Eigen 是基于C++開發(fā)代數(shù)的一個模板庫：矩陣控乾、矢量么介、數(shù)值解算器和相關(guān)算法娜遵。相比較Matlab，優(yōu)勢是利于基于c++的3D相機(jī)開發(fā)
(大部分3D相機(jī)SDK都支持c++)壤短，劣勢是語法較復(fù)雜设拟。本文目標(biāo)是針對3D相機(jī)手眼標(biāo)定過程中有關(guān)矩陣算術(shù)的eigen庫運(yùn)用進(jìn)行學(xué)習(xí)。
背景
Eigen 是基于C++開發(fā)代數(shù)的一個模板庫：矩陣久脯、矢量纳胧、數(shù)值解算器和相關(guān)算法。相比較Matlab帘撰，優(yōu)勢是利于c++開發(fā)跑慕，劣勢是語法較復(fù)雜
環(huán)境配置
1.下載eigen源碼包，可解壓到任意位置
2.新建vc++工程摧找，項(xiàng)目屬性 -> C/C++ -> 常規(guī) -> 附加包含目錄 -> 編輯 -> 新建路徑 -> 選擇eigen文件夾所在路徑 -> 運(yùn)行下面的demo
Demo代碼
https://www.cnblogs.com/winslam/p/12765822.html
Matrix類介紹
Matrix類模板6個參數(shù)
//共6個
Matrix int RowsAtCompileTime, int ColsAtCompileTime, int Options = 0, //默認(rèn)是按列存儲數(shù)據(jù)核行，可改成按行 int MaxRowsAtCompileTime = RowsAtCompileTime, //行數(shù)上限 int MaxColsAtCompileTime = ColsAtCompileTime> //列數(shù)上限 Matrinx類模板前三個參數(shù) 數(shù)據(jù)類型，行數(shù)蹬耘，列數(shù)芝雪，eigen已經(jīng)定義好了常用的，規(guī)律很好找 Matrix 示例 typedef Matrix typedef Matrix 特殊的Matrix類：Vector向量 是特殊的Matrix類综苔，只有一行或一列惩系，定義如下 typedef Matrix typedef Matrix 動態(tài)創(chuàng)建Matrix類對象 typedef Matrix typedef Matrix Matrix Matrix類構(gòu)造函數(shù) 默認(rèn)構(gòu)造函數(shù)不會動態(tài)分配內(nèi)存 對矩陣傳參總是優(yōu)先傳行數(shù) MatrixXf a(3,2); //3行2列 而給向量傳參 = 給向量傳大小： Vector2d a(5.0, 6.0); Vector3d b(5.0, 6.0, 7.0); Vector4d c(5.0, 6.0, 7.0, 8.0); 訪問/初始化 Matrix元素 Matrix數(shù)據(jù)存儲順序如筛，總是先列后行 逗號初始化堡牡，總是先行后列，但數(shù)據(jù)存儲順序還是不變 Vector無所謂杨刨，使用.transform()方法晤柄，即可轉(zhuǎn)換行列 盡量用固定大小的Matrix，內(nèi)存機(jī)制沒深究 Matrix&Vector 運(yùn)算 加減法 原則很簡單拭嫁，相同行列才能運(yùn)算： binary operator + as in a+b binary operator - as in a-b unary operator - as in -a compound operator += as in a+=b compound operator -= as in a-=b 注意：運(yùn)算符已被重載可免，不可2個以上的矩陣同時參與運(yùn)算抓于。此時用遍歷矩陣索引 VectorXf a(50),b(50),c(50),d(50); a = 3*b + 4*c + 5*d; 改成 for(int i = 0;i < 50; + + i){ a[i] = 3*b[i] + 4*c[i] + 5*d[i];} 系數(shù)乘除法 binary operator * as in matrix*scalar binary operator * as in scalar*matrix binary operator / as in matrix/scalar compound operator *= as in matrix*=scalar compound operator /= as in matrix/=scalar Matrix 轉(zhuǎn)置，共軛浇借，逆 //n階正交矩陣a特性： a*=aT,無共軛 Eigen::MatrixXd t = Eigen::MatrixXd::Random(3,3); cout << t.transpose() << endl; //a的轉(zhuǎn)置 cout << t.conjugate() << endl; //a的共軛 cout << t.adjoint() << endl; //a的逆 Matrix 乘法 代碼 #include #include int main() { using namespace Eigen; Matrix2d mat;mat << 1, 2, 3, 4;//運(yùn)算符重載捉撮，矩陣按行接收數(shù)據(jù)，但是儲存機(jī)制依舊優(yōu)先按列妇垢，即mat經(jīng)過上述賦值后為 [1,2 // 3,4]巾遭，但取值時，mat(1,0)值為3闯估，mat(0,1)值為2 Vector2d v1(-1, 1), v2(2, 0); //默認(rèn)列向量 cout << "mat * v1:" << endl<< mat * v1 << endl; cout << endl; cout << "mat * v2:" << endl << mat * v2 << endl; cout << endl; cout << "mat * mat:" << endl << mat * mat << endl; } 輸出 mat * v1: 1 1 mat * v2: 2 6 mat * mat: 7 10 15 22 Vector的點(diǎn)乘和叉乘 點(diǎn)乘灼舍。純代數(shù)運(yùn)算，適用與任意長度的向量涨薪，前提是2個向量長度相等 /* 計(jì)算公式: a(a1,a2,a3) , b(b1,b2,b3) a·b = a1*b1 + a2*b2 + a3*b3 幾何意義: a·b =|a|*|b|*cosθ */ 叉乘骑素。用于空間幾何，所以只適用于長度為3的向量 /* 幾何意義 axb =|a|*|b|*sinθ 注意：叉乘結(jié)果是向量刚夺，方向在z軸上献丑，θ表示向量a到向量b的角度，右手法則（從a到b）確定z朝向 */ 代碼 #include #include int main() { using namespace Eigen; Vector3d v1(1, 2, 3); Vector3d v2(1, 1, 2); //點(diǎn)積方法1：用同維度的向量做點(diǎn)積 cout << "v1 點(diǎn)乘 v2 :" << endl << v.dot(v2) << endl; //點(diǎn)積方法2：向量轉(zhuǎn)化為矩陣乘積來做點(diǎn)積 cout << "v1逆 點(diǎn)乘 v2 :" << endl << v.adjoint() * w << endl; //叉乘侠姑，外積 cout << "v1 叉乘 v2 :" << endl << v.cross(v2) << endl; return 1; } 輸出 v1 點(diǎn)乘 v2 : 9 v1 叉乘 v2 : 1 1 -1 v1逆 點(diǎn)乘 v2 : 9 Matrix內(nèi)部數(shù)據(jù)算術(shù) 代碼 #include #include using namespace std; int main() { Eigen::Matrix2d mat; mat << 1, 2, 3, 4; cout << "Here is mat.sum(): " << mat.sum() << endl;//求和 cout << "Here is mat.prod(): " << mat.prod() << endl;//連乘 cout << "Here is mat.mean(): " << mat.mean() << endl;//均值 cout << "Here is mat.minCoeff(): " << mat.minCoeff() << endl;//最小值 cout << "Here is mat.maxCoeff(): " << mat.maxCoeff() << endl;//最大值 //主對角線系數(shù)和 cout << "Here is mat.trace(): " << mat.trace() << endl; //某些函數(shù)可以重載 Matrix3f m = Matrix3f::Random(); std::ptrdiff_t i, j; float minOfM = m.minCoeff(&i,&j); cout << "Here is the matrix m:\n" << m << endl; cout << "Its minimum coefficient (" << minOfM << ") is at position (" << i << "," << j << ")\n\n"; RowVector4i v = RowVector4i::Random(); int maxOfV = v.maxCoeff(&i); cout << "Here is the vector v: " << v << endl; cout << "Its maximum coefficient (" << maxOfV << ") is at position " << i << endl; } 輸出 Here is mat.sum(): 10 Here is mat.prod(): 24 Here is mat.mean(): 2.5 Here is mat.minCoeff(): 1 Here is mat.maxCoeff(): 4 Here is mat.trace(): 5 Here is the matrix m: -1 -0.0827 -0.906 -0.737 0.0655 0.358 0.511 -0.562 0.359 Its minimum coefficient (-1) is at position (0,0) Here is the vector v: 9 -2 0 7 Its maximum coefficient (9) is at position 0 Matrix類與Array類互換 矩陣運(yùn)算用矩陣類创橄，系數(shù)運(yùn)算用數(shù)組類，互相轉(zhuǎn)換用.matrix()方法 和 .array()方法 求方陣的行列式值 Tatrix3d mat;mat << 1,2,3,4,5,6,7,8,9; //3階方陣 double result = matrix.determinant(); Matrix初始化方法 前面提到過的逗號初始化 .Zero()方法初始化所有系數(shù)為0 .Random()方法用隨機(jī)系數(shù)填充矩陣或數(shù)組 .Identity()方法初始化一個單位矩陣 用分塊矩陣構(gòu)造成一個大矩陣 代碼 const int size = 6; MatrixXd mat1(size, size); mat1.topLeftCorner(size/2, size/2) = MatrixXd::Zero(size/2, size/2); mat1.topRightCorner(size/2, size/2) = MatrixXd::Identity(size/2, size/2); mat1.bottomLeftCorner(size/2, size/2) = MatrixXd::Identity(size/2, size/2); mat1.bottomRightCorner(size/2, size/2) = MatrixXd::Zero(size/2, size/2); std::cout << mat1 << std::endl << std::endl; MatrixXd mat2(size, size); mat2.topLeftCorner(size/2, size/2).setZero(); mat2.topRightCorner(size/2, size/2).setIdentity(); mat2.bottomLeftCorner(size/2, size/2).setIdentity(); mat2.bottomRightCorner(size/2, size/2).setZero(); std::cout << mat2 << std::endl << std::endl; MatrixXd mat3(size, size); mat3 << MatrixXd::Zero(size/2, size/2), MatrixXd::Identity(size/2, size/2), MatrixXd::Identity(size/2, size/2), MatrixXd::Zero(size/2, size/2); std::cout << mat3 << std::endl; 輸出 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 構(gòu)造齊次矩陣（4*4） 旋轉(zhuǎn)向量莽红，旋轉(zhuǎn)矩陣妥畏，歐拉角，四元數(shù)4者互換 https://blog.csdn.net/yang__jing/article/details/82316093?utm_medium=distribute.pc_relevant.none-task-blog-BlogCommendFromMachineLearnPai2-2.control&depth_1-utm_source=distribute.pc_relevant.none-task-blog-BlogCommendFromMachineLearnPai2-2.control 四元數(shù)注意點(diǎn) 輸入順序是[w,x,y,z] 安吁，其中w是實(shí)數(shù)部分 儲存和輸出順序是[x,y,z,w] 醉蚁，其中w是實(shí)數(shù)部分 互為相反數(shù)的2組四元數(shù)效果一樣 歐拉角注意點(diǎn) pose轉(zhuǎn)matrix默認(rèn)用'gba'形式（ R = Rx * Ry * Rz），T1.rotation().eulerAngles(0, 1, 2) 線性代數(shù)求解Ax=b形式方程 QR分解法： 代碼 #include #include using namespace std; using namespace Eigen; int main() { Matrix3f A; Vector3f b; A << 1,2,3, 4,5,6, 7,8,10; b << 3, 3, 4; cout << "Here is the matrix A:\n" << A << endl; cout << "Here is the vector b:\n" << b << endl; Vector3f x = A.colPivHouseholderQr().solve(b); cout << "The solution is:\n" << x << endl; } 輸出 Here is the matrix A: 1 2 3 4 5 6 7 8 10 Here is the vector b: 3 3 4 The solution is: -2 1 1 bdcSVD分解法 最精確但速度最慢的求解方法柳畔，用于求解線性方程馍管，在沒有解的情況下能逼近 代碼 #include #include using namespace std; using namespace Eigen; int main() { MatrixXf A = MatrixXf::Random(3, 2); cout << "Here is the matrix A:\n" << A << endl; VectorXf b = VectorXf::Random(3); cout << "Here is the right hand side b:\n" << b << endl; cout << "The least-squares solution is:\n" << A.bdcSvd(ComputeThinU | ComputeThinV).solve(b) << endl; } 您可以通過我們的官方網(wǎng)站了解更多的產(chǎn)品信息，或直接來電咨詢4006-888-532薪韩。
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