c++ 的 eigen 类似于 python 的 numpy, 还有一个类似的库是 Armadillo, 当然还有 opencv.
Armadillo 与 matlab 在函数名称上更接近, 但是 TensorFlow 和 Ceres 使用了 eigen.
这里不讲究谁优谁劣, 入门阶段迅速掌握一个, 用起来就够了.
1. The Matrix Class
1) The first three template parameters of Matrix
Matrix<typename Scalar, int RowsAtCompileTime, int ColsAtCompileTime> typedef Matrix<float, , > Matrix4f;
2) Vectors
In Eigen, vectors are just a special case of matrices.
typedef Matrix<float, , > Vector3f; typedef Matrix<int, , > RowVector2i;
3) The special value dynamic
Matrices dimensions can be unknown at compile time in Eigen.
typedef Matrix<double, Dynamic, Dynamic> MatrixXd; typedef Matrix<int, Dynamic, > VectorXi; Matrix<float, , Dynamic>;
4) Constructors
Matrix3f a; MatrixXf b; MatrixXf a(, ); MatrixXf b(); Matrix3f a(, ); Vector2d a(5.0, 6.0); Vector3d b(5.0, 6.0, 7.0);
Matrix3f 声明的矩阵, 大小是固定的, 不能修改. 4维之下的矩阵可以使用形如 Matrix4f 的形式, 稍微增加编译时间, 但是执行更快.
5) Coefficient accessors
int main()
{
MatrixXd m(,);
m(,) = ;
m(,) = 2.5;
m(,) = -;
m(,) = m(,) + m(,);
std::cout << "Here is the matrix m:\n" << m << std::endl;
VectorXd v();
v() = ;
v() = v() - ;
std::cout << "Here is the vector v:\n" << v << std::endl;
}
m(index) 除了可以访问 vector, 还可以访问 matrix, matrix 在 Eigen 中默认按列存储. 在上例中, m(2) == -1.
[] 也被重载了, 类似 () 的功能, 但是只能有一个 index, 因为 c++ 的特性, matrix[i, j] == matrix[j].
推测, 访问单独或批量的列或行, 应该有专有的函数, 用 () 和 [] 只能访问单个值.
6) Comma-initialization
Matrix3f m;
m << , , ,
, , ,
, , ;
cout << m;
7) Resizing
int main()
{
MatrixXd m(,);
m.resize(,);
std::cout << "The matrix m is of size "
<< m.rows() << "x" << m.cols() << std::endl;
std::cout << "It has " << m.size() << " coefficients" << std::endl;
VectorXd v();
v.resize();
std::cout << "The vector v is of size " << v.size() << std::endl;
std::cout << "As a matrix, v is of size "
<< v.rows() << "x" << v.cols() << std::endl;
}
resize() 会覆盖之前矩阵的值, 如果要保留原有未知的值, 可以使用 conservativeResize().
再次强调, 形如 Matrix4d 声明的矩阵大小不可更改.
8) Assignment and resizing
MatrixXf a(,);
std::cout << "a is of size " << a.rows() << "x" << a.cols() << std::endl;
MatrixXf b(,);
a = b;
std::cout << "a is now of size " << a.rows() << "x" << a.cols() << std::endl;
如果等号左侧是 fixed size, 则 resizing 不被允许.
9) Convenience typedefs
MatrixNt for Matrix<type, N, N>
VectorNt for Matrix<type, N, 1>
RowVectorNt for Matrix<type, 1, N>
N can be 2, 3,... or X (Dynamic)
t can be i (int), f (float), d (double), cf (complex<float>), cd (complex<double>)
2. Matrix and vector arithmetic
1) Introduction
Eigen 重载了 matrix 之间的运算符号, 但是 matrix 和 scalar 之间的运算符号未重载.
2) Addition and subtraction
Eigen 不会对 scalar 进行自动类型转换.
a + b, a - b, -a, a += b, a -= b
int main()
{
Matrix2d a;
a << , ,
, ;
MatrixXd b(,);
b << , ,
, ;
std::cout << "a + b =\n" << a + b << std::endl;
std::cout << "a - b =\n" << a - b << std::endl;
std::cout << "Doing a += b;" << std::endl;
a += b;
std::cout << "Now a =\n" << a << std::endl;
Vector3d v(,,);
Vector3d w(,,);
std::cout << "-v + w - v =\n" << -v + w - v << std::endl;
}
3) Scalar multiplication and division
matrix * scalar, scalar * matrix, matrix / scalar, matrix *= scalar, matrix /= scalar
int main()
{
Matrix2d a;
a << , ,
, ;
Vector3d v(,,);
std::cout << "a * 2.5 =\n" << a * 2.5 << std::endl;
std::cout << "0.1 * v =\n" << 0.1 * v << std::endl;
std::cout << "Doing v *= 2;" << std::endl;
v *= ;
std::cout << "Now v =\n" << v << std::endl;
}
4) A note about expression templates
简单说就是 Eigen 对 arithmetic expression 有进行优化, 但这是它应该做的.
5) Transposition and conjugation
transpose() 转置
MatrixXcf a = MatrixXd::Random(,);
cout << "Here is the matrix a\n" << a << endl;
cout << "Here is the matrix a^T\n" << a.transpose() << endl;
不能使用 a = a.transpose(), a 的值会发生错乱, 要使用 a = a.transposeInPlace().
Matrix2i a; a << , , , ;
cout << "Here is the matrix a:\n" << a << endl;
a = a.transpose(); // !!! do NOT do this !!!
cout << "and the result of the aliasing effect:\n" << a << endl;
a.transposeInPlace();
cout << "and after being transposed:\n" << a << endl;
6) Matrix-matrix and matrix-vector multiplication
a * b, a *= b
对于 a *= b, 如果 size 会发生变化, 则 a 不能是 fixed size
int main()
{
Matrix2d mat;
mat << , ,
, ;
Vector2d u(-,), v(,);
std::cout << "Here is mat*mat:\n" << mat*mat << std::endl;
std::cout << "Here is mat*u:\n" << mat*u << std::endl;
std::cout << "Here is u^T*mat:\n" << u.transpose()*mat << std::endl;
std::cout << "Here is u^T*v:\n" << u.transpose()*v << std::endl;
std::cout << "Here is u*v^T:\n" << u*v.transpose() << std::endl;
std::cout << "Let's multiply mat by itself" << std::endl;
mat = mat*mat;
std::cout << "Now mat is mat:\n" << mat << std::endl;
}
7) Dot product and cross product
dot() 点积
cross() 叉积
int main()
{
Vector3d v(,,);
Vector3d w(,,);
cout << "Dot product: " << v.dot(w) << endl;
double dp = v.adjoint()*w; // automatic conversion of the inner product to a scalar
cout << "Dot product via a matrix product: " << dp << endl;
cout << "Cross product:\n" << v.cross(w) << endl;
}
8) Basic arithmetic reduction operations
sum(), prod(), mean(), minCoeff(), maxCoeff(), trace()
int main()
{
Eigen::Matrix2d mat;
mat << , ,
, ;
cout << "Here is mat.sum(): " << mat.sum() << endl;
cout << "Here is mat.prod(): " << mat.prod() << endl;
cout << "Here is mat.mean(): " << mat.mean() << endl;
cout << "Here is mat.minCoeff(): " << mat.minCoeff() << endl;
cout << "Here is mat.maxCoeff(): " << mat.maxCoeff() << endl;
cout << "Here is mat.trace(): " << mat.trace() << endl;
}
a.trace() == a.diagonal().sum()
minCoeff() 和 maxCoeff() 除了返回最小最大值, 还可以获取该值的索引
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;
9) Validity of operations
编译时和运行时都会检查运算的合法性
Matrix3f m;
Vector4f v;
v = m*v; // Compile-time error: YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES MatrixXf m(,);
VectorXf v();
v = m * v; // Run-time assertion failure here: "invalid matrix product"
3. The array class and coefficient-wise operations
1) Array types
Array<typename Scalar, int RowsAtCompileTime, int ColsAtCompileTime>
可以有多维数组
Array<float, Dynamic, 1> ArrayXf
Array<float, 3, 1> Array3f
Array<double, Dynamic, Dynamic> ArrayXXd
Array<double, 3, 3> Array33d
2) Accessing values inside an Array
int main()
{
ArrayXXf m(,); // assign some values coefficient by coefficient
m(,) = 1.0; m(,) = 2.0;
m(,) = 3.0; m(,) = m(,) + m(,); // print values to standard output
cout << m << endl << endl; // using the comma-initializer is also allowed
m << 1.0,2.0,
3.0,4.0; // print values to standard output
cout << m << endl;
}
3) Addition and subtraction
array + array, array - scalar
int main()
{
ArrayXXf a(,);
ArrayXXf b(,);
a << ,,,
,,,
,,;
b << ,,,
,,,
,,; // Adding two arrays
cout << "a + b = " << endl << a + b << endl << endl;
// Subtracting a scalar from an array
cout << "a - 2 = " << endl << a - << endl;
}
4) Array multiplicaiton
区分于矩阵乘法, 对应位置参数相乘.
int main()
{
ArrayXXf a(,);
ArrayXXf b(,);
a << ,,
,;
b << ,,
,;
cout << "a * b = " << endl << a * b << endl;
}
5) Other coefficient-wise operations
abs(), sqrt(), min() 两个 array 对应位置取最小
int main()
{
ArrayXf a = ArrayXf::Random();
a *= ;
cout << "a =" << endl
<< a << endl;
cout << "a.abs() =" << endl
<< a.abs() << endl;
cout << "a.abs().sqrt() =" << endl
<< a.abs().sqrt() << endl;
cout << "a.min(a.abs().sqrt()) =" << endl
<< a.min(a.abs().sqrt()) << endl;
}
6) Converting between array and matrix expressions
matrix 有一个 array() 方法, array 有一个 matrix 方法, .array() 和 .matrix() 的返回值可以用作左值或右值.
matrix 有一个 cwiseProduct() 方法, 用于两个 matrix 对应参数相乘.
Eigen 允许将 array expression 赋值给 matrix 变量.
int main()
{
MatrixXf m(,);
MatrixXf n(,);
MatrixXf result(,);
m << ,,
,;
n << ,,
,;
result = m * n;
cout << "-- Matrix m*n: --" << endl << result << endl << endl;
result = m.array() * n.array();
cout << "-- Array m*n: --" << endl << result << endl << endl;
result = m.cwiseProduct(n);
cout << "-- With cwiseProduct: --" << endl << result << endl << endl;
result = m.array() + ;
cout << "-- Array m + 4: --" << endl << result << endl << endl;
}
稍微复杂点的栗子
int main()
{
MatrixXf m(,);
MatrixXf n(,);
MatrixXf result(,);
m << ,,
,;
n << ,,
,; result = (m.array() + ).matrix() * m;
cout << "-- Combination 1: --" << endl << result << endl << endl;
result = (m.array() * n.array()).matrix() * m;
cout << "-- Combination 2: --" << endl << result << endl << endl;
}