N点标定-坐标系变换

文章目录

- 1：标定算法
- 2：外部链接

1：标定算法

vector_to_hom_mat2d(Px, Py, Qx, Qy, HomMat2D)
在这里插入图片描述
这里参考了halcon算子块的官方文档，使用的是最小二乘法，求HomMat2D矩阵。 -常用九点标定，求两个坐标系的坐标转换。。
下面个人实现原理，结果和上面算子算出来的结果一致，知识有限，仅供学习交流。
1：先来看一张图，图中矩阵为2行3列，最后一列为0,0,1；因为用到是齐次矩阵，所以展开就省略了。具体为什么是这样的矩阵形式，其实是一系列的变换，也就是平面二维仿射变换。

在这里插入图片描述

2：然后使用最小二乘法列出项（最小平方法）在这里插入图片描述其次在对每个未知量进行求偏导，图中是两个未知量，就是列出两个方程组，如果是三个未知量，那就是三个方程组

三个未知量进行求导，得出来的结果就是对应矩阵里的未知量。。

3：上面方法都是笔记上演示，如果放在程序里怎么做呢？用上面这种怎么求未知量呢？下面会有个人实现的源代码，这里求方程组使用克拉默法则，列出行列式进行求解，放在代码里，方便求解。

正确的做法就是求各个未知量的系数，比如：

在这里插入图片描述

4：核心代码演示

//保存从界面获取的坐标数据
struct _coordList{
    QList<QPair<double,double>> pixPoines;//保存像素坐标
    QList<QPair<double,double>> physicsPoines;//保存物理坐标
};

//保存最小二乘法三个未知量(a,b,c)的结果
struct _squareLaw{
    double aa=0;
    double bb=0;
    double cc=0;
    double ab=0;
    double ac=0;
    double bc=0;
    double _a=0;
    double _b=0;
    double _c=0;
};
//导数3x4
struct _differentialCoefficient{
    QList<double> diff_a;
    QList<double> diff_b;
    QList<double> diff_c;
};

//矩阵3X3
struct _mat{
    double a=0;
    double b=0;
    double c=0;
    double d=0;
    double e=0;
    double f=0;
    double g=0;
    double h=0;
    double i=0;
};

//标定
_mat MainWindow::calibration(_coordList coordL)
{
    //保存未知量（a,b,c）的系数
    _squareLaw squareLaw_x;
    _squareLaw squareLaw_y;
    for(int i=0;i<coordL.pixPoines.count();i++) {
        squareLaw_x.aa += pow(coordL.pixPoines.at(i).first,2)*2;
        squareLaw_x.bb += pow(coordL.pixPoines.at(i).second,2)*2;
        squareLaw_x.cc += 1*2;
        squareLaw_x.ab += coordL.pixPoines.at(i).first*coordL.pixPoines.at(i).second*2;
        squareLaw_x.ac += coordL.pixPoines.at(i).first*2;
        squareLaw_x.bc += coordL.pixPoines.at(i).second*2;
        squareLaw_x._a += coordL.pixPoines.at(i).first*coordL.physicsPoines.at(i).first*2;
        squareLaw_x._b += coordL.pixPoines.at(i).second*coordL.physicsPoines.at(i).first*2;
        squareLaw_x._c += coordL.physicsPoines.at(i).first*2;
    }

    for(int i=0;i<coordL.pixPoines.count();i++) {
        squareLaw_y.aa += pow(coordL.pixPoines.at(i).first,2)*2;
        squareLaw_y.bb += pow(coordL.pixPoines.at(i).second,2)*2;
        squareLaw_y.cc += 1*2;
        squareLaw_y.ab += coordL.pixPoines.at(i).first*coordL.pixPoines.at(i).second*2;
        squareLaw_y.ac += coordL.pixPoines.at(i).first*2;
        squareLaw_y.bc += coordL.pixPoines.at(i).second*2;
        squareLaw_y._a += coordL.pixPoines.at(i).first*coordL.physicsPoines.at(i).second*2;
        squareLaw_y._b += coordL.pixPoines.at(i).second*coordL.physicsPoines.at(i).second*2;
        squareLaw_y._c += coordL.physicsPoines.at(i).second*2;
    }

    //求导数  3元方程组系数
    _differentialCoefficient diff_x;
    _differentialCoefficient diff_y;
    diff_x.diff_a.append(squareLaw_x.aa);
    diff_x.diff_a.append(squareLaw_x.ab);
    diff_x.diff_a.append(squareLaw_x.ac);
    diff_x.diff_a.append(squareLaw_x._a);
    diff_x.diff_b.append(squareLaw_x.ab);
    diff_x.diff_b.append(squareLaw_x.bb);
    diff_x.diff_b.append(squareLaw_x.bc);
    diff_x.diff_b.append(squareLaw_x._b);
    diff_x.diff_c.append(squareLaw_x.ac);
    diff_x.diff_c.append(squareLaw_x.bc);
    diff_x.diff_c.append(squareLaw_x.cc);
    diff_x.diff_c.append(squareLaw_x._c);

    diff_y.diff_a.append(squareLaw_y.aa);
    diff_y.diff_a.append(squareLaw_y.ab);
    diff_y.diff_a.append(squareLaw_y.ac);
    diff_y.diff_a.append(squareLaw_y._a);
    diff_y.diff_b.append(squareLaw_y.ab);
    diff_y.diff_b.append(squareLaw_y.bb);
    diff_y.diff_b.append(squareLaw_y.bc);
    diff_y.diff_b.append(squareLaw_y._b);
    diff_y.diff_c.append(squareLaw_y.ac);
    diff_y.diff_c.append(squareLaw_y.bc);
    diff_y.diff_c.append(squareLaw_y.cc);
    diff_y.diff_c.append(squareLaw_y._c);

    //解方程 使用克莱默法则解方程
    //a11a22a33+a12a23a31+a13a21a32 −a13a22a31−a11a23a32−a12a21a33

    double D_x,Da,Db,Dc, D_y,Dd,De,Df;
    D_x = diff_x.diff_a.at(0)*diff_x.diff_b.at(1)*diff_x.diff_c.at(2) + diff_x.diff_a.at(1)*diff_x.diff_b.at(2)*diff_x.diff_c.at(0) +diff_x.diff_a.at(2)*diff_x.diff_b.at(0)*diff_x.diff_c.at(1)
            -diff_x.diff_a.at(2)*diff_x.diff_b.at(1)*diff_x.diff_c.at(0) - diff_x.diff_a.at(0)*diff_x.diff_b.at(2)*diff_x.diff_c.at(1) -diff_x.diff_a.at(1)*diff_x.diff_b.at(0)*diff_x.diff_c.at(2);
    Da = diff_x.diff_a.at(3)*diff_x.diff_b.at(1)*diff_x.diff_c.at(2) + diff_x.diff_a.at(1)*diff_x.diff_b.at(2)*diff_x.diff_c.at(3) +diff_x.diff_a.at(2)*diff_x.diff_b.at(3)*diff_x.diff_c.at(1)
            -diff_x.diff_a.at(2)*diff_x.diff_b.at(1)*diff_x.diff_c.at(3) - diff_x.diff_a.at(3)*diff_x.diff_b.at(2)*diff_x.diff_c.at(1) -diff_x.diff_a.at(1)*diff_x.diff_b.at(3)*diff_x.diff_c.at(2);
    Db = diff_x.diff_a.at(0)*diff_x.diff_b.at(3)*diff_x.diff_c.at(2) + diff_x.diff_a.at(3)*diff_x.diff_b.at(2)*diff_x.diff_c.at(0) +diff_x.diff_a.at(2)*diff_x.diff_b.at(0)*diff_x.diff_c.at(3)
            -diff_x.diff_a.at(2)*diff_x.diff_b.at(3)*diff_x.diff_c.at(0) - diff_x.diff_a.at(0)*diff_x.diff_b.at(2)*diff_x.diff_c.at(3) -diff_x.diff_a.at(3)*diff_x.diff_b.at(0)*diff_x.diff_c.at(2);
    Dc = diff_x.diff_a.at(0)*diff_x.diff_b.at(1)*diff_x.diff_c.at(3) + diff_x.diff_a.at(1)*diff_x.diff_b.at(3)*diff_x.diff_c.at(0) +diff_x.diff_a.at(3)*diff_x.diff_b.at(0)*diff_x.diff_c.at(1)
            -diff_x.diff_a.at(3)*diff_x.diff_b.at(1)*diff_x.diff_c.at(0) - diff_x.diff_a.at(0)*diff_x.diff_b.at(3)*diff_x.diff_c.at(1) -diff_x.diff_a.at(1)*diff_x.diff_b.at(0)*diff_x.diff_c.at(3);
    D_y = diff_y.diff_a.at(0)*diff_y.diff_b.at(1)*diff_y.diff_c.at(2) + diff_y.diff_a.at(1)*diff_y.diff_b.at(2)*diff_y.diff_c.at(0) +diff_y.diff_a.at(2)*diff_y.diff_b.at(0)*diff_y.diff_c.at(1)
            -diff_y.diff_a.at(2)*diff_y.diff_b.at(1)*diff_y.diff_c.at(0) - diff_y.diff_a.at(0)*diff_y.diff_b.at(2)*diff_y.diff_c.at(1) -diff_y.diff_a.at(1)*diff_y.diff_b.at(0)*diff_y.diff_c.at(2);
    Dd = diff_y.diff_a.at(3)*diff_y.diff_b.at(1)*diff_y.diff_c.at(2) + diff_y.diff_a.at(1)*diff_y.diff_b.at(2)*diff_y.diff_c.at(3) +diff_y.diff_a.at(2)*diff_y.diff_b.at(3)*diff_y.diff_c.at(1)
            -diff_y.diff_a.at(2)*diff_y.diff_b.at(1)*diff_y.diff_c.at(3) - diff_y.diff_a.at(3)*diff_y.diff_b.at(2)*diff_y.diff_c.at(1) -diff_y.diff_a.at(1)*diff_y.diff_b.at(3)*diff_y.diff_c.at(2);
    De = diff_y.diff_a.at(0)*diff_y.diff_b.at(3)*diff_y.diff_c.at(2) + diff_y.diff_a.at(3)*diff_y.diff_b.at(2)*diff_y.diff_c.at(0) +diff_y.diff_a.at(2)*diff_y.diff_b.at(0)*diff_y.diff_c.at(3)
            -diff_y.diff_a.at(2)*diff_y.diff_b.at(3)*diff_y.diff_c.at(0) - diff_y.diff_a.at(0)*diff_y.diff_b.at(2)*diff_y.diff_c.at(3) -diff_y.diff_a.at(3)*diff_y.diff_b.at(0)*diff_y.diff_c.at(2);
    Df = diff_y.diff_a.at(0)*diff_y.diff_b.at(1)*diff_y.diff_c.at(3) + diff_y.diff_a.at(1)*diff_y.diff_b.at(3)*diff_y.diff_c.at(0) +diff_y.diff_a.at(3)*diff_y.diff_b.at(0)*diff_y.diff_c.at(1)
            -diff_y.diff_a.at(3)*diff_y.diff_b.at(1)*diff_y.diff_c.at(0) - diff_y.diff_a.at(0)*diff_y.diff_b.at(3)*diff_y.diff_c.at(1) -diff_y.diff_a.at(1)*diff_y.diff_b.at(0)*diff_y.diff_c.at(3);

    _mat mat;
    mat.a = Da/D_x;
    mat.b = Db/D_x;
    mat.c = Dc/D_x;
    mat.d = Dd/D_y;
    mat.e = De/D_y;
    mat.f = Df/D_y;
    mat.g = 0;
    mat.h = 0;
    mat.i = 1;
    return mat;
}
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完整代码实现xxx.h

#ifndef MAINWINDOW_H
#define MAINWINDOW_H

#include 

class QTableWidgetItem;
QT_BEGIN_NAMESPACE
namespace Ui { class MainWindow; }
QT_END_NAMESPACE

//保存从界面获取的坐标数据
struct _coordList{
    QList<QPair<double,double>> pixPoines;//保存像素坐标
    QList<QPair<double,double>> physicsPoines;//保存物理坐标
};

//保存最小二乘法三个未知量(a,b,c)的结果
struct _squareLaw{
    double aa=0;
    double bb=0;
    double cc=0;
    double ab=0;
    double ac=0;
    double bc=0;
    double _a=0;
    double _b=0;
    double _c=0;
};
//导数3x4
struct _differentialCoefficient{
    QList<double> diff_a;
    QList<double> diff_b;
    QList<double> diff_c;
};

//矩阵3X3
struct _mat{
    double a=0;
    double b=0;
    double c=0;
    double d=0;
    double e=0;
    double f=0;
    double g=0;
    double h=0;
    double i=0;
};

class MainWindow : public QMainWindow
{
    Q_OBJECT

public:
    MainWindow(QWidget *parent = nullptr);
    ~MainWindow();

    _mat calibration(_coordList);//标定

private slots:

    void on_pushButton_clicked();

    void on_pushButton_2_clicked();

    void on_tableWidget_itemChanged(QTableWidgetItem *item);

    //void on_tableWidget_itemDoubleClicked(QTableWidgetItem *item);

    void on_pushButton_3_clicked();

private:
    Ui::MainWindow *ui;

    QString old_text;

};
#endif // MAINWINDOW_H

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xxx.cpp

#include "mainwindow.h"
#include "ui_mainwindow.h"

#include 
#include 
#include 
#include 
#include 
MainWindow::MainWindow(QWidget *parent)
    : QMainWindow(parent)
    , ui(new Ui::MainWindow)
{
    ui->setupUi(this);

    //设置列数
    ui->tableWidget->setColumnCount(4);
    //设置标题
    //表头标题用QStringList来表示
    QStringList headerText;
    headerText<<"像素X坐标"<<"像素Y坐标"<<"物理X坐标"<<"物理Y坐标";
    ui->tableWidget->setHorizontalHeaderLabels(headerText);
    old_text = "";
}

MainWindow::~MainWindow()
{
    delete ui;
}
//设置n点标定
void MainWindow::on_pushButton_clicked()
{
    //获取标定的个数
    quint32 transitionCount = ui->spinBox->value();

    int oldRow = ui->tableWidget->rowCount();
    //设置行数
    ui->tableWidget->setRowCount(transitionCount);

    int newRow = ui->tableWidget->rowCount();
    //生成表格，默认值为0
    if(newRow>oldRow) {
        for(int i = oldRow;i<newRow;i++) {
            for (int j=0;j<4;j++) {
                ui->tableWidget->setItem(i,j,new QTableWidgetItem("0"));
            }
        }
    }

}
//标定
_mat MainWindow::calibration(_coordList coordL)
{
    //保存未知量（a,b,c）的系数
    _squareLaw squareLaw_x;
    _squareLaw squareLaw_y;
    for(int i=0;i<coordL.pixPoines.count();i++) {
        squareLaw_x.aa += pow(coordL.pixPoines.at(i).first,2)*2;
        squareLaw_x.bb += pow(coordL.pixPoines.at(i).second,2)*2;
        squareLaw_x.cc += 1*2;
        squareLaw_x.ab += coordL.pixPoines.at(i).first*coordL.pixPoines.at(i).second*2;
        squareLaw_x.ac += coordL.pixPoines.at(i).first*2;
        squareLaw_x.bc += coordL.pixPoines.at(i).second*2;
        squareLaw_x._a += coordL.pixPoines.at(i).first*coordL.physicsPoines.at(i).first*2;
        squareLaw_x._b += coordL.pixPoines.at(i).second*coordL.physicsPoines.at(i).first*2;
        squareLaw_x._c += coordL.physicsPoines.at(i).first*2;
    }

    for(int i=0;i<coordL.pixPoines.count();i++) {
        squareLaw_y.aa += pow(coordL.pixPoines.at(i).first,2)*2;
        squareLaw_y.bb += pow(coordL.pixPoines.at(i).second,2)*2;
        squareLaw_y.cc += 1*2;
        squareLaw_y.ab += coordL.pixPoines.at(i).first*coordL.pixPoines.at(i).second*2;
        squareLaw_y.ac += coordL.pixPoines.at(i).first*2;
        squareLaw_y.bc += coordL.pixPoines.at(i).second*2;
        squareLaw_y._a += coordL.pixPoines.at(i).first*coordL.physicsPoines.at(i).second*2;
        squareLaw_y._b += coordL.pixPoines.at(i).second*coordL.physicsPoines.at(i).second*2;
        squareLaw_y._c += coordL.physicsPoines.at(i).second*2;
    }

    //求导数  3元方程组系数
    _differentialCoefficient diff_x;
    _differentialCoefficient diff_y;
    diff_x.diff_a.append(squareLaw_x.aa);
    diff_x.diff_a.append(squareLaw_x.ab);
    diff_x.diff_a.append(squareLaw_x.ac);
    diff_x.diff_a.append(squareLaw_x._a);
    diff_x.diff_b.append(squareLaw_x.ab);
    diff_x.diff_b.append(squareLaw_x.bb);
    diff_x.diff_b.append(squareLaw_x.bc);
    diff_x.diff_b.append(squareLaw_x._b);
    diff_x.diff_c.append(squareLaw_x.ac);
    diff_x.diff_c.append(squareLaw_x.bc);
    diff_x.diff_c.append(squareLaw_x.cc);
    diff_x.diff_c.append(squareLaw_x._c);

    diff_y.diff_a.append(squareLaw_y.aa);
    diff_y.diff_a.append(squareLaw_y.ab);
    diff_y.diff_a.append(squareLaw_y.ac);
    diff_y.diff_a.append(squareLaw_y._a);
    diff_y.diff_b.append(squareLaw_y.ab);
    diff_y.diff_b.append(squareLaw_y.bb);
    diff_y.diff_b.append(squareLaw_y.bc);
    diff_y.diff_b.append(squareLaw_y._b);
    diff_y.diff_c.append(squareLaw_y.ac);
    diff_y.diff_c.append(squareLaw_y.bc);
    diff_y.diff_c.append(squareLaw_y.cc);
    diff_y.diff_c.append(squareLaw_y._c);

    //解方程 使用克莱默法则解方程
    //a11a22a33+a12a23a31+a13a21a32 −a13a22a31−a11a23a32−a12a21a33

    double D_x,Da,Db,Dc, D_y,Dd,De,Df;
    D_x = diff_x.diff_a.at(0)*diff_x.diff_b.at(1)*diff_x.diff_c.at(2) + diff_x.diff_a.at(1)*diff_x.diff_b.at(2)*diff_x.diff_c.at(0) +diff_x.diff_a.at(2)*diff_x.diff_b.at(0)*diff_x.diff_c.at(1)
            -diff_x.diff_a.at(2)*diff_x.diff_b.at(1)*diff_x.diff_c.at(0) - diff_x.diff_a.at(0)*diff_x.diff_b.at(2)*diff_x.diff_c.at(1) -diff_x.diff_a.at(1)*diff_x.diff_b.at(0)*diff_x.diff_c.at(2);
    Da = diff_x.diff_a.at(3)*diff_x.diff_b.at(1)*diff_x.diff_c.at(2) + diff_x.diff_a.at(1)*diff_x.diff_b.at(2)*diff_x.diff_c.at(3) +diff_x.diff_a.at(2)*diff_x.diff_b.at(3)*diff_x.diff_c.at(1)
            -diff_x.diff_a.at(2)*diff_x.diff_b.at(1)*diff_x.diff_c.at(3) - diff_x.diff_a.at(3)*diff_x.diff_b.at(2)*diff_x.diff_c.at(1) -diff_x.diff_a.at(1)*diff_x.diff_b.at(3)*diff_x.diff_c.at(2);
    Db = diff_x.diff_a.at(0)*diff_x.diff_b.at(3)*diff_x.diff_c.at(2) + diff_x.diff_a.at(3)*diff_x.diff_b.at(2)*diff_x.diff_c.at(0) +diff_x.diff_a.at(2)*diff_x.diff_b.at(0)*diff_x.diff_c.at(3)
            -diff_x.diff_a.at(2)*diff_x.diff_b.at(3)*diff_x.diff_c.at(0) - diff_x.diff_a.at(0)*diff_x.diff_b.at(2)*diff_x.diff_c.at(3) -diff_x.diff_a.at(3)*diff_x.diff_b.at(0)*diff_x.diff_c.at(2);
    Dc = diff_x.diff_a.at(0)*diff_x.diff_b.at(1)*diff_x.diff_c.at(3) + diff_x.diff_a.at(1)*diff_x.diff_b.at(3)*diff_x.diff_c.at(0) +diff_x.diff_a.at(3)*diff_x.diff_b.at(0)*diff_x.diff_c.at(1)
            -diff_x.diff_a.at(3)*diff_x.diff_b.at(1)*diff_x.diff_c.at(0) - diff_x.diff_a.at(0)*diff_x.diff_b.at(3)*diff_x.diff_c.at(1) -diff_x.diff_a.at(1)*diff_x.diff_b.at(0)*diff_x.diff_c.at(3);
    D_y = diff_y.diff_a.at(0)*diff_y.diff_b.at(1)*diff_y.diff_c.at(2) + diff_y.diff_a.at(1)*diff_y.diff_b.at(2)*diff_y.diff_c.at(0) +diff_y.diff_a.at(2)*diff_y.diff_b.at(0)*diff_y.diff_c.at(1)
            -diff_y.diff_a.at(2)*diff_y.diff_b.at(1)*diff_y.diff_c.at(0) - diff_y.diff_a.at(0)*diff_y.diff_b.at(2)*diff_y.diff_c.at(1) -diff_y.diff_a.at(1)*diff_y.diff_b.at(0)*diff_y.diff_c.at(2);
    Dd = diff_y.diff_a.at(3)*diff_y.diff_b.at(1)*diff_y.diff_c.at(2) + diff_y.diff_a.at(1)*diff_y.diff_b.at(2)*diff_y.diff_c.at(3) +diff_y.diff_a.at(2)*diff_y.diff_b.at(3)*diff_y.diff_c.at(1)
            -diff_y.diff_a.at(2)*diff_y.diff_b.at(1)*diff_y.diff_c.at(3) - diff_y.diff_a.at(3)*diff_y.diff_b.at(2)*diff_y.diff_c.at(1) -diff_y.diff_a.at(1)*diff_y.diff_b.at(3)*diff_y.diff_c.at(2);
    De = diff_y.diff_a.at(0)*diff_y.diff_b.at(3)*diff_y.diff_c.at(2) + diff_y.diff_a.at(3)*diff_y.diff_b.at(2)*diff_y.diff_c.at(0) +diff_y.diff_a.at(2)*diff_y.diff_b.at(0)*diff_y.diff_c.at(3)
            -diff_y.diff_a.at(2)*diff_y.diff_b.at(3)*diff_y.diff_c.at(0) - diff_y.diff_a.at(0)*diff_y.diff_b.at(2)*diff_y.diff_c.at(3) -diff_y.diff_a.at(3)*diff_y.diff_b.at(0)*diff_y.diff_c.at(2);
    Df = diff_y.diff_a.at(0)*diff_y.diff_b.at(1)*diff_y.diff_c.at(3) + diff_y.diff_a.at(1)*diff_y.diff_b.at(3)*diff_y.diff_c.at(0) +diff_y.diff_a.at(3)*diff_y.diff_b.at(0)*diff_y.diff_c.at(1)
            -diff_y.diff_a.at(3)*diff_y.diff_b.at(1)*diff_y.diff_c.at(0) - diff_y.diff_a.at(0)*diff_y.diff_b.at(3)*diff_y.diff_c.at(1) -diff_y.diff_a.at(1)*diff_y.diff_b.at(0)*diff_y.diff_c.at(3);

    _mat mat;
    mat.a = Da/D_x;
    mat.b = Db/D_x;
    mat.c = Dc/D_x;
    mat.d = Dd/D_y;
    mat.e = De/D_y;
    mat.f = Df/D_y;
    mat.g = 0;
    mat.h = 0;
    mat.i = 1;
    return mat;
}
//执行标定
void MainWindow::on_pushButton_2_clicked()
{
    _coordList coordL;
    //获取界面坐标数据
    for(int i=0;i<ui->tableWidget->rowCount();i++) {
        QPair<double,double> pairPix;
        QPair<double,double> pairPhy;

        for(int j=0;j<4;j++) {
            switch (j) {
            case 0:
                pairPix.first = ui->tableWidget->item(i,j)->text().toDouble();
                break;
            case 1:
                pairPix.second = ui->tableWidget->item(i,j)->text().toDouble();
                break;
            case 2:
                pairPhy.first = ui->tableWidget->item(i,j)->text().toDouble();
                break;
            case 3:
                pairPhy.second = ui->tableWidget->item(i,j)->text().toDouble();
                break;
            default:
                break;

            }
        }
        coordL.pixPoines.append(pairPix);
        coordL.physicsPoines.append(pairPhy);
    }
    //执行标定，获取结果
    _mat mat = calibration(coordL);


    //显示标定结果
    ui->textEdit->append(QString::number(mat.a,'e',15)+" , "+QString::number(mat.b,'e',15)+" , "+QString::number(mat.c,'e',15));
    ui->textEdit->append(QString::number(mat.d,'e',15)+" , "+QString::number(mat.e,'e',15)+" , "+QString::number(mat.f,'e',15));
    ui->textEdit->append(QString::number(mat.g)+" , "+QString::number(mat.h)+" , "+QString::number(mat.i));
    ui->textEdit->append("\r\n");

}

//item没有字符时，双击触发
void MainWindow::on_tableWidget_itemChanged(QTableWidgetItem *item)
{

    //2、匹配正负整数、正负浮点数
    QString Pattern("(-?[1-9][0-9]+)|(-?[0-9])|(-?[1-9]\\d+\\.\\d+)|(-?[0-9]\\.\\d+)");
    QRegExp  reg(Pattern);

    //3.获取修改的新的单元格内容
    QString str=item->text();

    if(str.isEmpty()) {
        return;
    }
    //匹配失败，返回原来的字符
    if(!reg.exactMatch(str)){
        QMessageBox::information(this,"匹配失败","请输入小数和整数!");
        item->setText("0");  //更换之前的内容
    }
    //1、记录旧的单元格内容
    old_text = item->text();

}



//打开文件-》从文件读取坐标
void MainWindow::on_pushButton_3_clicked()
{
    QString fileName = QFileDialog::getOpenFileName(this,tr("文件对话框！"), "",tr("文件(*.csv *.txt)"));
    if(!fileName.isEmpty()) {
        QFile file(fileName);
        quint32 line=0;
        if(file.open(QIODevice::ReadOnly|QIODevice::Text)) {
            for (;;) {
                QString data = file.readLine();
                if(data.isEmpty()) {
                    QMessageBox::information(this,"完成","数据读取完成!");
                    return;
                }
                QStringList sL = data.split(",");
                if(sL.count()!=4) {
                    QMessageBox::information(this,"错误","请确定每行数据为4个并且以,分割!");
                    return;
                }
                for(int i=0;i<sL.count();i++) {
                    bool isOk;
                    sL.at(i).toDouble(&isOk);
                    sL.at(i).toULongLong(&isOk);
                    if(isOk) {
                        //设置行数
                        ui->tableWidget->setRowCount(line+1);
                        ui->tableWidget->setItem(line,i,new QTableWidgetItem(sL.at(i)));
                    }else {
                        QMessageBox::information(this,"错误","数据转换double失败!");
                    }
                }

                line++;
            }


        }

    }

}

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在这里插入图片描述

2：外部链接

最小二乘法
 仿射变换
 克拉默法则
 行列式计算方法
 2D坐标系下的点的转换矩阵

在这里插入图片描述

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原文地址：https://blog.csdn.net/weixin_43763292/article/details/126881396