如何从 OpenCV 的 fitEllipse 函数中获取椭圆系数? [英] How can I get ellipse coefficient from fitEllipse function of OpenCV?
问题描述
我想从一张图片中提取红球并得到图片中检测到的椭圆矩阵.
这是我的例子:
我对图片进行阈值处理，使用 findContour() 函数找到红球的轮廓，并使用 fitEllipse() 拟合椭圆.
但我想要的是得到这个椭圆的系数.因为fitEllipse()返回的是一个旋转矩形(RotatedRect)，所以需要重新编写这个函数.
一个椭圆可以表示为Ax^2 + By^2 + Cxy + Dx + Ey + F = 0；所以我想得到 u=(A,B,C,D,E,F) 或 u=(A,B,C,D,E) 如果 F 是 1(构造一个椭圆矩阵).
我看了fitEllipse()的源码，一共有三个SVD过程，我想我可以从这三个SVD过程的结果中得到上面的系数.但我很困惑每个 SVD 过程的每个结果(变量 cv::Mat x)代表什么，为什么这里有三个 SVD?
这是这个函数:
cv::RotatedRect cv::fitEllipse( InputArray _points ){垫点 = _points.getMat();int i, n = points.checkVector(2);int depth = points.depth();CV_Assert(n >= 0 && (depth == CV_32F || depth == CV_32S));旋转矩形框；如果( n <5 )CV_Error( CV_StsBadSize, "应该至少有 5 个点来拟合椭圆");//新的拟合椭圆算法，由 Daniel Weiss 博士贡献Point2f c(0,0);双 gfp[5], rp[5], t;const double min_eps = 1e-8;bool is_float = depth == CV_32F;const Point* ptsi = points.ptr();const Point2f* ptsf = points.ptr();AutoBuffer_广告(n*5), _bd(n);double *Ad = _Ad, *bd = _bd;//首先适合参数 A - E垫 A( n, 5, CV_64F, Ad );垫 b( n, 1, CV_64F, bd );垫 x( 5, 1, CV_64F, gfp );for( i = 0; i min_eps)t = gfp[2]/sin(-2.0 * rp[4]);else//椭圆旋转了 pi/2 的整数倍t = gfp[1] - gfp[0];rp[2] = fabs(gfp[0] + gfp[1] - t);如果( rp[2] > min_eps )rp[2] = std::sqrt(2.0/rp[2]);rp[3] = fabs(gfp[0] + gfp[1] + t);如果( rp[3] > min_eps )rp[3] = std::sqrt(2.0/rp[3]);box.center.x = (float)rp[0] + c.x;box.center.y = (float)rp[1] + c.y;box.size.width = (float)(rp[2]*2);box.size.height = (float)(rp[3]*2);if( box.size.width > box.size.height ){浮动 tmp;CV_SWAP( box.size.width, box.size.height, tmp );box.angle = (float)(90 + rp[4]*180/CV_PI);}如果(框.角度 <-180 )box.angle += 360;if(box.angle > 360 )box.angle -= 360;退货箱；} 源码链接:
因此，您可以从RotatedRect 中获取您需要的参数，而无需更改函数fitEllipse.solve 函数用于求解线性系统或最小二乘法问题.使用 SVD 分解方法，系统可以被过度定义和/或矩阵 src1 可以是奇异的.
有关算法的更多详细信息，您可以查看Fitzgibbon 的论文 提出了这种拟合椭圆方法.
I want to extract the red ball from one picture and get the detected ellipse matrix in picture.
Here is my example:
I threshold the picture, find the contour of red ball by using findContour() function and use fitEllipse() to fit an ellipse.
But what I want is to get coefficient of this ellipse. Because the fitEllipse() return a rotation rectangle (RotatedRect), so I need to re-write this function.
One Ellipse can be expressed as Ax^2 + By^2 + Cxy + Dx + Ey + F = 0; So I want to get u=(A,B,C,D,E,F) or u=(A,B,C,D,E) if F is 1 (to construct an ellipse matrix).
I read the source code of fitEllipse(), there are totally three SVD process, I think I can get the above coefficients from the results of those three SVD process. But I am quite confused what does each result (variable cv::Mat x) of each SVD process represent and why there are three SVD here?
Here is this function:
cv::RotatedRect cv::fitEllipse( InputArray _points )
{
Mat points = _points.getMat();
int i, n = points.checkVector(2);
int depth = points.depth();
CV_Assert( n >= 0 && (depth == CV_32F || depth == CV_32S));
RotatedRect box;
if( n < 5 )
CV_Error( CV_StsBadSize, "There should be at least 5 points to fit the ellipse" );
// New fitellipse algorithm, contributed by Dr. Daniel Weiss
Point2f c(0,0);
double gfp[5], rp[5], t;
const double min_eps = 1e-8;
bool is_float = depth == CV_32F;
const Point* ptsi = points.ptr<Point>();
const Point2f* ptsf = points.ptr<Point2f>();
AutoBuffer<double> _Ad(n*5), _bd(n);
double *Ad = _Ad, *bd = _bd;
// first fit for parameters A - E
Mat A( n, 5, CV_64F, Ad );
Mat b( n, 1, CV_64F, bd );
Mat x( 5, 1, CV_64F, gfp );
for( i = 0; i < n; i++ )
{
Point2f p = is_float ? ptsf[i] : Point2f((float)ptsi[i].x, (float)ptsi[i].y);
c += p;
}
c.x /= n;
c.y /= n;
for( i = 0; i < n; i++ )
{
Point2f p = is_float ? ptsf[i] : Point2f((float)ptsi[i].x, (float)ptsi[i].y);
p -= c;
bd[i] = 10000.0; // 1.0?
Ad[i*5] = -(double)p.x * p.x; // A - C signs inverted as proposed by APP
Ad[i*5 + 1] = -(double)p.y * p.y;
Ad[i*5 + 2] = -(double)p.x * p.y;
Ad[i*5 + 3] = p.x;
Ad[i*5 + 4] = p.y;
}
solve(A, b, x, DECOMP_SVD);
// now use general-form parameters A - E to find the ellipse center:
// differentiate general form wrt x/y to get two equations for cx and cy
A = Mat( 2, 2, CV_64F, Ad );
b = Mat( 2, 1, CV_64F, bd );
x = Mat( 2, 1, CV_64F, rp );
Ad[0] = 2 * gfp[0];
Ad[1] = Ad[2] = gfp[2];
Ad[3] = 2 * gfp[1];
bd[0] = gfp[3];
bd[1] = gfp[4];
solve( A, b, x, DECOMP_SVD );
// re-fit for parameters A - C with those center coordinates
A = Mat( n, 3, CV_64F, Ad );
b = Mat( n, 1, CV_64F, bd );
x = Mat( 3, 1, CV_64F, gfp );
for( i = 0; i < n; i++ )
{
Point2f p = is_float ? ptsf[i] : Point2f((float)ptsi[i].x, (float)ptsi[i].y);
p -= c;
bd[i] = 1.0;
Ad[i * 3] = (p.x - rp[0]) * (p.x - rp[0]);
Ad[i * 3 + 1] = (p.y - rp[1]) * (p.y - rp[1]);
Ad[i * 3 + 2] = (p.x - rp[0]) * (p.y - rp[1]);
}
solve(A, b, x, DECOMP_SVD);
// store angle and radii
rp[4] = -0.5 * atan2(gfp[2], gfp[1] - gfp[0]); // convert from APP angle usage
if( fabs(gfp[2]) > min_eps )
t = gfp[2]/sin(-2.0 * rp[4]);
else // ellipse is rotated by an integer multiple of pi/2
t = gfp[1] - gfp[0];
rp[2] = fabs(gfp[0] + gfp[1] - t);
if( rp[2] > min_eps )
rp[2] = std::sqrt(2.0 / rp[2]);
rp[3] = fabs(gfp[0] + gfp[1] + t);
if( rp[3] > min_eps )
rp[3] = std::sqrt(2.0 / rp[3]);
box.center.x = (float)rp[0] + c.x;
box.center.y = (float)rp[1] + c.y;
box.size.width = (float)(rp[2]*2);
box.size.height = (float)(rp[3]*2);
if( box.size.width > box.size.height )
{
float tmp;
CV_SWAP( box.size.width, box.size.height, tmp );
box.angle = (float)(90 + rp[4]*180/CV_PI);
}
if( box.angle < -180 )
box.angle += 360;
if( box.angle > 360 )
box.angle -= 360;
return box;
}
The source code link: https://github.com/Itseez/opencv/blob/master/modules/imgproc/src/shapedescr.cpp
The function fitEllipse returns a RotatedRect that contains all the parameters of the ellipse.
An ellipse is defined by 5 parameters:
- xc : x coordinate of the center
- yc : y coordinate of the center
- a : major semi-axis
- b : minor semi-axis
- theta : rotation angle
You can obtain these parameters like:
RotatedRect e = fitEllipse(points);
float xc = e.center.x;
float yc = e.center.y;
float a = e.size.width / 2; // width >= height
float b = e.size.height / 2;
float theta = e.angle; // in degrees
You can draw an ellipse with the function ellipse using the RotatedRect:
ellipse(image, e, Scalar(0,255,0));
or, equivalently using the ellipse parameters:
ellipse(res, Point(xc, yc), Size(a, b), theta, 0.0, 360.0, Scalar(0,255,0));
If you need the values of the coefficients of the implicit equation, you can do like (from Wikipedia):
So, you can get the parameters you need from the RotatedRect, and you don't need to change the function fitEllipse.
The solve function is used to solve linear systems or least-squares problems. Using the SVD decomposition method the system can be over-defined and/or the matrix src1 can be singular.
For more details on the algorithm, you can see the paper of Fitzgibbon that proposed this fit ellipse method.
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