人脸姿态的随机采样一致性（RANSAC）

Random Sample Consensus: A Paradigm for Model Fitting with Apphcatlons to Image Analysis and Automated Cartography

（早就听说过随机采样一致性，但论文还是第一次读，居然是一辆1981年的老爷车。）

在实现人脸姿态估计时，通过opencv自带的DLT（solvepnp）解算出来的值误差很大，而且不稳定。

经过一系列思考和验证，发现实现上并无差错。那么是什么造成了呢？

这是二阶段或多阶段模型的共性错误：通常为分类误差引起的gross error。按照论文所述，分类误差是不服从于正态分布的，所以难以平滑。但可以使用随机采样一致性解决。

随机采样一致性的大致步骤，可以用伪代码表示：

Given:
    data – a set of observed data points
    model – a model that can be fitted to data points
    n – the minimum number of data values required to fit the model
    k – the maximum number of iterations allowed in the algorithm
    t – a threshold value for determining when a data point fits a model
    d – the number of close data values required to assert that a model fits well to data
 
Return:
    bestfit – model parameters which best fit the data (or nul if no good model is found)
 
iterations = 0
bestfit = nul
besterr = something really large
while iterations < k {
    maybeinliers = n randomly selected values from data
    maybemodel = model parameters fitted to maybeinliers
    alsoinliers = empty set
    for every point in data not in maybeinliers {
        if point fits maybemodel with an error smaller than t
             add point to alsoinliers
    }
    if the number of elements in alsoinliers is > d {
        % this implies that we may have found a good model
        % now test how good it is
        bettermodel = model parameters fitted to all points in maybeinliers and alsoinliers
        thiserr = a measure of how well model fits these points
        if thiserr < besterr {
            bestfit = bettermodel
            besterr = thiserr
        }
    }
    increment iterations
}
return bestfit

论文简单讨论了随机采样一致性的k、t值，t值应该取测量平均误差的两到三倍。

而k值，则给出了公式：

z表示的是可信度，b则表示随机选取一次，得到合适模型的概率。

当真实值淹没在噪音中，也可以使用该方法来找出群内点，但是如果b值过低，该方法是无效的。

蓝线为预测的鼻头指向，左图为单纯的dlt方法，右图加入了ransac。显然右图的预测效果要好一些。

另外给出人脸姿态的预测代码：

#ifndef CAL_ROT
#define CAL_ROT
#include <dlfcn.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <sys/time.h>
#include <iostream>
#include <vector>
#include "opencv2/core/core.hpp"
#include "opencv2/opencv.hpp"
#include "opencv2/highgui/highgui.hpp"
 
using namespace std;
using namespace cv;
#define FOCAL 112
#define PI 3.1415<br>
static vector<cv::Point3d> facePoints6p={
cv::Point3d(0,0,0),//nose tip 30
cv::Point3d(0.0,-330.0,65.0),//chin 8
cv::Point3d(-225.0,170.0,135.0),//36
cv::Point3d(225.0,170.0,135.0),//45
cv::Point3d(-150.0,-150.0,125.0),//48
cv::Point3d(150.0,-150.0,125.0),//54
};
cv::Scalar color[3]={cv::Scalar(255,0,0),cv::Scalar(0,255,0),cv::Scalar(0,0,255)};
 
void calRotate6P(cv::Mat &img,cv::Point2f land_mark[])
{
     
    double focal_length = FOCAL; // Approximate focal length.
    cv::Point2d center = cv::Point2d(FOCAL*0.5,FOCAL*0.5);
    cv::Mat camera_matrix = (cv::Mat_<double>(3, 3) << focal_length, 0, center.x, 0, focal_length, center.y, 0, 0, 1);
    cv::Mat dist_coeffs = cv::Mat::zeros(4, 1, cv::DataType<double>::type); // Assuming no lens distortion
 
    cv::Mat rotation_vector; // Rotation in axis-angle form
    cv::Mat translation_vector;
 
    vector<Point2d> landmark;
    landmark.push_back(land_mark[30]);
    landmark.push_back(land_mark[8]);
    landmark.push_back(land_mark[36]);
    landmark.push_back(land_mark[45]);
    landmark.push_back(land_mark[48]);
    landmark.push_back(land_mark[54]);
    // Solve for pose
    //cv::solvePnP(facePoints, landmark, camera_matrix, dist_coeffs, rotation_vector, translation_vector);
    cv::solvePnPRansac(facePoints6p, landmark, camera_matrix, dist_coeffs, rotation_vector, translation_vector);
    //rotation trans
    vector<Point3d> nose_end_point3d=
    {cv::Point3d(0,0,-100.0),cv::Point3d(0,100,0),cv::Point3d(100,0,0)};
    vector<Point2d> nose_end_point2d;
    //draw rotate
    cv::projectPoints(nose_end_point3d,rotation_vector,translation_vector,camera_matrix,dist_coeffs,nose_end_point2d);
    //cv::circle(img,cv::Point(10,10),1,cv::Scalar(0,0,0),-1);
    for(int i=0;i<nose_end_point2d.size();i++)
    {
    std::cout<<"point:"<<nose_end_point2d[i].x<<" "<<nose_end_point2d[i].y<<std::endl;
      cv::line(img,landmark[0],nose_end_point2d[i],color[i]);
    }
     
    //计算欧拉角
    double r = norm(rotation_vector, NORM_L2);
    double w = cos(r / 2);
    double x = sin(r / 2)*(*rotation_vector.ptr<double>(0, 0))/r;
    double y = sin(r / 2)*(*rotation_vector.ptr<double>(1, 0))/r;
    double z = sin(r / 2)*(*rotation_vector.ptr<double>(2, 0))/r;
    //x
    double t0 = 2 * (w*x + y * z);
    double t1 = 1 - 2 * (x*x + y * y);
    double pitch = atan2(t0, t1) / PI * 180;
    //y
    double t2 = 2 * (w*y - z * x);
    if (t2 > 1)
    {
        t2 = 1;
    }
    else if (t2 < -1)
    {
        t2 = -1;
    }
    double yaw = asin(t2) / PI * 180;
 
    //z
    double t3 = 2 * (w*z + x * y);
    double t4 = 1 - 2 * (y*y + z * z);
    double roll = atan2(t3, t4)/PI*180;
     
    cout<<r<<"\txangle："<<pitch<<"\tyangle："<<yaw<<"\tzangle"<<roll<<endl;
}
#endif