8点算法计算基础矩阵(计算机视觉)
源自OpenCV中的代码：
int CvFMEstimator::run8Point( const CvMat* _m1, const CvMat* _m2, CvMat* _fmatrix ) { double a[9*9], w[9], v[9*9]; CvMat W = cvMat( 1, 9, CV_64F, w ); CvMat V = cvMat( 9, 9, CV_64F, v ); CvMat A = cvMat( 9, 9, CV_64F, a ); CvMat U, F0, TF; CvPoint2D64f m0c = {0,0}, m1c = {0,0}; double t, scale0 = 0, scale1 = 0; const CvPoint2D64f* m1 = (const CvPoint2D64f*)_m1->data.ptr; const CvPoint2D64f* m2 = (const CvPoint2D64f*)_m2->data.ptr; double* fmatrix = _fmatrix->data.db; int i, j, k, count = _m1->cols*_m1->rows; // compute centers and average distances for each of the two point sets for( i = 0; i < count; i++ ) { double x = m1[i].x, y = m1[i].y; m0c.x += x; m0c.y += y; x = m2[i].x, y = m2[i].y; m1c.x += x; m1c.y += y; } // calculate the normalizing transformations for each of the point sets: // after the transformation each set will have the mass center at the coordinate origin // and the average distance from the origin will be ~sqrt(2). t = 1./count; m0c.x *= t; m0c.y *= t; m1c.x *= t; m1c.y *= t; for( i = 0; i < count; i++ ) { double x = m1[i].x - m0c.x, y = m1[i].y - m0c.y; scale0 += sqrt(x*x + y*y); x = fabs(m2[i].x - m1c.x), y = fabs(m2[i].y - m1c.y); scale1 += sqrt(x*x + y*y); } scale0 *= t; scale1 *= t; if( scale0 < FLT_EPSILON || scale1 < FLT_EPSILON ) return 0; scale0 = sqrt(2.)/scale0; scale1 = sqrt(2.)/scale1; cvZero( &A ); // form a linear system Ax=0: for each selected pair of points m1 & m2, // the row of A(=a) represents the coefficients of equation: (m2, 1)'*F*(m1, 1) = 0 // to save computation time, we compute (At*A) instead of A and then solve (At*A)x=0. for( i = 0; i < count; i++ ) { double x0 = (m1[i].x - m0c.x)*scale0; double y0 = (m1[i].y - m0c.y)*scale0; double x1 = (m2[i].x - m1c.x)*scale1; double y1 = (m2[i].y - m1c.y)*scale1; double r[9] = { x1*x0, x1*y0, x1, y1*x0, y1*y0, y1, x0, y0, 1 }; for( j = 0; j < 9; j++ ) for( k = 0; k < 9; k++ ) a[j*9+k] += r[j]*r[k]; } cvSVD( &A, &W, 0, &V, CV_SVD_MODIFY_A + CV_SVD_V_T ); for( i = 0; i < 8; i++ ) { if( fabs(w[i]) < DBL_EPSILON ) break; } if( i < 7 ) return 0; F0 = cvMat( 3, 3, CV_64F, v + 9*8 ); // take the last column of v as a solution of Af = 0 // make F0 singular (of rank 2) by decomposing it with SVD, // zeroing the last diagonal element of W and then composing the matrices back. // use v as a temporary storage for different 3x3 matrices W = U = V = TF = F0; W.data.db = v; U.data.db = v + 9; V.data.db = v + 18; TF.data.db = v + 27; cvSVD( &F0, &W, &U, &V, CV_SVD_MODIFY_A + CV_SVD_U_T + CV_SVD_V_T ); W.data.db[8] = 0.; // F0 <- U*diag([W(1), W(2), 0])*V' cvGEMM( &U, &W, 1., 0, 0., &TF, CV_GEMM_A_T ); cvGEMM( &TF, &V, 1., 0, 0., &F0, 0/*CV_GEMM_B_T*/ ); // apply the transformation that is inverse // to what we used to normalize the point coordinates { double tt0[] = { scale0, 0, -scale0*m0c.x, 0, scale0, -scale0*m0c.y, 0, 0, 1 }; double tt1[] = { scale1, 0, -scale1*m1c.x, 0, scale1, -scale1*m1c.y, 0, 0, 1 }; CvMat T0, T1; T0 = T1 = F0; T0.data.db = tt0; T1.data.db = tt1; // F0 <- T1'*F0*T0 cvGEMM( &T1, &F0, 1., 0, 0., &TF, CV_GEMM_A_T ); F0.data.db = fmatrix; cvGEMM( &TF, &T0, 1., 0, 0., &F0, 0 ); // make F(3,3) = 1 if( fabs(F0.data.db[8]) > FLT_EPSILON ) cvScale( &F0, &F0, 1./F0.data.db[8] ); } return 1; }