to step S4.1.2. The input line passes through the matrix After transformation:

[0054]

[0055] in, is the matrix The direction vector of the straight line obtained after transformation , The lane lines of each image input are transformed and made parallel to each other to obtain two sets of constraints.

[0056]

[0057]

[0058] in ,in , .

[0059] Step S4.2.3, using the above equation, we can get and The analytical solution is:

[0060]

[0061]

[0062] in,

[0063]

[0064]

[0065]

[0066]

[0067]

[0068]

[0069] Step S4.3: Input multiple lane lines. In this example, five images are input:

[0070] Step S4.3.1: Input multiple sets of parallel lines that are not parallel in the BEV perspective, where Group No. straight lines ,in , . Can be optimized and . Set the rotation matrix according to the geometric relationship for:

[0071]

[0072] The rotation matrix Indicates first winding Axis rotation , then go around Axis rotation .

[0073] Step S4.3.2: This step is similar to S4.2.2. The optimization goal is to minimize the angle difference between the two transformed straight line direction vectors. In multiple images, fit a set of parallel lines in each image to obtain Group of straight lines.

[0074]

[0075] in, for Through the matrix The direction vector of the straight line obtained after transformation, , .

[0076] Step S4.3.3: From a three-dimensional perspective, a step-by-step optimization method is used. First, fix , optimized based on the following formula horn:

[0077]

[0078]

[0079] in, is the loss function; , Indicates the The angle between the direction vectors of the two transformed lines in the image.

[0080] S4.3.4. Obtain optimized After the corner, Substituting these into the formula in S4.3.2 as known quantities, we obtain ( As the optimized parameter), the calculation method of S4.3.3 can be used again to optimize horn.

[0081] S5. Based on the internal parameter matrix obtained above and the rotation matrix , calculate the transformation matrix .

[0082] The pixel coordinates of the image pass through the matrix After the transformation, the coordinate range does not fall within the imaging interval. Through the following transformation matrix Perform scaling and translation transformations on the coordinates to make the pixel coordinates fall within the camera imaging range.

[0083]

[0084] in, Indicates the stretching transformation size of pixels. and Indicates the pixel coordinates in Axis and The size of the translation transformation of the axis. It is a transformation matrix that has both scaling and translation functions. By offsetting the spatial position of the image, the final image falls within the imaging range of the pixel coordinate system.

[0085] Transformation Matrix The specific calculation method is to rewrite the pixel coordinates of the image into homogeneous coordinates according to the width and height of the input image, which is then matrix After the multiplication transformation, all the pixel coordinates after the transformation are normalized. According to the coordinates of the pixel after the transformation, its position in and Boundaries in direction:

[0086]

[0087]

[0088] in, Represents the matrix The coordinates of the normalized pixel after transformation.

[0089] Next, in order to make the transformed image fully displayed within the imaging interval, the scaling factor is calculated and translation bias :

[0090]

[0091]

[0092]

[0093]

[0094]

[0095] in, The width and height of the input image.

[0096] Step S6: Based on the above experiment, the internal parameter matrix determined in step S3 is 、The rotation matrix solved in step S4 And the boundary adjustment matrix obtained in step S5 , construct the homography matrix through matrix multiplication: ;

[0097] Step S7: The pixel coordinates of the image are transformed through the homography matrix Map it to the target view coordinate system and use bilinear interpolation to generate the BEV image. You can use the warpperspective function of OpenCV to perform the transformation to get the BEV view conversion result.

[0098] The homography matrix to be constructed Input the warpPerspective function of the OpenCV library to convert the original image to the BEV perspective. The calling format of this function is:

[0099]

[0100] in: The image to be converted after being cropped in step S1; Define the mapping rules of pixel coordinates for the homography matrix calculated in step S6; The size of the BEV output image is usually set to a rectangle proportional to the width and height of the original image ( ); Specify the bilinear interpolation algorithm to improve the smoothness of the transformed image. Through the perspective transformation operation of this function, the lane lines in the original image are restored to a parallel state under the BEV perspective, and the road area is stretched according to the bird's-eye view ratio. The final output is This is the BEV conversion result that eliminates perspective distortion.

Claims

1. A BEV conversion method based on lane line constraints without the need for internal and external parameters, characterized in that: The following steps are involved: Step S1, image preprocessing: input Crop the image to remove irrelevant background around the edges of the image and retain the area containing the road surface and lane lines; Step S2: After preprocessing all images, the lane line detection model is used to perform lane line pixel-level detection. The detected lane line pixels are fitted into straight lines using the least squares method. Two lane lines are extracted from each image. These two lane lines are parallel in the BEV perspective. Picture No. The equation of the straight line in the pixel coordinate system is ;in, , , is the horizontal coordinate in the pixel plane coordinate system, is the vertical coordinate in the pixel plane coordinate system, is the parameter result of straight line fitting; Step S3: According to the width and height parameters of the input image Set the camera's intrinsic matrix ; Step S4: Solve the rotation matrix based on the parameter results fitted in S2 and the orthogonality and rotation characteristics of the camera extrinsic rotation matrix ; Category 3 The solution is as follows: Step S4.1: Input a single image and construct a vector map of the direction of the parallel line projection using the direction vector relationship of the parallel line projection in perspective projection. The geometric equation of Axis rotation angle ; Step S4.2: Input two images. When the lane lines in these two images are not parallel to each other from the BEV perspective, set the Euler angle rotation order to , combining the two sets of straight line direction vector constraints to construct the equations, and solve them to get Axis rotation angle , Axis rotation angle ; Step S4.3: For multiple sets of parallel line scenes formed by multiple frames of data, the sum of the squares of the angle differences between the direction vectors of each set of straight lines in the BEV space is used as the optimization target. The lane lines are fitted on each image to obtain Set straight line parameters and gradually optimize them using gradient descent method and , until the sum of the angle differences converges to a preset threshold; Step S5: The internal parameter matrix obtained in step S3 and the rotation matrix solved in step S4 , perform rotation transformation on the transformed image to correct the orientation deviation, and adjust the image position by translation operation so that the effective area of ​​the BEV perspective falls completely within the image coordinate system, and obtain the transformation matrix for boundary adjustment ; Step S6: Constructing the homography matrix ; Homography matrix By transformation matrix , the rotation matrix and internal inverse moment Cascade structure, the calculation formula is , this matrix realizes the direct mapping from the pixel coordinate system to the BEV coordinate system; Step S7: The pixel coordinates of the image are transformed through the homography matrix Map it to the target view coordinate system and use bilinear interpolation to generate the BEV image.

2. The BEV conversion method based on lane line constraints without requiring internal and external parameters according to claim 1, characterized in that: In step S3, the internal parameter matrix By camera focal length ,center constitute, ; Internal parameter matrix for: 。 3. The BEV conversion method based on lane line constraints without requiring internal and external parameters according to claim 1, characterized in that: The rotation matrix in step S4 The calculation process is as follows: First, the Euler angle rotation order of the camera external parameters is specified as ,set up The clockwise rotation angle of the axis is , The clockwise rotation angle of the axis is ; The spatial relationship between the camera coordinate system and the ground is as follows: the origin of the camera coordinate system is located at the optical center of the lens, The axis points horizontally to the right side of the vehicle. The axis points vertically upwards towards the vehicle. The axis points to the front of the camera along the optical axis, and the ground is in the camera coordinate system. Below the plane, by winding Axis rotation angle , around Axis rotation angle Eliminate perspective effects; The relationship between the camera coordinate system and the ground is as follows: ； ； in, Matrix represents the three-dimensional coordinate axis around Axis rotation clockwise , Matrix represents the three-dimensional coordinate axis around Axis rotation clockwise ; Based on the lane line parameters extracted in step S2 and the constraint that the lane lines must remain parallel in the normalized coordinate system, a rotation matrix is ​​constructed. Geometric relationship with the straight line direction vector: When the lane line is parallel from the BEV perspective, it is rotated in the normalized coordinate system by the matrix The transformed direction vector must satisfy the parallel constraint condition; use this constraint condition to establish the and equation, and then solve the rotation matrix Specific parameters.

4. The BEV conversion method based on lane line constraints without requiring internal and external parameters according to claim 1, characterized in that: Three categories in step S4 The solution is as follows: Step S4.1: Input a set of lane lines. That is, input an image and a set of lane lines: Step S4.1.1: Input a set of lines , ,in and are the homogeneous coordinates of the two lane lines in the picture; in this case, the rotation matrix only contains the rotation around The rotation parameters of the axis are set according to the geometric relationship, and the rotation matrix is ​​set ; ； Step S4.1.2: Define the matrix , The matrix simulates the camera reprojection process to calculate the rotation matrix of the external parameters ; A set of straight lines input through the matrix After transformation: ； ； in, yes Through the matrix Then the homogeneous coordinates of the lane lines from the BEV perspective are obtained, The direction vector is: ； ； Step S4.1.3: Based on the constraint that lane lines are parallel from the BEV perspective, we can get The analytical solution is: ； Step S4.2: Input two sets of lane lines. That is, input two images. If the lane lines in these two images are not parallel to each other from the BEV's perspective: Step S4.2.1: Input two sets of parallel lines that are not parallel in the BEV perspective, that is, input ,in are the two lane lines in the first picture, These are the two lane lines in the second picture, constraining and ; Set the rotation matrix according to the geometric relationship for: ； The rotation matrix Indicates first winding Axis rotation , then go around Axis rotation ; Step S4.2.2: The input straight line passes through the matrix After transformation: ； in, is the matrix The direction vector of the straight line obtained after transformation , ;Use the lane lines of each image input to transform them into parallel ones, and get two sets of constraints; ； ； in ,in , ; Step S4.2.3, using the above equation, we can get and The analytical solution is: ； ； in, ； ； ； ； ； ； Step S4.3: Input multiple lane lines: Step S4.3.1: Input multiple lane lines, each of which is parallel from the BEV perspective. Group No. straight lines Optimize and ; Set the rotation matrix according to the geometric relationship for: ； The rotation matrix Indicates first winding Axis rotation , then go around Axis rotation ; Step S4.3.2, the optimization goal is to minimize the angle difference between the two transformed straight line direction vectors; in multiple images, fit a set of parallel lines in each image to obtain Group straight line; ； in, for Through the matrix The direction vector of the straight line obtained after transformation; Step S4.3.3: From a three-dimensional perspective, a step-by-step optimization method is used; first, fix , optimized based on the following formula horn: ； ； in, is the loss function; Indicates the The angle between the two transformed straight line direction vectors in the image; S4.3.

4. Obtain optimized After the corner, Substituting these into the formula in S4.3.2 as known quantities, we obtain , the calculation method of S4.3.3 can be used again to optimize horn.

5. The BEV conversion method based on lane line constraints without requiring internal and external parameters according to claim 1, characterized in that: In step S5, the pixel coordinates of the image are processed by the matrix After the transformation, the coordinate range does not fall within the imaging interval. Through the following transformation matrix Perform scaling and translation transformation on the coordinates to make the pixel coordinates fall into the camera imaging range: ； in, Indicates the stretching transformation size of pixels. and Indicates the pixel coordinates in Axis and The magnitude of the translation transformation of the axis; Transformation Matrix The specific calculation method is to rewrite the pixel coordinates of the image into homogeneous coordinates according to the width and height of the input image, which is then matrix After the multiplication transformation, the coordinates of all the transformed pixels are normalized; according to the coordinates of the transformed pixels, their and Boundaries in direction: ； ； in, Represents the matrix The coordinates of the normalized pixel points after transformation; Next, in order to make the transformed image fully displayed within the imaging interval, the scaling factor is calculated and translation bias : ； ； ； ； ； in, The width and height of the input image.

Images