eigen 笔记1

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;
}