图像几何变换之透视变换

1. 基本原理

透视变换（Perspective Transformation）的本质是将图像投影到一个新的视平面，其通用变换公式为：

（u，v）为原始图像像素坐标，（x=x’/w’，y=y’/w’）为变换之后的图像像素坐标。透视变换矩阵图解如下：

仿射变换（Affine Transformation）可以理解为透视变换的特殊形式。透视变换的数学表达式为：

所以，给定透视变换对应的四对像素点坐标，即可求得透视变换矩阵；反之，给定透视变换矩阵，即可对图像或像素点坐标完成透视变换，如下图所示：

2. OpenCV透视变换函数

Mat getPerspectiveTransform(const Point2f* src, const Point2f* dst)
// Calculate a perspective transform from four pairs of the corresponding points.
// src – Coordinates of quadrangle vertices in the source image.
// dst – Coordinates of the corresponding quadrangle vertices in the destination image.

void warpPerspective(InputArray src, OutputArray dst, InputArray M, Size dsize, int flags=INTER_LINEAR, int borderMode=BORDER_CONSTANT, const Scalar& borderValue=Scalar())
// Apply a perspective transform to an image.
// src – Source image.
// dst – Destination image that has the size dsize and the same type as src.
// M – 3*3 transformation matrix.
// dsize – Size of the destination image.
// flags – Combination of interpolation methods and the optional flag WARP_INVERSE_MAP that means that M is the inverse transformation (dst?src).
// borderMode – Pixel extrapolation method. When borderMode=BORDER_TRANSPARENT, it means that the pixels in the destination image that corresponds to the “outliers” in the source image are not modified by the function.
// borderValue – Value used in case of a constant border. By default, it is 0.

3. 程序

#include <iostream>

#include "highgui.h"
#include "opencv2/imgproc/imgproc.hpp"

int main()
{
	// get original image.
	cv::Mat originalImage = cv::imread("road.png");

	// perspective image.
	cv::Mat perspectiveImage;

	// perspective transform
	cv::Point2f objectivePoints[4], imagePoints[4];

	// original image points.
	imagePoints[0].x = 10.0; imagePoints[0].y = 457.0;
	imagePoints[1].x = 395.0; imagePoints[1].y = 291.0;
	imagePoints[2].x = 624.0; imagePoints[2].y = 291.0;
	imagePoints[3].x = 1000.0; imagePoints[3].y = 457.0;

	// objective points of perspective image.
	// move up the perspective image : objectivePoints.y - value .
	// move left the perspective image : objectivePoints.x - value.
	double moveValueX = 0.0;
	double moveValueY = 0.0;

	objectivePoints[0].x = 46.0 + moveValueX; objectivePoints[0].y = 920.0 + moveValueY;
	objectivePoints[1].x = 46.0 + moveValueX; objectivePoints[1].y = 100.0 + moveValueY;
	objectivePoints[2].x = 600.0 + moveValueX; objectivePoints[2].y = 100.0 + moveValueY;
	objectivePoints[3].x = 600.0 + moveValueX; objectivePoints[3].y = 920.0 + moveValueY;

	cv::Mat transform = cv::getPerspectiveTransform(objectivePoints, imagePoints);

	// perspective.
	cv::warpPerspective(originalImage,
		                perspectiveImage,
						transform,
						cv::Size(originalImage.rows, originalImage.cols),
						cv::INTER_LINEAR | cv::WARP_INVERSE_MAP);

	// cv::imshow("perspective image", perspectiveImage);
	// cvWaitKey(0);

	cv::imwrite("perspectiveImage.png", perspectiveImage);

	return 0;
}

原始图像及其透视变换结果：