Eigen之矩阵、向量、标量的操作运算

本文详细介绍了Eigen库中矩阵和向量的各种运算方法,包括加减运算、标量乘除、矩阵乘法、转置、点乘和叉乘等,并提供了丰富的示例代码。

摘要生成于 C知道 ,由 DeepSeek-R1 满血版支持, 前往体验 >

介绍

Eigen是通过中重载C++操作运算符如+、-、*或通过dot()、cross()等来实现矩阵/向量的操作运算

加法和减法

  • 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

例子:

#include <iostream>
#include <Eigen/Dense>
using namespace Eigen;
int main()
{
  Matrix2d a;
  a << 1, 2,
       3, 4;
  MatrixXd b(2,2);
  b << 2, 3,
       1, 4;
  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(1,2,3);
  Vector3d w(1,0,0);
  std::cout << "-v + w - v =\n" << -v + w - v << std::endl;
}

Output:

a + b =
3 5
4 8
a - b =
-1 -1
 2  0
Doing a += b;
Now a =
3 5
4 8
-v + w - v =
-1
-4
-6

标量的乘除

  • 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
    例子:
#include <iostream>
#include <Eigen/Dense>
using namespace Eigen;
int main()
{
  Matrix2d a;
  a << 1, 2,
       3, 4;
  Vector3d v(1,2,3);
  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 *= 2;
  std::cout << "Now v =\n" << v << std::endl;
}

Output:

a * 2.5 =
2.5   5
7.5  10
0.1 * v =
0.1
0.2
0.3
Doing v *= 2;
Now v =
2
4
6

A note about expression templates

这是我们在这个页面上解释的一个高级主题,但是现在只提到它是很有用的。在Eigen,算术运算符,比如运算符+不执行任何计算,他们只是返回一个描述计算的“表达式对象”。实际的计算发生在稍后,当整个表达式被求值时,通常在操作符=。虽然这听起来可能很重,但是任何现代的优化编译器都能够优化抽象,结果是完美的优化代码。例如,当你这样做的时候:

VectorXf a(50), b(50), c(50), d(50);
...
a = 3*b + 4*c + 5*d;

Eigen将其编译为一个for循环,这样数组就只遍历一次。简化(例如忽略SIMD优化),这个循环是这样的:

for(int i = 0; i < 50; ++i)
{
  a[i] = 3*b[i] + 4*c[i] + 5*d[i];
}

因此,你不应该害怕使用Eigen的相对大的算术表达式:这只给了Eigen更多的机会来优化。

矩阵的转置、共轭及伴随

例:

MatrixXcf a = MatrixXcf::Random(2,2);
cout << "Here is the matrix a\n" << a << endl;
cout << "Here is the matrix a^T\n" << a.transpose() << endl;
cout << "Here is the conjugate of a\n" << a.conjugate() << endl;
cout << "Here is the matrix a^*\n" << a.adjoint() << endl;

结果:

Here is the matrix a
 (-0.211,0.68) (-0.605,0.823)
 (0.597,0.566)  (0.536,-0.33)
Here is the matrix a^T
 (-0.211,0.68)  (0.597,0.566)
(-0.605,0.823)  (0.536,-0.33)
Here is the conjugate of a
 (-0.211,-0.68) (-0.605,-0.823)
 (0.597,-0.566)    (0.536,0.33)
Here is the matrix a^*
 (-0.211,-0.68)  (0.597,-0.566)
(-0.605,-0.823)    (0.536,0.33)

注:对于一个实数矩阵,conjugate()是没有进行什么操作的,因此adjoint() = transpose()。
对于基本运算操作,transpose()和adjoint() 都仅是返回一个中间变量而没有进行实际的值改变。如果运算时b = a.transpose(),则结果值会赋值给b,但是如果进行的运算为a = a.transpose(),则Eigen会在转置完成之前将值赋值给a,因此无法达到期望的值。
例:

Matrix2i a; a << 1, 2, 3, 4;
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;

结果:

Here is the matrix a:
1 2
3 4
and the result of the aliasing effect:
1 2
2 4

这种现象被称为混叠问题(aliasing issue)
为解决该问题,可以使用transposeInPlace()函数来完成:

MatrixXf a(2,3); a << 1, 2, 3, 4, 5, 6;
cout << "Here is the initial matrix a:\n" << a << endl;
a.transposeInPlace();
cout << "and after being transposed:\n" << a << endl;

结果:

Here is the initial matrix a:
1 2 3
4 5 6
and after being transposed:
1 4
2 5
3 6

矩阵与矩阵、矩阵与向量间的乘法

  • binary operator * as in a*b
  • compound operator = as in a=b (this multiplies on the right: a*=b
    is equivalent to a = a*b)
    例:
#include <iostream>
#include <Eigen/Dense>
using namespace Eigen;
int main()
{
  Matrix2d mat;
  mat << 1, 2,
         3, 4;
  Vector2d u(-1,1), v(2,0);
  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;
}

结果:

Here is mat*mat:
 7 10
15 22
Here is mat*u:
1
1
Here is u^T*mat:
2 2
Here is u^T*v:
-2
Here is u*v^T:
-2 -0
 2  0
Let's multiply mat by itself
Now mat is mat:
 7 10
15 22

提示:对于矩阵的乘法不会出现上述的混叠问题,因为Eigen将乘法作为一个特例

tmp = m*m;
m = tmp;

点乘和叉乘

Eigen中通过dot()和cross()来实现点乘和叉乘
例:

#include <iostream>
#include <Eigen/Dense>
using namespace Eigen;
using namespace std;
int main()
{
  Vector3d v(1,2,3);
  Vector3d w(0,1,2);
  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;
}

结果:

Dot product: 8
Dot product via a matrix product: 8
Cross product:
 1
-2
 1

其他基础操作

Eigen还提供了一些函数实现一些其他基本操作,如sum()进行求和,prod()实现矩阵内值的积,maxCoeff()求矩阵内最大值等。
例:

#include <iostream>
#include <Eigen/Dense>
using namespace std;
int main()
{
  Eigen::Matrix2d mat;
  mat << 1, 2,
         3, 4;
  cout << "Here is mat.sum():       " << mat.sum()       << endl;
  cout << "Here is mat.prod():      " << mat.prod()      << endl;
  cout << "Here is mat.mean():      " << mat.mean()      << endl;  // 求均值
  cout << "Here is mat.minCoeff():  " << mat.minCoeff()  << endl;
  cout << "Here is mat.maxCoeff():  " << mat.maxCoeff()  << endl;
  cout << "Here is mat.trace():     " << mat.trace()     << endl;   // 求对角和
}

结果:

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

minCoeff()函数和maxCoeff()函数的变换使用:

 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;

结果:

Here is the matrix m:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
Its minimum coefficient (-0.605) is at position (2,1)

Here is the vector v:  1  0  3 -3
Its maximum coefficient (3) is at position 2

【参考】http://eigen.tuxfamily.org/dox/group__TutorialMatrixArithmetic.html

### RT-DETRv3 网络结构分析 RT-DETRv3 是一种基于 Transformer 的实时端到端目标检测算法,其核心在于通过引入分层密集正监督方法以及一系列创新性的训练策略,解决了传统 DETR 模型收敛慢和解码器训练不足的问题。以下是 RT-DETRv3 的主要网络结构特点: #### 1. **基于 CNN 的辅助分支** 为了增强编码器的特征表示能力,RT-DETRv3 引入了一个基于卷积神经网络 (CNN) 的辅助分支[^3]。这一分支提供了密集的监督信号,能够与原始解码器协同工作,从而提升整体性能。 ```python class AuxiliaryBranch(nn.Module): def __init__(self, in_channels, out_channels): super(AuxiliaryBranch, self).__init__() self.conv = nn.Conv2d(in_channels, out_channels, kernel_size=3, padding=1) self.bn = nn.BatchNorm2d(out_channels) def forward(self, x): return F.relu(self.bn(self.conv(x))) ``` 此部分的设计灵感来源于传统的 CNN 架构,例如 YOLO 系列中的 CSPNet 和 PAN 结构[^2],这些技术被用来优化特征提取效率并减少计算开销。 --- #### 2. **自注意力扰动学习策略** 为解决解码器训练不足的问题,RT-DETRv3 提出了一种名为 *self-att 扰动* 的新学习策略。这种策略通过对多个查询组中阳性样本的标签分配进行多样化处理,有效增加了阳例的数量,进而提高了模型的学习能力和泛化性能。 具体实现方式是在训练过程中动态调整注意力权重分布,确保更多的高质量查询可以与真实标注 (Ground Truth) 进行匹配。 --- #### 3. **共享权重解编码器分支** 除了上述改进外,RT-DETRv3 还引入了一个共享权重的解编码器分支,专门用于提供密集的正向监督信号。这一设计不仅简化了模型架构,还显著降低了参数量和推理时间,使其更适合实时应用需求。 ```python class SharedDecoderEncoder(nn.Module): def __init__(self, d_model, nhead, num_layers): super(SharedDecoderEncoder, self).__init__() decoder_layer = nn.TransformerDecoderLayer(d_model=d_model, nhead=nhead) self.decoder = nn.TransformerDecoder(decoder_layer, num_layers=num_layers) def forward(self, tgt, memory): return self.decoder(tgt=tgt, memory=memory) ``` 通过这种方式,RT-DETRv3 实现了高效的目标检测流程,在保持高精度的同时大幅缩短了推理延迟。 --- #### 4. **与其他模型的关系** 值得一提的是,RT-DETRv3 并未完全抛弃经典的 CNN 技术,而是将其与 Transformer 结合起来形成混合架构[^4]。例如,它采用了 YOLO 系列中的 RepNCSP 模块替代冗余的多尺度自注意力层,从而减少了不必要的计算负担。 此外,RT-DETRv3 还借鉴了 DETR 的一对一匹配策略,并在此基础上进行了优化,进一步提升了小目标检测的能力。 --- ### 总结 综上所述,RT-DETRv3 的网络结构主要包括以下几个关键组件:基于 CNN 的辅助分支、自注意力扰动学习策略、共享权重解编码器分支以及混合编码器设计。这些技术创新共同推动了实时目标检测领域的发展,使其在复杂场景下的表现更加出色。 ---
评论
添加红包

请填写红包祝福语或标题

红包个数最小为10个

红包金额最低5元

当前余额3.43前往充值 >
需支付:10.00
成就一亿技术人!
领取后你会自动成为博主和红包主的粉丝 规则
hope_wisdom
发出的红包
实付
使用余额支付
点击重新获取
扫码支付
钱包余额 0

抵扣说明:

1.余额是钱包充值的虚拟货币,按照1:1的比例进行支付金额的抵扣。
2.余额无法直接购买下载,可以购买VIP、付费专栏及课程。

余额充值