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I. Introduction
The essence of image perspective transformation (Perspective Transformation) is to project an image from one geometric plane to another. Perspective transformation ensures that the points of the same straight line are still on the same straight line, but are no longer guaranteed to be parallel. Because this is a two-dimensional image that undergoes a three-dimensional transformation, and then is mapped to another two-dimensional space, the two-dimensional space of the two-dimensional image is different from the two-dimensional space after the mapping, if the same is the affine transformation.
Similar to the " https://blog.csdn.net/LaoYuanPython/article/details/113788380 Image Affine Transformation Principle 3: Affine Transformation Type and Transformation Matrix Detailed Explanation " introduced the affine matrix of affine transformation, the perspective transformation is also based on Based on the matrix operation, the pixel correspondence between the old and the new image can be quickly found through the matrix operation.
The perspective transformation is described by the following homogeneous coordinate formula: the
above 3rd order square matrix is the perspective transformation matrix, this matrix is a homography matrix, after the matrix is determined, the mapping relationship between the old and the new image can be confirmed, and in OpenCV -In Python, the function getPerspectiveTransform can get the perspective transformation matrix through four pairs of points.
2. Analysis
The homogeneous coordinate formula of perspective transformation also corresponds to the linear equation system:
convert homogeneous coordinates (X,Y,Z) to （x',y',1）(for those interested in the explanation of homogeneous coordinates, please refer to the old monkey paid column " https://blog.csdn.net/LaoYuanPython /article/details/113743213 Principle of Image Affine Transformation 1: Homogeneous coordinate ins and outs "), you can know:
from the above formula, you can know that to confirm the 9 elements of a perspective transformation matrix, you must pass a certain number of known The coordinates of the source image (x, y) and the coordinates (x', y') of the result image are used to solve the values of these 9 elements. So how many points can solve these 9 element values? This number is 4 pairs, which are 4 points where any three points in the source image are not on a straight line and 4 points corresponding to the corresponding positions of the target image. Why are there 4 pairs of points? This is because:
- 4 points correspond to 4 sets of (x, y), (x', y') values, put them into the above formula, you can get 8 equations;
- Since the above formula is based on homogeneous coordinates, the following two formulas are equivalent
to
and According to the knowledge of matrix multiplication and matrix number multiplication, there are:
Set k=1/m 33 , then k×m 33 =1, and set k×m ij = M ij , then the final perspective matrix has only 8 variables.
Three, summary
This article analyzes the principle of getPerspectiveTransform in OpenCV-Python to confirm the perspective transformation matrix through 4 pairs of points, which will help everyone understand the knowledge of perspective transformation.
Four, reference knowledge
- Regarding homogeneous coordinates, you can check the relevant information yourself, or refer to the old ape paid column article " https://blog.csdn.net/LaoYuanPython/article/details/113743213 Image Affine Transformation Principle 1: Homogeneous coordinate ins and outs detailed explanation "Introduction;
- The perspective matrix is actually a homography matrix. For the discussion about the homography matrix 4 pairs of points can be obtained, you can also refer to CSDN blogger lyhbkz " Understanding and solving the homography matrix 3 "
- For the understanding of the perspective transformation matrix, you can refer to the previous article on the principle of affine matrix " https://blog.csdn.net/LaoYuanPython/article/details/113924892 The principle of affine transformation and its OpenCV-Python implementation knowledge summary ", Although there are differences between the two, it is helpful to understand the role of matrix in geometric transformation
- For matrix-related knowledge, please refer to " https://blog.csdn.net/LaoYuanPython/article/details/112460899 Artificial Intelligence Mathematical Foundation-Linear Algebra 4: Matrix and Matrix Operations "
For more image processing introduction, please refer to the column " OpenCV-Python graphics and image processing https://blog.csdn.net/laoyuanpython/category_9979286.html " and " https://blog.csdn.net/laoyuanpython/category_10581071.html OpenCV -Python Beginners Difficult Questions Collection " related articles.
For more basic mathematical knowledge of image processing, please refer to the column " Artificial Intelligence Mathematical Foundation https://blog.csdn.net/laoyuanpython/category_10382948.html "
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