霍夫变换是一种用于寻找简单几何形状的特征提取方法。
1 寻找直线
极坐标与直角坐标的变换关系：
线在极坐标系中的表式：(r=xcos heta +ysin heta)
其中，(r)是从原点到直线的距离，( heta)是(vec{r})和(x)轴的夹角
给定一个点((x_{0},y_{0}))，通过该点的所有直线的参数((r, heta))的集合，会在((r, heta))平面上形成一个三角函数。
[r=x_{0}cos heta +y_{0}sin heta
Rightarrow r={sqrt {x_{0}^{2}+y_{0}^{2}}}left({frac {x_{0}}{sqrt {x_{0}^{2}+y_{0}^{2}}}}cos heta +{frac {y_{0}}{sqrt {x_{0}^{2}+y_{0}^{2}}}}sin heta
ight)
Rightarrow r={sqrt {x_{0}^{2}+y_{0}^{2}}}left(cos phi cos heta +sin phi sin heta
ight)
Rightarrow r={sqrt {x_{0}^{2}+y_{0}^{2}}}cos( heta -phi )
]
给定很多点，判断这些点是否共线的问题，经由上述变换之后，变成判断一堆曲线是否在平面((r, heta))上相交于同一点的问题。
注：之所以使用极坐标系是为了计算方便，因为直角坐标系中当直线趋近于垂直(x)轴，它的斜率趋近于无限大
1.1 在OpenCV中使用霍夫变换寻找直线
// Read the image as gray-scale
Mat img = imread('lanes.jpg', IMREAD_COLOR);
// Convert to gray-scale
Mat gray = cvtColor(img, COLOR_BGR2GRAY);
// Store the edges
Mat edges;
// Find the edges in the image using canny detector
Canny(gray, edges, 50, 200);
// Create a vector to store lines of the image
vector<Vec4i> lines;
// Apply Hough Transform
HoughLinesP(edges, lines, 1, CV_PI/180, thresh, 10, 250);
// Draw lines on the image
for (size_t i=0; i<lines.size(); i++) {
Vec4i l = lines[i];
line(src, Point(l[0], l[1]), Point(l[2], l[3]), Scalar(255, 0, 0), 3, LINE_AA);
}
// Show result image
imshow("Result Image", img);
2 寻找圆
圆的表达式：((x-x_0)^2 + (y-y_0)^2 = r^2)
2.1 在OpenCV中使用霍夫变换寻找圆
// Read the image as gray-scale
img = imread("circles.png", IMREAD_COLOR);
// Convert to gray-scale
gray = cvtColor(img, COLOR_BGR2GRAY);
// Blur the image to reduce noise
Mat img_blur;
medianBlur(gray, img_blur, 5);
// Create a vector for detected circles
vector<Vec3f> circles;
// Apply Hough Transform
HoughCircles(img_blur, circles, HOUGH_GRADIENT, 1, img.rows/64, 200, 10, 5, 30);
// Draw detected circles
for(size_t i=0; i<circles.size(); i++) {
Point center(cvRound(circles[i][0]), cvRound(circles[i][1]));
int radius = cvRound(circles[i][2]);
circle(img, center, radius, Scalar(255, 255, 255), 2, 8, 0);
}