opencv焦距估计函数cvInitIntrinsicParams2D原理解析
背景介绍
在进行相机标定时，通常会包括内参初始化估计、外参初始化估计，以及非线性优化参数。在opencv中内参数估计主要在cvInitIntrinsicParams2D函数中实现，外参数估计在cvFindExtrinsicCameraParams2函数中实现。本文对内参数估计原理进行解析，外参数估计在我之前的博客中已经做了较为全面的原理解析。
算法原理
opencv中对内参矩阵的定义为
式中的s表示CCD传感器横纵分布不垂直程度的系数，也就是传感器理想是矩形，但其实可能是具有剪切变形，是直角边不垂直的平行四边形。而Cx和Cy表示主点在图像中的位置，包含了主点偏差。在标定初始时，通常是考虑所有畸变量为零，也就是说此时的内参矩阵为
[ f x 0 u 0 0 f y v 0 0 0 1 ] (1) \begin{bmatrix} f_x & 0 & u_0 \\ 0 & f_y & v_0 \\ 0 & 0 & 1\end{bmatrix}\tag{1} ⎣ ⎡fx000fy0u0v01⎦ ⎤(1)式中， u 0 , v 0 u_0, v_0 u0,v0为图像宽度和高度的一半，单位为像素，含义是从像素坐标系转换到像平面坐标的偏移量。cvInitIntrinsicParams2D就是用于估计式（1）中焦距的值，前提是用于平面标定板。
参考张正友的文献【1】 假设世界坐标系在平面标定板上，Z轴垂直于标定平面，那么标定板上点的Z=0。根据投影仪方程表达为 λ [ u v 1 ] = A [ r 1 r 2 r 3 t ] [ X Y 0 1 ] = A [ r 1 r 2 t ] [ X Y 1 ] (2) \lambda \begin{bmatrix} u \\ v\\ 1\end{bmatrix}=\bf{A} \begin{bmatrix} \bf{r_1} & \bf{r_2}&\bf{r_3} &\bf{t}\end{bmatrix} \begin{bmatrix} X \\ Y\\0\\ 1\end{bmatrix}=\bf{A} \begin{bmatrix} \bf{r_1} & \bf{r_2}&\bf{t}\end{bmatrix} \begin{bmatrix} X \\ Y\\1\end{bmatrix}\tag{2} λ⎣ ⎡uv1⎦ ⎤=A[r1r2r3t]⎣ ⎡XY01⎦ ⎤=A[r1r2t]⎣ ⎡XY1⎦ ⎤(2)式中， A \bf{A} A表示式（1）的内参矩阵， r 1 , r 2 , r 3 , t \bf{r_1}, \bf{r_2},\bf{r_3} ,\bf{t} r1,r2,r3,t是外参矩阵中的列向量。由于标定板上三维点和图像上二维点各自都近似在平面上，利用平面单应性关系有 λ [ u v 1 ] = H [ X Y 1 ] (3) \lambda \begin{bmatrix} u \\ v\\ 1\end{bmatrix} = \bf{H}\begin{bmatrix} X \\ Y\\ 1\end{bmatrix}\tag{3} λ⎣ ⎡uv1⎦ ⎤=H⎣ ⎡XY1⎦ ⎤(3)在每个拍摄角度下，都能解算出一个单应性矩阵H，根据式（2）和式（3）可知， H = [ H 1 , H 2 , H 3 ] = λ A [ r 1 r 2 t ] (4) \bf{H} = \begin{bmatrix} \bf{H_1},\bf{H_2},\bf{H_3}\end{bmatrix} = \lambda \bf{A} \begin{bmatrix} \bf{r_1} & \bf{r_2}&\bf{t}\end{bmatrix} \tag{4} H=[H1,H2,H3]=λA[r1r2t](4)式中，H1，H2和H3表示单应性矩阵的三列，特别注意的是，后面的符号中， H 12 H_{12} H12不是表示的H矩阵第1行2列的元素，而是列向量 H 1 \bf{H_1} H1中的第二个元素，在矩阵中的位置是第1列第2行，这里和常规的表示不太一样。
根据式（4）可以得到外参矩阵中的列向量r1、r2，根据旋转矩阵单位正交的性质可知
r 1 T ⋅ r 2 = 0 (5) \bf{r_1}^T \cdot \bf{r_2}=0 \tag{5} r1T⋅r2=0(5) r 1 T ⋅ r 1 = r 2 T ⋅ r 2 = 1 (6) \bf{r_1}^T \cdot \bf{r_1}= \bf{r_2}^T \cdot \bf{r_2} = 1\tag{6} r1T⋅r1=r2T⋅r2=1(6)根据式（4）得到r1和r2，并带入上面两个式子得到
( A − 1 H 1 ) T ⋅ ( A − 1 H 2 ) = 0 (7) (\bf{A}^{-1}\bf{H_1})^T \cdot (\bf{A}^{-1}\bf{H_2})=0\tag{7} (A−1H1)T⋅(A−1H2)=0(7) ( A − 1 H 1 ) T ⋅ ( A − 1 H 1 ) = ( A − 1 H 2 ) T ⋅ ( A − 1 H 2 ) (8) (\bf{A}^{-1}\bf{H_1})^T \cdot (\bf{A}^{-1}\bf{H_1})= (\bf{A}^{-1}\bf{H_2})^T \cdot (\bf{A}^{-1}\bf{H_2})\tag{8} (A−1H1)T⋅(A−1H1)=(A−1H2)T⋅(A−1H2)(8)进一步简化得到 H 1 T A − T A − 1 H 2 = 0 (9) \bf{H_1^T } \bf{A}^{-T} \bf{A}^{-1}\bf{H_2}=0\tag{9} H1TA−TA−1H2=0(9) H 1 T A − T A − 1 H 1 = H 2 T A − T A − 1 H 2 (10) \bf{H_1^T } \bf{A}^{-T} \bf{A}^{-1}\bf{H_1} = \bf{H_2^T } \bf{A}^{-T} \bf{A}^{-1}\bf{H_2} \tag{10} H1TA−TA−1H1=H2TA−TA−1H2(10)式中， A − T \bf{A}^{-T} A−T表示 ( A − 1 ) T (\bf{A}^{-1})^T (A−1)T. 接下来就是利用式（9）和（10）来解算焦距 f x , f y f_x,f_y fx,fy。为了和代码对应，接下来的推导过程可能比较慢，公式比较长，这一段其实就是上述两个式子进一步带入具体的元素，来构建 A x = b Ax=b Ax=b的解算式。
文献【1】中令
在所有畸变量初始为零的情况下，上式简化为 B = [ 1 α 2 0 − u 0 α 2 0 1 β 2 − v 0 β 2 − u 0 α 2 − v 0 β 2 u 0 2 α 2 + v 0 2 β 2 + 1 ] (11) B=\begin{bmatrix} \frac{1}{\alpha^2} & 0 & -\frac{u_0}{\alpha^2} \\ 0 & \frac{1}{\beta^2} & -\frac{v_0}{\beta^2} \\ -\frac{u_0}{\alpha^2} & -\frac{v_0}{\beta^2} & \frac{u_0^2}{\alpha^2}+ \frac{v_0^2}{\beta^2}+1\end{bmatrix}\tag{11} B=⎣ ⎡α210−α2u00β21−β2v0−α2u0−β2v0α2u02+β2v02+1⎦ ⎤(11)论文中 α , β \alpha,\beta α,β对应我这里的 f x , f y f_x,f_y fx,fy，为了后面推导表达方便，令 B 1 = 1 α 2 , B 2 = 1 β 2 B_1=\frac{1}{\alpha^2},B_2=\frac{1}{\beta^2} B1=α21,B2=β21, 下面分两部分分别对式（9）和式（10）详细展开
式（9）的展开
[ H 11 H 12 H 13 ] [ B 1 0 − u 0 B 1 0 B 2 − v 0 B 2 − u 0 B 1 − v 0 B 2 u 0 2 B 1 + v 0 2 B 2 + 1 ] [ H 21 H 22 H 23 ] = 0 (12) \begin{bmatrix} H_{11} & H_{12}& H_{13} \end{bmatrix} \begin{bmatrix} B_1 & 0 & -u_0B_1 \\ 0 &B_2 & -v_0B_2 \\ -u_0B_1 & -v_0B_2 & u_0^2B_1+v_0^2B_2+1\end{bmatrix} \begin{bmatrix} H_{21} \\ H_{22}\\H_{23} \end{bmatrix}=0\tag{12} [H11H12H13]⎣ ⎡B10−u0B10B2−v0B2−u0B1−v0B2u02B1+v02B2+1⎦ ⎤⎣ ⎡H21H22H23⎦ ⎤=0(12) [ H 11 B 1 − u 0 H 13 B 1 H 12 B 2 − v 0 H 13 B 2 − u 0 H 11 B 1 − v 0 H 12 B 2 + u 0 2 H 13 B 1 + v 0 2 H 13 B 2 + H 13 ] [ H 21 H 22 H 23 ] = 0 (13) \begin{bmatrix} H_{11}B_1-u_0H_{13}B_1 & H_{12}B_2-v_0H_{13}B_2&-u_0H_{11}B_1-v_0H_{12}B_2+u_0^2H_{13}B_1+v_0^2H_{13}B_2+H_{13}\end{bmatrix} \begin{bmatrix} H_{21} \\ H_{22}\\H_{23} \end{bmatrix}=0\tag{13} [H11B1−u0H13B1H12B2−v0H13B2−u0H11B1−v0H12B2+u02H13B1+v02H13B2+H13]⎣ ⎡H21H22H23⎦ ⎤=0(13) ( H 21 H 11 B 1 − u 0 H 21 H 13 B 1 ) + ( H 22 H 12 B 2 − v 0 H 22 H 13 B 2 ) + ( − u 0 H 23 H 11 B 1 − v 0 H 23 H 12 B 2 + u 0 2 H 23 H 13 B 1 + v 0 2 H 23 H 13 B 2 + H 23 H 13 ) = 0 (14) (H_{21}H_{11}B_1-u_0H_{21}H_{13}B_1) \\ + (H_{22}H_{12}B_2-v_0H_{22}H_{13}B_2) \\ + (-u_0H_{23}H_{11}B_1-v_0H_{23}H_{12}B_2+u_0^2H_{23}H_{13}B_1+v_0^2H_{23}H_{13}B_2+H_{23}H_{13})=0\tag{14} (H21H11B1−u0H21H13B1)+(H22H12B2−v0H22H13B2)+(−u0H23H11B1−v0H23H12B2+u02H23H13B1+v02H23H13B2+H23H13)=0(14) B 1 ( H 21 H 11 − u 0 H 21 H 13 − u 0 H 23 H 11 + u 0 2 H 23 H 13 ) + B 2 ( H 22 H 12 − v 0 H 22 H 13 − v 0 H 23 H 12 + v 0 2 H 23 H 13 ) + H 23 H 13 = 0 (15) B_1(H_{21}H_{11}-u_0H_{21}H_{13}-u_0H_{23}H_{11}+u_0^2H_{23}H_{13}) \\ +B_2 (H_{22}H_{12}-v_0H_{22}H_{13}-v_0H_{23}H_{12}+v_0^2H_{23}H_{13}) \\ +H_{23}H_{13}=0\tag{15} B1(H21H11−u0H21H13−u0H23H11+u02H23H13)+B2(H22H12−v0H22H13−v0H23H12+v02H23H13)+H23H13=0(15) B 1 ( H 11 − u 0 H 13 ) ( H 21 − u 0 H 23 ) + B 2 ( H 12 − v 0 H 13 ) ( H 22 − v 0 H 23 ) + H 23 H 13 = 0 (16) B_1(H_{11}-u_0H_{13})(H_{21}-u_0H_{23}) \\ + B_2(H_{12}-v_0H_{13})(H_{22}-v_0H_{23}) \\ +H_{23}H_{13}=0\tag{16} B1(H11−u0H13)(H21−u0H23)+B2(H12−v0H13)(H22−v0H23)+H23H13=0(16)
式（10）的展开
和式（9）的推导相似，对式（16）可以直接替换下标，结果为
B 1 ( H 11 − u 0 H 13 ) ( H 11 − u 0 H 13 ) + B 2 ( H 12 − v 0 H 13 ) ( H 12 − v 0 H 13 ) + H 13 H 13 − B 1 ( H 21 − u 0 H 23 ) ( H 21 − u 0 H 23 ) − B 2 ( H 22 − v 0 H 23 ) ( H 22 − v 0 H 23 ) − H 23 H 23 = 0 (17) B_1(H_{11}-u_0H_{13})(H_{11}-u_0H_{13}) \\ + B_2(H_{12}-v_0H_{13})(H_{12}-v_0H_{13}) \\ +H_{13}H_{13} \\ -B_1(H_{21}-u_0H_{23})(H_{21}-u_0H_{23}) \\ - B_2(H_{22}-v_0H_{23})(H_{22}-v_0H_{23}) \\ -H_{23}H_{23}=0\tag{17} B1(H11−u0H13)(H11−u0H13)+B2(H12−v0H13)(H12−v0H13)+H13H13−B1(H21−u0H23)(H21−u0H23)−B2(H22−v0H23)(H22−v0H23)−H23H23=0(17) B 1 [ ( H 11 − u 0 H 13 ) 2 − ( H 21 − u 0 H 23 ) 2 ] + B 2 ( H 12 − v 0 H 13 ) 2 − ( H 22 − v 0 H 23 ) 2 + H 13 2 − H 23 2 = 0 (18) B_1[(H_{11}-u_0H_{13})^2-(H_{21}-u_0H_{23})^2] \\ + B_2(H_{12}-v_0H_{13})^2-(H_{22}-v_0H_{23})^2 \\ +H_{13}^2-H_{23}^2=0\tag{18} B1[(H11−u0H13)2−(H21−u0H23)2]+B2(H12−v0H13)2−(H22−v0H23)2+H132−H232=0(18)
式（9）和式（10）推导出的最终形式，联合式（16）和（18）并写出矩阵形式
[ ( H 11 − u 0 H 13 ) ( H 21 − u 0 H 23 ) ( H 12 − v 0 H 13 ) ( H 22 − v 0 H 23 ) ( H 11 − u 0 H 13 ) 2 − ( H 21 − u 0 H 23 ) 2 ( H 12 − v 0 H 13 ) 2 − ( H 22 − v 0 H 23 ) 2 ] [ B 1 B 2 ] = [ − H 23 H 13 − ( H 13 2 − H 23 2 ) ] (19) \begin{bmatrix} (H_{11}-u_0H_{13})(H_{21}-u_0H_{23}) &(H_{12}-v_0H_{13})(H_{22}-v_0H_{23}) \\ (H_{11}-u_0H_{13})^2-(H_{21}-u_0H_{23})^2 &(H_{12}-v_0H_{13})^2-(H_{22}-v_0H_{23})^2 \end{bmatrix} \begin{bmatrix} B_1\\B_2 \end{bmatrix} = \begin{bmatrix} -H_{23}H_{13} \\ -(H_{13}^2-H_{23}^2)\end{bmatrix} \tag{19} [(H11−u0H13)(H21−u0H23)(H11−u0H13)2−(H21−u0H23)2(H12−v0H13)(H22−v0H23)(H12−v0H13)2−(H22−v0H23)2][B1B2]=[−H23H13−(H132−H232)](19)这是典型的形如“Ax=b"的形式，能够容易的求解B1和B2，也就是得到了焦距
f x = 1 / B 1 , f y = 1 / B 2 (20) f_x=\sqrt{1/B_1}, f_y=\sqrt{1/B_2} \tag{20} fx=1/B1,fy=1/B2(20)为了和Opencv代码中符号一致，各变量整理为 h = t 0 = [ H 11 − u 0 H 13 H 12 − v 0 H 13 H 13 ] (21) \bf{h} = t_0=\begin{bmatrix} H_{11}-u_0H_{13} \\ H_{12}-v_0H_{13} \\ H_{13} \end{bmatrix}\tag{21} h=t0=⎣ ⎡H11−u0H13H12−v0H13H13⎦ ⎤(21) v = t 1 = [ H 21 − u 0 H 23 H 22 − v 0 H 23 H 23 ] (22) \bf{v}=t_1=\begin{bmatrix} H_{21}-u_0H_{23} \\ H_{22}-v_0H_{23} \\ H_{23} \end{bmatrix}\tag{22} v=t1=⎣ ⎡H21−u0H23H22−v0H23H23⎦ ⎤(22) d 1 = t 0 + t 1 = [ ( H 11 − u 0 H 13 ) + ( H 21 − u 0 H 23 ) ( H 12 − v 0 H 13 ) + ( H 22 − v 0 H 23 ) H 13 + H 23 ] (23) \bf{d_1}=t_0+t_1=\begin{bmatrix} (H_{11}-u_0H_{13}) + (H_{21}-u_0H_{23}) \\ (H_{12}-v_0H_{13}) + (H_{22}-v_0H_{23}) \\ H_{13} + H_{23} \end{bmatrix}\tag{23} d1=t0+t1=⎣ ⎡(H11−u0H13)+(H21−u0H23)(H12−v0H13)+(H22−v0H23)H13+H23⎦ ⎤(23) d 2 = t 0 − t 1 = [ ( H 11 − u 0 H 13 ) − ( H 21 − u 0 H 23 ) ( H 12 − v 0 H 13 ) − ( H 22 − v 0 H 23 ) H 13 − H 23 ] (24) \bf{d_2}=t_0-t_1=\begin{bmatrix} (H_{11}-u_0H_{13}) - (H_{21}-u_0H_{23}) \\ (H_{12}-v_0H_{13}) - (H_{22}-v_0H_{23}) \\ H_{13} - H_{23} \end{bmatrix}\tag{24} d2=t0−t1=⎣ ⎡(H11−u0H13)−(H21−u0H23)(H12−v0H13)−(H22−v0H23)H13−H23⎦ ⎤(24)使用式(21)~(24)的符号带入式（19）
[ h [ 0 ] ⋅ v [ 0 ] h [ 1 ] ⋅ v [ 1 ] d 1 [ 0 ] ⋅ d 2 [ 0 ] d 1 [ 1 ] ⋅ d 2 [ 1 ] ] [ B 1 B 2 ] = [ − h [ 2 ] ⋅ v [ 2 ] − d 1 [ 2 ] ⋅ d 2 [ 2 ] ] (25) \begin{bmatrix} \bf{h}[0]\cdot v[0] &\bf{h}[1]\cdot v[1] \\ \bf{d_1}[0] \cdot \bf{d_2}[0] &\bf{d_1}[1] \cdot \bf{d_2}[1] \end{bmatrix} \begin{bmatrix} B_1\\B_2 \end{bmatrix} = \begin{bmatrix} -\bf{h}[2]\cdot v[2] \\ -\bf{d_1}[2] \cdot \bf{d_2}[2] \end{bmatrix} \tag{25} [h[0]⋅v[0]d1[0]⋅d2[0]h[1]⋅v[1]d1[1]⋅d2[1]][B1B2]=[−h[2]⋅v[2]−d1[2]⋅d2[2]](25)令 A p = [ h [ 0 ] ⋅ v [ 0 ] h [ 1 ] ⋅ v [ 1 ] d 1 [ 0 ] ⋅ d 2 [ 0 ] d 1 [ 1 ] ⋅ d 2 [ 1 ] ] (26) \bf{Ap} = \begin{bmatrix} \bf{h}[0]\cdot v[0] &\bf{h}[1]\cdot v[1] \\ \bf{d_1}[0] \cdot \bf{d_2}[0] &\bf{d_1}[1] \cdot \bf{d_2}[1] \end{bmatrix} \tag{26} Ap=[h[0]⋅v[0]d1[0]⋅d2[0]h[1]⋅v[1]d1[1]⋅d2[1]](26) b p = [ − h [ 2 ] ⋅ v [ 2 ] − d 1 [ 2 ] ⋅ d 2 [ 2 ] ] (27) \bf{bp} = \begin{bmatrix} -\bf{h}[2]\cdot v[2] \\ -\bf{d_1}[2] \cdot \bf{d_2}[2] \end{bmatrix} \tag{27} bp=[−h[2]⋅v[2]−d1[2]⋅d2[2]](27)以上推导过程中与opencv不同的有两点，1是代码中对 h \bf{h} h， v \bf{v} v, d 1 \bf{d_1} d1和 d 2 \bf{d_2} d2做了归一化处理，应该是防止系数矩阵元素之间差别太大，导致病态问题；2是 d 1 = ( t 0 + t 1 ) ∗ 0.5 , d 2 = ( t 0 − t 1 ) ∗ 0.5 \bf{d_1}=(t_0+t_1)*0.5, \bf{d_2}=(t_0-t_1)*0.5 d1=(t0+t1)∗0.5,d2=(t0−t1)∗0.5这里的0.5在我推导中目前没有找到合理的解释。
代码注释
CV_IMPL void cvInitIntrinsicParams2D( const CvMat* objectPoints,
const CvMat* imagePoints, const CvMat* npoints,
CvSize imageSize, CvMat* cameraMatrix,
double aspectRatio )
{
Ptr<CvMat> matA, _b, _allH;
int i, j, pos, nimages, ni = 0;
// **a是内参矩阵
double a[9] = { 0, 0, 0, 0, 0, 0, 0, 0, 1 };
double H[9] = {0}, f[2] = {0};
CvMat _a = cvMat( 3, 3, CV_64F, a );
CvMat matH = cvMat( 3, 3, CV_64F, H );
CvMat _f = cvMat( 2, 1, CV_64F, f );
assert( CV_MAT_TYPE(npoints->type) == CV_32SC1 &&
CV_IS_MAT_CONT(npoints->type) );
nimages = npoints->rows + npoints->cols - 1;
if( (CV_MAT_TYPE(objectPoints->type) != CV_32FC3 &&
CV_MAT_TYPE(objectPoints->type) != CV_64FC3) ||
(CV_MAT_TYPE(imagePoints->type) != CV_32FC2 &&
CV_MAT_TYPE(imagePoints->type) != CV_64FC2) )
CV_Error( CV_StsUnsupportedFormat, "Both object points and image points must be 2D" );
if( objectPoints->rows != 1 || imagePoints->rows != 1 )
CV_Error( CV_StsBadSize, "object points and image points must be a single-row matrices" );
matA.reset(cvCreateMat( 2*nimages, 2, CV_64F ));
_b.reset(cvCreateMat( 2*nimages, 1, CV_64F ));
a[2] = (!imageSize.width) ? 0.5 : (imageSize.width)*0.5;
a[5] = (!imageSize.height) ? 0.5 : (imageSize.height)*0.5;
_allH.reset(cvCreateMat( nimages, 9, CV_64F ));
// extract vanishing points in order to obtain initial value for the focal length
for( i = 0, pos = 0; i < nimages; i++, pos += ni )
{
double* Ap = matA->data.db + i*4;
double* bp = _b->data.db + i*2;
ni = npoints->data.i[i];
double h[3], v[3], d1[3], d2[3];
double n[4] = {0,0,0,0};
CvMat _m, matM;
cvGetCols( objectPoints, &matM, pos, pos + ni );
cvGetCols( imagePoints, &_m, pos, pos + ni );
// **求解单应性矩阵H
cvFindHomography( &matM, &_m, &matH );
memcpy( _allH->data.db + i*9, H, sizeof(H) );
// **见式(21),(22)
H[0] -= H[6]*a[2]; H[1] -= H[7]*a[2]; H[2] -= H[8]*a[2];
H[3] -= H[6]*a[5]; H[4] -= H[7]*a[5]; H[5] -= H[8]*a[5];
for( j = 0; j < 3; j++ )
{
double t0 = H[j*3], t1 = H[j*3+1];
// **见式(21),(22),(23),(24)
h[j] = t0; v[j] = t1;
d1[j] = (t0 + t1)*0.5;
d2[j] = (t0 - t1)*0.5;
n[0] += t0*t0; n[1] += t1*t1;
n[2] += d1[j]*d1[j]; n[3] += d2[j]*d2[j];
}
// **求h,v,d1和d2的模
for( j = 0; j < 4; j++ )
n[j] = 1./std::sqrt(n[j]);
// **h,v,d1和d2归一化
for( j = 0; j < 3; j++ )
{
h[j] *= n[0]; v[j] *= n[1];
d1[j] *= n[2]; d2[j] *= n[3];
}
// **见式(25),(26),(27)
Ap[0] = h[0]*v[0]; Ap[1] = h[1]*v[1];
Ap[2] = d1[0]*d2[0]; Ap[3] = d1[1]*d2[1];
bp[0] = -h[2]*v[2]; bp[1] = -d1[2]*d2[2];
}
// **求解式(19),或者说(25)
cvSolve( matA, _b, &_f, CV_NORMAL + CV_SVD );
// **求解式(20)
a[0] = std::sqrt(fabs(1./f[0]));
a[4] = std::sqrt(fabs(1./f[1]));
// **若定义CCD传感器横纵分布不垂直程度的系数不为零，修正焦距
if( aspectRatio != 0 )
{
double tf = (a[0] + a[4])/(aspectRatio + 1.);
a[0] = aspectRatio*tf;
a[4] = tf;
}
cvConvert( &_a, cameraMatrix );
}
参考文献
[1] Zhang Z . A flexible new technique for camera calibration[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22.