圖像幾何變換之透視變換

1. 基本原理

  透視變換（Perspective Transformation）的本質是將圖像投影到一個新的視平面，其通用變換公式為：

  

  （u，v）為原始圖像像素坐標，（x=x’/w’，y=y’/w’）為變換之后的圖像像素坐標。透視變換矩陣圖解如下：

  

  仿射變換（Affine Transformation）可以理解為透視變換的特殊形式。透視變換的數學表達式為：

  

  所以，給定透視變換對應的四對像素點坐標，即可求得透視變換矩陣；反之，給定透視變換矩陣，即可對圖像或像素點坐標完成透視變換，如下圖所示：

  

2. OpenCV透視變換函數

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;
}

  原始圖像及其透視變換結果：