You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


/*
 * PnPProblem.cpp
 *  2d-3d点对匹配 Rt变换求解器
 */
#include <iostream>
#include <sstream>

#include "PnPProblem.h"
#include "Mesh.h"

#include <opencv2/calib3d/calib3d.hpp>

/* Functions headers */
cv::Point3f CROSS(cv::Point3f v1, cv::Point3f v2);//向量叉乘
double DOT(cv::Point3f v1, cv::Point3f v2);//向量点乘法 
cv::Point3f SUB(cv::Point3f v1, cv::Point3f v2);//向量减法
// 找最近的3d点
cv::Point3f get_nearest_3D_point(std::vector<cv::Point3f> &points_list, cv::Point3f origin);
//=================向量 叉乘===============================
// Functions for Möller–Trumbore intersection algorithm
// (v1.x v1.y v1.z)  
// 得到向量　　叉乘 
// (v2.x v2.y v2.z)
// ====>
//     v1.y * v2.z - v1.z * v2.y
//     v1.z * v2.x - v1.x * v2.z
//     v1.x * v2.y - v1.y * v2.x
//=====>写成叉乘矩阵　
//
//     0    -v1.z   v1.y       v2.x
//    v1.z    0    -v1.x    *  v2.y
//    -v1.y v1.x    0          v2.z
cv::Point3f CROSS(cv::Point3f v1, cv::Point3f v2)
{
  cv::Point3f tmp_p;
  tmp_p.x =  v1.y*v2.z - v1.z*v2.y;
  tmp_p.y =  v1.z*v2.x - v1.x*v2.z;
  tmp_p.z =  v1.x*v2.y - v1.y*v2.x;
  return tmp_p;
}
//=========向量点乘=====================
//得到常数　　对应元素相乘后加在一起
double DOT(cv::Point3f v1, cv::Point3f v2)
{
  return v1.x*v2.x + v1.y*v2.y + v1.z*v2.z;
}
//========向量相减====================
//得到向量　　对应元素相减
cv::Point3f SUB(cv::Point3f v1, cv::Point3f v2)
{
  cv::Point3f tmp_p;
  tmp_p.x =  v1.x - v2.x;
  tmp_p.y =  v1.y - v2.y;
  tmp_p.z =  v1.z - v2.z;
  return tmp_p;
}

//End functions for Möller–Trumbore intersection algorithm
// 在给定的3d点容器里(就存储了两个点)　找一个与　特定3d点　最相邻　的点　
// Function to get the nearest 3D point to the Ray origin
// 注意这里　3d点容器　为引用类型　避免拷贝
cv::Point3f get_nearest_3D_point(std::vector<cv::Point3f> &points_list, cv::Point3f origin)
{
  cv::Point3f p1 = points_list[0];//　第一个点
  cv::Point3f p2 = points_list[1];//　第二个点

  double d1 = std::sqrt( std::pow(p1.x-origin.x, 2) + std::pow(p1.y-origin.y, 2) + std::pow(p1.z-origin.z, 2) );
  double d2 = std::sqrt( std::pow(p2.x-origin.x, 2) + std::pow(p2.y-origin.y, 2) + std::pow(p2.z-origin.z, 2) );

  if(d1 < d2)//与第一个点最近
  {
      return p1;
  }
  else//与第二个点最近
  {
      return p2;
  }
}

// Custom constructor given the intrinsic camera parameters
// PnP求解器类　默认构造函数
PnPProblem::PnPProblem(const double params[])
{
  // PnP求解器参数初始化
  // 相机内参数矩阵
  _A_matrix = cv::Mat::zeros(3, 3, CV_64FC1);   // intrinsic camera parameters
  _A_matrix.at<double>(0, 0) = params[0];       //      [ fx   0  cx ]
  _A_matrix.at<double>(1, 1) = params[1];       //      [  0  fy  cy ]
  _A_matrix.at<double>(0, 2) = params[2];       //      [  0   0   1 ]
  _A_matrix.at<double>(1, 2) = params[3];
  _A_matrix.at<double>(2, 2) = 1;
  // 旋转矩阵  R   需要求解的
  _R_matrix = cv::Mat::zeros(3, 3, CV_64FC1);   // rotation matrix
  // 平移向量　t
  _t_matrix = cv::Mat::zeros(3, 1, CV_64FC1);   // translation matrix
  // 变换矩阵　P = [R t]
  _P_matrix = cv::Mat::zeros(3, 4, CV_64FC1);   // rotation-translation matrix

}

// PnP求解器类　默认析构函数
PnPProblem::~PnPProblem()
{
  // TODO Auto-generated destructor stub
}

// 由　旋转矩阵R　和　平移向量t　　构造变换矩阵P = [R t]　
void PnPProblem::set_P_matrix( const cv::Mat &R_matrix, const cv::Mat &t_matrix)
{
  // Rotation-Translation Matrix Definition
  _P_matrix.at<double>(0,0) = R_matrix.at<double>(0,0);
  _P_matrix.at<double>(0,1) = R_matrix.at<double>(0,1);
  _P_matrix.at<double>(0,2) = R_matrix.at<double>(0,2);
  _P_matrix.at<double>(1,0) = R_matrix.at<double>(1,0);
  _P_matrix.at<double>(1,1) = R_matrix.at<double>(1,1);
  _P_matrix.at<double>(1,2) = R_matrix.at<double>(1,2);
  _P_matrix.at<double>(2,0) = R_matrix.at<double>(2,0);
  _P_matrix.at<double>(2,1) = R_matrix.at<double>(2,1);
  _P_matrix.at<double>(2,2) = R_matrix.at<double>(2,2);
  _P_matrix.at<double>(0,3) = t_matrix.at<double>(0);
  _P_matrix.at<double>(1,3) = t_matrix.at<double>(1);
  _P_matrix.at<double>(2,3) = t_matrix.at<double>(2);
}

// 2D-3D　单个点对　估计变换矩阵　cv::solvePnP() 得到　旋转向量 ----> 旋转矩阵
// Estimate the pose given a list of 2D/3D correspondences and the method to use
bool PnPProblem::estimatePose( const std::vector<cv::Point3f> &list_points3d,
                               const std::vector<cv::Point2f> &list_points2d,
                               int flags)
{
  cv::Mat distCoeffs = cv::Mat::zeros(4, 1, CV_64FC1);// distCoeffs是4×1、1×4、5×1或1×5的相机畸变向量
  cv::Mat rvec = cv::Mat::zeros(3, 1, CV_64FC1);// 3*1 旋转向量 方向表示旋转轴　　大小表示　旋转角度
  cv::Mat tvec = cv::Mat::zeros(3, 1, CV_64FC1);// 3*1 平移向量
  bool useExtrinsicGuess = false;//按初始值不断优化
  // 变量useExtrinsicGuess的值为true，函数将使用所提供的rvec和tvec向量作为旋转和平移的初始值，然后会不断优化它们。
  // 函数按最小化重投影误差来计算摄像机的变换，即让所得的投影向量imagePoints与被投影向量之间的距离平方和最小。
  // Pose estimation
  bool correspondence = cv::solvePnP( list_points3d,// objectPoints 对象所在空间的三维点　 
				      list_points2d,// imagePoints是相应图像点（或投影）
				      _A_matrix,    // cameraMatrix是3×3的摄像机内部参数矩阵
				      distCoeffs,   // distCoeffs是4×1、1×4、5×1或1×5的相机畸变向量
				      rvec,    
// rvec是输出的旋转向量，该向量将点由模型坐标系统 转换为 摄像机坐标系统  	worid 到camera相机
				      tvec,        // tvec是输出的平移向量
				      useExtrinsicGuess, 
				      flags);
// 欧拉角的表示方式里，三个旋转轴一般是随着刚体在运动 row pitch yaw
// 旋转向量　转到 旋转矩阵(旋转矩阵　也叫　方向余弦矩阵(Direction Cosine Matrix)，简称DCM　)
// 直观来讲，一个四维向量(theta,x,y,z)就可以表示出三维空间任意的旋转。
// 注意，这里的三维向量(x,y,z)只是用来表示旋转轴axis的方向朝向，
// 因此更紧凑的表示方式是用一个单位向量来表示方向axis，
// 而用该三维向量的长度来表示角度值theta。
// 这样以来，可以用一个三维向量(theta*x, theta*y, theta*z)就可以表示出三维空间任意的旋转，
// 前提是其中(x,y,z)是单位向量。这就是旋转向量(Rotation Vector)的表示方式，
// ===============四元素==========================================
// 同上，假设(x,y,z)是axis方向的单位向量，theta是绕axis转过的角度，
// 那么四元数可以表示为[cos(theta/2), x*sin(theta/2), y*sin(theta/2), z*sin(theta/2)]。
  Rodrigues(rvec,_R_matrix);// 旋转向量 ----> 旋转矩阵
  _t_matrix = tvec;// 平移向量

  // 调用类的函数　设置　变换矩阵　Set projection matrix
  this->set_P_matrix(_R_matrix, _t_matrix);

  return correspondence;
}

// 随机序列采样　一致性　估计　变换矩阵
// 得到符合变换的内点外点数量找到　内点比例最大的　一个变换
// Estimate the pose given a list of 2D/3D correspondences with RANSAC and the method to use
void PnPProblem::estimatePoseRANSAC( const std::vector<cv::Point3f> &list_points3d,   // 对象所在空间的三维点　　　列表
                                     const std::vector<cv::Point2f> &list_points2d,   // 相应图像二维点（或投影）　列表
                                     int flags, cv::Mat &inliers, int iterationsCount,// PnP method; inliers container
                                     float reprojectionError, double confidence )     // Ransac parameters
{
  cv::Mat distCoeffs = cv::Mat::zeros(4, 1, CV_64FC1);// distCoeffs是4×1、1×4、5×1或1×5的相机畸变向量
  cv::Mat rvec = cv::Mat::zeros(3, 1, CV_64FC1);      // 3*1 旋转向量 方向表示旋转轴　　大小表示　旋转角度
  cv::Mat tvec = cv::Mat::zeros(3, 1, CV_64FC1);      // 3*1 平移向量
  bool useExtrinsicGuess = false;
  // 变量useExtrinsicGuess的值为true，函数将使用所提供的rvec和tvec向量作为旋转和平移的初始值，然后会不断优化它们。
  // 函数按最小化重投影误差来计算摄像机的变换，即让所得的投影向量imagePoints与被投影向量之间的距离平方和最小。
  cv::solvePnPRansac( list_points3d, // 对象所在空间的三维点　　　列表
                      list_points2d, // 相应图像二维点（或投影）　列表
                      _A_matrix,     // cameraMatrix是3×3的摄像机内部参数矩阵
                      distCoeffs,    // distCoeffs是4×1、1×4、5×1或1×5的相机畸变向量
                      rvec, // rvec是输出的旋转向量，该向量将点由模型坐标系统 转换为 摄像机坐标系统  worid 到camera相机
                      tvec, // tvec是输出的平移向量
                      useExtrinsicGuess, // 使用初始优化？
                      iterationsCount,   // 随机采样迭代计数
                      reprojectionError, // 投影误差
                      confidence,        // 内点所占比例　　可信度
		      inliers,           // 内点数量
		      flags );

  Rodrigues(rvec,_R_matrix);// 旋转向量 ----> 旋转矩阵
  _t_matrix = tvec;         // 平移向量
  // 调用类的函数　设置　变换矩阵　Set projection matrix
  this->set_P_matrix(_R_matrix, _t_matrix); // set rotation-translation matrix
}
//　网格文件　顶点　三维坐标　--> 2d　
// Given the mesh, backproject the 3D points to 2D to verify the pose estimation
std::vector<cv::Point2f> PnPProblem::verify_points(Mesh *mesh)
{
  std::vector<cv::Point2f> verified_points_2d;
  for( int i = 0; i < mesh->getNumVertices(); i++)
  {
    cv::Point3f point3d = mesh->getVertex(i);//　网格　顶点　三维坐标
    cv::Point2f point2d = this->backproject3DPoint(point3d);//3d点反投影到2d点
    verified_points_2d.push_back(point2d);//3d --> 2d
  }
  return verified_points_2d;
}

// 反投影　3d --> 2d　　　[u v 1]'　＝　K * P * [x y z 1]'　　再归一化　　得到二维像素点坐标[u v]'
// Backproject a 3D point to 2D using the estimated pose parameters
cv::Point2f PnPProblem::backproject3DPoint(const cv::Point3f &point3d)
{
  // 三维点　的　齐次表达式　3D point vector [x y z 1]'
  cv::Mat point3d_vec = cv::Mat(4, 1, CV_64FC1);
  point3d_vec.at<double>(0) = point3d.x;
  point3d_vec.at<double>(1) = point3d.y;
  point3d_vec.at<double>(2) = point3d.z;
  point3d_vec.at<double>(3) = 1;

  // 二维点　的　齐次表达式2D point vector [u v 1]'
  cv::Mat point2d_vec = cv::Mat(4, 1, CV_64FC1);
  point2d_vec = _A_matrix * _P_matrix * point3d_vec;

  // 将第三个量归一化　得到2d 点　Normalization of [u v]'
  cv::Point2f point2d;
  point2d.x = (float)(point2d_vec.at<double>(0) / point2d_vec.at<double>(2));
  point2d.y = (float)(point2d_vec.at<double>(1) / point2d_vec.at<double>(2));

  return point2d;
}

// 反投影　2d -> 3d 查看是否在　为物体网格模型的平面上
// Back project a 2D point to 3D and returns if it's on the object surface
bool PnPProblem::backproject2DPoint(const Mesh *mesh, const cv::Point2f &point2d, cv::Point3f &point3d)
{
  // 三角形面 顶点 索引  n*3
  std::vector<std::vector<int> > triangles_list = mesh->getTrianglesList();

  double lambda = 8;
  double u = point2d.x;// 二维像素点坐标[u v]'
  double v = point2d.y;

  // 二维点　的　齐次表达式
  cv::Mat point2d_vec = cv::Mat::ones(3, 1, CV_64F); // 3x1
  point2d_vec.at<double>(0) = u * lambda;
  point2d_vec.at<double>(1) = v * lambda;
  point2d_vec.at<double>(2) = lambda;

  // camera相机  coordinates
  cv::Mat X_c = _A_matrix.inv() * point2d_vec ; // 3x1

  // Point in world coordinates
  cv::Mat X_w = _R_matrix.inv() * ( X_c - _t_matrix ); // 3x1

  // Center of projection
  cv::Mat C_op = cv::Mat(_R_matrix.inv()).mul(-1) * _t_matrix; // 3x1

  // Ray direction vector
  cv::Mat ray = X_w - C_op; // 3x1
  ray = ray / cv::norm(ray); // 3x1

  // Set up Ray
  Ray R((cv::Point3f)C_op, (cv::Point3f)ray);

  // A vector to store the intersections found
  std::vector<cv::Point3f> intersections_list;

  // Loop for all the triangles and check the intersection
  for (unsigned int i = 0; i < triangles_list.size(); i++)
  {
    cv::Point3f V0 = mesh->getVertex(triangles_list[i][0]);
    cv::Point3f V1 = mesh->getVertex(triangles_list[i][1]);
    cv::Point3f V2 = mesh->getVertex(triangles_list[i][2]);

    Triangle T(i, V0, V1, V2);

    double out;
    if(this->intersect_MollerTrumbore(R, T, &out))
    {
      cv::Point3f tmp_pt = R.getP0() + out*R.getP1(); // P = O + t*D
      intersections_list.push_back(tmp_pt);
    }
  }

  // If there are intersection, find the nearest one
  if (!intersections_list.empty())
  {
    point3d = get_nearest_3D_point(intersections_list, R.getP0());
    return true;
  }
  else
  {
    return false;
  }
}

// Möller–Trumbore intersection algorithm
bool PnPProblem::intersect_MollerTrumbore(Ray &Ray, Triangle &Triangle, double *out)
{
  const double EPSILON = 0.000001;

  cv::Point3f e1, e2;
  cv::Point3f P, Q, T;
  double det, inv_det, u, v;
  double t;

  cv::Point3f V1 = Triangle.getV0();  // Triangle vertices
  cv::Point3f V2 = Triangle.getV1();
  cv::Point3f V3 = Triangle.getV2();

  cv::Point3f O = Ray.getP0(); // Ray origin
  cv::Point3f D = Ray.getP1(); // Ray direction

  //Find vectors for two edges sharing V1
  e1 = SUB(V2, V1);
  e2 = SUB(V3, V1);

  // Begin calculation determinant - also used to calculate U parameter
  P = CROSS(D, e2);

  // If determinant is near zero, ray lie in plane of triangle
  det = DOT(e1, P);

  //NOT CULLING
  if(det > -EPSILON && det < EPSILON) return false;
  inv_det = 1.f / det;

  //calculate distance from V1 to ray origin
  T = SUB(O, V1);

  //Calculate u parameter and test bound
  u = DOT(T, P) * inv_det;

  //The intersection lies outside of the triangle
  if(u < 0.f || u > 1.f) return false;

  //Prepare to test v parameter
  Q = CROSS(T, e1);

  //Calculate V parameter and test bound
  v = DOT(D, Q) * inv_det;

  //The intersection lies outside of the triangle
  if(v < 0.f || u + v  > 1.f) return false;

  t = DOT(e2, Q) * inv_det;

  if(t > EPSILON) { //ray intersection
    *out = t;
    return true;
  }

  // No hit, no win
  return false;
}