RANSAC(随机采样一致算法)原理及openCV代码实现



本文转自：http://blog.csdn.net/yihaizhiyan/article/details/5973729

http://blog.csdn.net/Sway_2012/article/details/37765765

http://blog.csdn.net/zouwen198317/article/details/38494149

1.什么是RANSAC?

RANSAC是RANdom SAmple Consensus（随机抽样一致性）的缩写。它是从一个观察数据集合中，估计模型参数（模型拟合）的迭代方法。它是一种随机的不确定算法，每次运算求出的结果可能不相同，但总能给出一个合理的结果，为了提高概率必须提高迭代次数。

2.算法详解

给定两个点p1与p2的坐标，确定这两点所构成的直线，要求对于输入的任意点p3，都可以判断它是否在该直线上。初中解析几何知识告诉我们，判断一个点在直线上，只需其与直线上任意两点点斜率都相同即可。实际操作当中，往往会先根据已知的两点算出直线的表达式（点斜式、截距式等等），然后通过向量计算即可方便地判断p3是否在该直线上。 

生产实践中的数据往往会有一定的偏差。例如我们知道两个变量X与Y之间呈线性关系，Y=aX+b，我们想确定参数a与b的具体值。通过实验，可以得到一组X与Y的测试值。虽然理论上两个未知数的方程只需要两组值即可确认，但由于系统误差的原因，任意取两点算出的a与b的值都不尽相同。我们希望的是，最后计算得出的理论模型与测试值的误差最小。大学的高等数学课程中，详细阐述了最小二乘法的思想。通过计算最小均方差关于参数a、b的偏导数为零时的值。事实上，在很多情况下，最小二乘法都是线性回归的代名词。 

遗憾的是，最小二乘法只适合与误差较小的情况。试想一下这种情况，假使需要从一个噪音较大的数据集中提取模型（比方说只有20%的数据时符合模型的）时，最小二乘法就显得力不从心了。例如下图，肉眼可以很轻易地看出一条直线（模式），但算法却找错了。 



RANSAC算法的输入是一组观测数据（往往含有较大的噪声或无效点），一个用于解释观测数据的参数化模型以及一些可信的参数。RANSAC通过反复选择数据中的一组随机子集来达成目标。被选取的子集被假设为局内点，并用下述方法进行验证： 

• 有一个模型适应于假设的局内点，即所有的未知参数都能从假设的局内点计算得出。
• 用1中得到的模型去测试所有的其它数据，如果某个点适用于估计的模型，认为它也是局内点。
• 如果有足够多的点被归类为假设的局内点，那么估计的模型就足够合理。
• 然后，用所有假设的局内点去重新估计模型（譬如使用最小二乘法），因为它仅仅被初始的假设局内点估计过。
• 最后，通过估计局内点与模型的错误率来评估模型。
• 上述过程被重复执行固定的次数，每次产生的模型要么因为局内点太少而被舍弃，要么因为比现有的模型更好而被选用。

整个过程可参考下图： 

3.代码实现

随机一致性采样RANSAC是一种鲁棒的模型拟合算法，能够从有外点的数据中拟合准确的模型。

RANSAC过程中用到的参数

N-- 拟合模型所需要的最少的样本个数

K--算法的迭代次数

t--用于判断数据是否是内点

d--判定模型是否符合使用于数据集，也就是判断是否是好的模型

RANSAC算法过程

1  for K 次迭代

2     从数据中均匀随机采样N个点

3     利用采样的N个点拟合你个模型

4     for 对于除采样点外的每一个样本点

5          利用t检测样本点到模型的距离，如果小于t则认为是一致，否则认为是外点

6     end

7     如果有d或者更多的一致点，则认为拟合的模型是好的

8 end

9 使用拟合误差作为标准，选择最好的拟合模型

迭代次数的计算

假设 r = 内点个数/所有点的个数

 则：

   p0 = pow(r, N) 表示采样的N个点全为内点，也就是是一次有效采样的概率

   p1 = 1 - pow(r, N) 表示采样的N个点中至少有一个外点，即一次无效采样的概率

   p2 = pow(p1, K) 表示K次无效采样的概率

假设p表示K次采样中至少一次采样是有效采样，则有1-p = pow(p1, K), 两边取对数

则有 K = log(1- p )/log(1-p1).

 附一份来自google 的RANSAC的代码框架

#ifndef FVISION_RANSAC_H_
#define FVISION_RANSAC_H_

#include <fvision/utils/random_utils.h>
#include <fvision/utils/misc.h>

#include <vector>
#include <iostream>
#include <cassert>

namespace fvision {

class RANSAC_SamplesNumber {
public:
        RANSAC_SamplesNumber(int modelSampleSize) {
                this->s = modelSampleSize;
                this->p = 0.99;
        }
        ~RANSAC_SamplesNumber(void) {}

public:
        long calcN(int inliersNumber, int samplesNumber) {
                double e = 1 - (double)inliersNumber / samplesNumber;
                //cout<<"e: "<<e<<endl;
                if (e > 0.9) e = 0.9;
                //cout<<"pow: "<<pow((1 - e), s)<<endl;
                //cout<<log(1 - pow((1 - e), s))<<endl;
                long N = (long)(log(1 - p) / log(1 - pow((1 - e), s)));
                if (N < 0) return (long)1000000000;
                else return N;
        }

private:
        int s;      //samples size for fitting a model
        double p;   //probability that at least one of the random samples if free from outliers
                    //usually 0.99
};

//fit a model to a set of samples
template <typename M, typename S>
class GenericModelCalculator {
public:
        typedef std::vector<S> Samples;
        virtual M compute(const Samples& samples) = 0;

        virtual ~GenericModelCalculator<M, S>() {}

        //the model calculator may only use a subset of the samples for computing
        //default return empty for both
        virtual const std::vector<int>& getInlierIndices() const { return defaultInlierIndices; };
        virtual const std::vector<int>& getOutlierIndices() const { return defaultOutlierIndices; };

        // if the subclass has a threshold parameter, it need to override the following three functions
        // this is used for algorithms which have a normalization step on input samples
        virtual bool hasThreshold() const { return false; }
        virtual void setThreshold(double threshold) {}
        virtual double getThreshold() const { return 0; }

protected:
        std::vector<int> defaultInlierIndices;
        std::vector<int> defaultOutlierIndices;
};

//evaluate a model to samples
//using a threshold to distinguish inliers and outliers
template <typename M, typename S>
class GenericErrorCaclculator {
public:
        virtual ~GenericErrorCaclculator<M, S>() {}

        typedef std::vector<S> Samples;

        virtual double compute(const M& model, const S& sample) const = 0;

        double computeAverage(const M& model, const Samples& samples) const {
                int n = (int)samples.size();
                if (n == 0) return 0;
                double sum = 0;
                for (int i = 0; i < n; i++) {
                        sum += compute(model, samples[i]);
                }
                return sum / n;
        }

        double computeInlierAverage(const M& model, const Samples& samples) const {
                int n = (int)samples.size();
                if (n == 0) return 0;
                double sum = 0;
                double error = 0;
                int inlierNum = 0;
                for (int i = 0; i < n; i++) {
                        error = compute(model, samples[i]);
                        if (error <= threshold) {
                                sum += error;
                                inlierNum++;
                        }
                }
                if (inlierNum == 0) return 1000000;
                return sum / inlierNum;
        }

public:

        /** set a threshold for classify inliers and outliers
         */
        void setThreshold(double v) { threshold = v; }

        double getThreshold() const { return threshold; }

        /** classify all samples to inliers and outliers
         *
         */
        void classify(const M& model, const Samples& samples, Samples& inliers, Samples& outliers) const {
                inliers.clear();
                outliers.clear();
                Samples::const_iterator iter = samples.begin();
                for (; iter != samples.end(); ++iter) {
                        if (isInlier(model, *iter)) inliers.push_back(*iter);
                        else outliers.push_back(*iter);
                }
        }

        /** classify all samples to inliers and outliers, output indices
         *
         */
        void classify(const M& model, const Samples& samples, std::vector<int>& inlierIndices, std::vector<int>& outlierIndices) const {
                inlierIndices.clear();
                outlierIndices.clear();
                Samples::const_iterator iter = samples.begin();
                int i = 0;
                for (; iter != samples.end(); ++iter, ++i) {
                        if (isInlier(model, *iter)) inlierIndices.push_back(i);
                        else outlierIndices.push_back(i);
                }
        }

        /** classify all samples to inliers and outliers
         *
         */
        void classify(const M& model, const Samples& samples,
                std::vector<int>& inlierIndices, std::vector<int>& outlierIndices,
                Samples& inliers, Samples& outliers) const {

                inliers.clear();
                outliers.clear();
                inlierIndices.clear();
                outlierIndices.clear();
                Samples::const_iterator iter = samples.begin();
                int i = 0;
                for (; iter != samples.end(); ++iter, ++i) {
                        if (isInlier(model, *iter)) {
                                inliers.push_back(*iter);
                                inlierIndices.push_back(i);
                        }
                        else {
                                outliers.push_back(*iter);
                                outlierIndices.push_back(i);
                        }
                }
        }

        int calcInliersNumber(const M& model, const Samples& samples) const {
                int n = 0;
                for (int i = 0; i < (int)samples.size(); i++) {
                        if (isInlier(model, samples[i])) ++n;
                }
                return n;
        }

        bool isInlier(const M& model, const S& sample) const {
                return (compute(model, sample) <= threshold);
        }

private:
        double threshold;
};

/** generic RANSAC framework
 * make use of a model calculator and an error calculator
 * M is the model type, need to support copy assignment operator and default constructor
 * S is the sample type.
 *
 * Interface:
 *  M compute(samples); input a set of samples, output a model.
 *  after compute, inliers and outliers can be retrieved
 *
 */
template <typename M, typename S>
class Ransac : public GenericModelCalculator<M, S> {
public:
        typedef std::vector<S> Samples;

        /** Constructor
         *
         * @param pmc a GenericModelCalculator object
         * @param modelSampleSize how much samples are used to fit a model
         * @param pec a GenericErrorCaclculator object
         */
        Ransac(GenericModelCalculator<M, S>* pmc, int modelSampleSize, GenericErrorCaclculator<M, S>* pec) {
                this->pmc = pmc;
                this->modelSampleSize = modelSampleSize;
                this->pec = pec;
                this->maxSampleCount = 500;
                this->minInliersNum = 1000000;

                this->verbose = false;
        }

        const GenericErrorCaclculator<M, S>* getErrorCalculator() const { return pec; }

        virtual ~Ransac() {
                delete pmc;
                delete pec;
        }

        void setMaxSampleCount(int n) {
                this->maxSampleCount = n;
        }

        void setMinInliersNum(int n) {
                this->minInliersNum = n;
        }

        virtual bool hasThreshold() const { return true; }

        virtual void setThreshold(double threshold) {
                pec->setThreshold(threshold);
        }

        virtual double getThreshold() const {
                return pec->getThreshold();
        }

public:
        /** Given samples, compute a model that has most inliers. Assume the samples size is larger or equal than model sample size
         * inliers, outliers, inlierIndices and outlierIndices are stored
         *
         */
        M compute(const Samples& samples) {
                clear();

                int pointsNumber = (int)samples.size();

                assert(pointsNumber >= modelSampleSize);

                long N = 100000;
                int sampleCount = 0;
                RANSAC_SamplesNumber ransac(modelSampleSize);

                M bestModel;
                int maxInliersNumber = 0;

                bool stop = false;
                while (sampleCount < N && sampleCount < maxSampleCount && !stop) {

                        Samples nsamples;
                        randomlySampleN(samples, nsamples, modelSampleSize);

                        M sampleModel = pmc->compute(nsamples);
                        if (maxInliersNumber == 0) bestModel = sampleModel;  //init bestModel

                        int inliersNumber = pec->calcInliersNumber(sampleModel, samples);
                        if (verbose) std::cout<<"inliers number: "<<inliersNumber<<std::endl;

                        if (inliersNumber > maxInliersNumber) {
                                bestModel = sampleModel;
                                maxInliersNumber = inliersNumber;
                                N = ransac.calcN(inliersNumber, pointsNumber);
                                if (maxInliersNumber > minInliersNum) stop = true;
                        }

                        if (verbose) std::cout<<"N: "<<N<<std::endl;

                        sampleCount ++;
                }

                if (verbose) std::cout<<"sampleCount: "<<sampleCount<<std::endl;

                finalModel = computeUntilConverge(bestModel, maxInliersNumber, samples);

                pec->classify(finalModel, samples, inlierIndices, outlierIndices, inliers, outliers);

                inliersRate = (double)inliers.size() / samples.size();

                return finalModel;
        }

        const Samples& getInliers() const { return inliers; }
        const Samples& getOutliers() const { return outliers; }

        const std::vector<int>& getInlierIndices() const { return inlierIndices; }
        const std::vector<int>& getOutlierIndices() const { return outlierIndices; }

        double getInliersAverageError() const {
                return pec->computeAverage(finalModel, inliers);
        }

        double getInliersRate() const {
                return inliersRate;
        }

        void setVerbose(bool v) {
                verbose = v;
        }

private:
        void randomlySampleN(const Samples& samples, Samples& nsamples, int sampleSize) {
                std::vector<int> is = ranis((int)samples.size(), sampleSize);
                for (int i = 0; i < sampleSize; i++) {
                        nsamples.push_back(samples[is[i]]);
                }
        }

        /** from initial model, iterate to find the best model.
         *
         */
        M computeUntilConverge(M initModel, int initInliersNum, const Samples& samples) {
                if (verbose) {
                        std::cout<<"iterate until converge...."<<std::endl;
                        std::cout<<"init inliers number: "<<initInliersNum<<std::endl;
                }

                M bestModel = initModel;
                M newModel = initModel;

                int lastInliersNum = initInliersNum;

                Samples newInliers, newOutliers;
                pec->classify(initModel, samples, newInliers, newOutliers);
                double lastInlierAverageError = pec->computeAverage(initModel, newInliers);

                if (verbose) std::cout<<"init inlier average error: "<<lastInlierAverageError<<std::endl;

                while (true && (int)newInliers.size() >= modelSampleSize) {

                        //update new model with new inliers, the new model does not necessarily have more inliers
                        newModel = pmc->compute(newInliers);

                        pec->classify(newModel, samples, newInliers, newOutliers);

                        int newInliersNum = (int)newInliers.size();
                        double newInlierAverageError = pec->computeAverage(newModel, newInliers);

                        if (verbose) {
                                std::cout<<"new inliers number: "<<newInliersNum<<std::endl;
                                std::cout<<"new inlier average error: "<<newInlierAverageError<<std::endl;
                        }
                        if (newInliersNum < lastInliersNum) break;
                        if (newInliersNum == lastInliersNum && newInlierAverageError >= lastInlierAverageError) break;

                        //update best model with the model has more inliers
                        bestModel = newModel;

                        lastInliersNum = newInliersNum;
                        lastInlierAverageError = newInlierAverageError;
                }

                return bestModel;
        }

        void clear() {
                inliers.clear();
                outliers.clear();
                inlierIndices.clear();
                outlierIndices.clear();
        }

private:
        GenericModelCalculator<M, S>* pmc;
        GenericErrorCaclculator<M, S>* pec;
        int modelSampleSize;

        int maxSampleCount;
        int minInliersNum;

        M finalModel;

        Samples inliers;
        Samples outliers;

        std::vector<int> inlierIndices;
        std::vector<int> outlierIndices;

        double inliersRate;

private:
        bool verbose;

};

}
#endif // FVISION_RANSAC_H_

实例2

1. #include <math.h>  
2. #include "LineParamEstimator.h"  
3.   
4. LineParamEstimator::LineParamEstimator(double delta) : m_deltaSquared(delta*delta) {}  
5. /*****************************************************************************/  
6. /* 
7.  * Compute the line parameters  [n_x,n_y,a_x,a_y] 
8.  * 通过输入的两点来确定所在直线，采用法线向量的方式来表示，以兼容平行或垂直的情况 
9.  * 其中n_x,n_y为归一化后，与原点构成的法线向量，a_x,a_y为直线上任意一点 
10.  */  
11. void LineParamEstimator::estimate(std::vector<Point2D *> &data,   
12.                                                                     std::vector<double> ¶meters)  
13. {  
14.     parameters.clear();  
15.     if(data.size()<2)  
16.         return;  
17.     double nx = data[1]->y - data[0]->y;  
18.     double ny = data[0]->x - data[1]->x;// 原始直线的斜率为K，则法线的斜率为-1/k  
19.     double norm = sqrt(nx*nx + ny*ny);  
20.       
21.     parameters.push_back(nx/norm);  
22.     parameters.push_back(ny/norm);  
23.     parameters.push_back(data[0]->x);  
24.     parameters.push_back(data[0]->y);          
25. }  
26. /*****************************************************************************/  
27. /* 
28.  * Compute the line parameters  [n_x,n_y,a_x,a_y] 
29.  * 使用最小二乘法，从输入点中拟合出确定直线模型的所需参量 
30.  */  
31. void LineParamEstimator::leastSquaresEstimate(std::vector<Point2D *> &data,   
32.                                                                                             std::vector<double> ¶meters)  
33. {  
34.     double meanX, meanY, nx, ny, norm;  
35.     double covMat11, covMat12, covMat21, covMat22; // The entries of the symmetric covarinace matrix  
36.     int i, dataSize = data.size();  
37.   
38.     parameters.clear();  
39.     if(data.size()<2)  
40.         return;  
41.   
42.     meanX = meanY = 0.0;  
43.     covMat11 = covMat12 = covMat21 = covMat22 = 0;  
44.     for(i=0; i<dataSize; i++) {  
45.         meanX +=data[i]->x;  
46.         meanY +=data[i]->y;  
47.   
48.         covMat11    +=data[i]->x * data[i]->x;  
49.         covMat12    +=data[i]->x * data[i]->y;  
50.         covMat22    +=data[i]->y * data[i]->y;  
51.     }  
52.   
53.     meanX/=dataSize;  
54.     meanY/=dataSize;  
55.   
56.     covMat11 -= dataSize*meanX*meanX;  
57.         covMat12 -= dataSize*meanX*meanY;  
58.     covMat22 -= dataSize*meanY*meanY;  
59.     covMat21 = covMat12;  
60.   
61.     if(covMat11<1e-12) {  
62.         nx = 1.0;  
63.             ny = 0.0;  
64.     }  
65.     else {      //lamda1 is the largest eigen-value of the covariance matrix   
66.                //and is used to compute the eigne-vector corresponding to the smallest  
67.                //eigenvalue, which isn't computed explicitly.  
68.         double lamda1 = (covMat11 + covMat22 + sqrt((covMat11-covMat22)*(covMat11-covMat22) + 4*covMat12*covMat12)) / 2.0;  
69.         nx = -covMat12;  
70.         ny = lamda1 - covMat22;  
71.         norm = sqrt(nx*nx + ny*ny);  
72.         nx/=norm;  
73.         ny/=norm;  
74.     }  
75.     parameters.push_back(nx);  
76.     parameters.push_back(ny);  
77.     parameters.push_back(meanX);  
78.     parameters.push_back(meanY);  
79. }  
80. /*****************************************************************************/  
81. /* 
82.  * Given the line parameters  [n_x,n_y,a_x,a_y] check if 
83.  * [n_x, n_y] dot [data.x-a_x, data.y-a_y] < m_delta 
84.  * 通过与已知法线的点乘结果，确定待测点与已知直线的匹配程度；结果越小则越符合，为 
85.  * 零则表明点在直线上 
86.  */  
87. bool LineParamEstimator::agree(std::vector<double> ¶meters, Point2D &data)  
88. {  
89.     double signedDistance = parameters[0]*(data.x-parameters[2]) + parameters[1]*(data.y-parameters[3]);   
90.     return ((signedDistance*signedDistance) < m_deltaSquared);  
91. }  

92. #include <math.h>
    #include "LineParamEstimator.h"
    
    LineParamEstimator::LineParamEstimator(double delta) : m_deltaSquared(delta*delta) {}
    /*****************************************************************************/
    /*
     * Compute the line parameters  [n_x,n_y,a_x,a_y]
     * 通过输入的两点来确定所在直线，采用法线向量的方式来表示，以兼容平行或垂直的情况
     * 其中n_x,n_y为归一化后，与原点构成的法线向量，a_x,a_y为直线上任意一点
     */
    void LineParamEstimator::estimate(std::vector<Point2D *> &data, 
    																	std::vector<double> ¶meters)
    {
    	parameters.clear();
    	if(data.size()<2)
    		return;
    	double nx = data[1]->y - data[0]->y;
    	double ny = data[0]->x - data[1]->x;// 原始直线的斜率为K，则法线的斜率为-1/k
    	double norm = sqrt(nx*nx + ny*ny);
    	
    	parameters.push_back(nx/norm);
    	parameters.push_back(ny/norm);
    	parameters.push_back(data[0]->x);
    	parameters.push_back(data[0]->y);		
    }
    /*****************************************************************************/
    /*
     * Compute the line parameters  [n_x,n_y,a_x,a_y]
     * 使用最小二乘法，从输入点中拟合出确定直线模型的所需参量
     */
    void LineParamEstimator::leastSquaresEstimate(std::vector<Point2D *> &data, 
    																							std::vector<double> ¶meters)
    {
    	double meanX, meanY, nx, ny, norm;
    	double covMat11, covMat12, covMat21, covMat22; // The entries of the symmetric covarinace matrix
    	int i, dataSize = data.size();
    
    	parameters.clear();
    	if(data.size()<2)
    		return;
    
    	meanX = meanY = 0.0;
    	covMat11 = covMat12 = covMat21 = covMat22 = 0;
    	for(i=0; i<dataSize; i++) {
    		meanX +=data[i]->x;
    		meanY +=data[i]->y;
    
    		covMat11	+=data[i]->x * data[i]->x;
    		covMat12	+=data[i]->x * data[i]->y;
    		covMat22	+=data[i]->y * data[i]->y;
    	}
    
    	meanX/=dataSize;
    	meanY/=dataSize;
    
    	covMat11 -= dataSize*meanX*meanX;
            covMat12 -= dataSize*meanX*meanY;
    	covMat22 -= dataSize*meanY*meanY;
    	covMat21 = covMat12;
    
    	if(covMat11<1e-12) {
    		nx = 1.0;
    	        ny = 0.0;
    	}
    	else {	    //lamda1 is the largest eigen-value of the covariance matrix 
    	           //and is used to compute the eigne-vector corresponding to the smallest
    	           //eigenvalue, which isn't computed explicitly.
    		double lamda1 = (covMat11 + covMat22 + sqrt((covMat11-covMat22)*(covMat11-covMat22) + 4*covMat12*covMat12)) / 2.0;
    		nx = -covMat12;
    		ny = lamda1 - covMat22;
    		norm = sqrt(nx*nx + ny*ny);
    		nx/=norm;
    		ny/=norm;
    	}
    	parameters.push_back(nx);
    	parameters.push_back(ny);
    	parameters.push_back(meanX);
    	parameters.push_back(meanY);
    }
    /*****************************************************************************/
    /*
     * Given the line parameters  [n_x,n_y,a_x,a_y] check if
     * [n_x, n_y] dot [data.x-a_x, data.y-a_y] < m_delta
     * 通过与已知法线的点乘结果，确定待测点与已知直线的匹配程度；结果越小则越符合，为
     * 零则表明点在直线上
     */
    bool LineParamEstimator::agree(std::vector<double> ¶meters, Point2D &data)
    {
    	double signedDistance = parameters[0]*(data.x-parameters[2]) + parameters[1]*(data.y-parameters[3]); 
    	return ((signedDistance*signedDistance) < m_deltaSquared);
    }

RANSAC寻找匹配的代码如下：

[cpp] view plain copy print ?

/*****************************************************************************/
template<class T, class S>
double Ransac<T,S>::compute(std::vector<S> ¶meters,
                                                      ParameterEsitmator<T,S> *paramEstimator ,
                                                    std::vector<T> &data,
                                                    int numForEstimate)
{
    std::vector<T *> leastSquaresEstimateData;
    int numDataObjects = data.size();
    int numVotesForBest = -1;
    int *arr = new int[numForEstimate];// numForEstimate表示拟合模型所需要的最少点数，对本例的直线来说，该值为2
    short *curVotes = new short[numDataObjects];  //one if data[i] agrees with the current model, otherwise zero
    short *bestVotes = new short[numDataObjects];  //one if data[i] agrees with the best model, otherwise zero


              //there are less data objects than the minimum required for an exact fit
    if(numDataObjects < numForEstimate)
        return 0;
        // 计算所有可能的直线，寻找其中误差最小的解。对于100点的直线拟合来说，大约需要100*99*0.5=4950次运算，复杂度无疑是庞大的。一般采用随机选取子集的方式。
    computeAllChoices(paramEstimator,data,numForEstimate,
                                        bestVotes, curVotes, numVotesForBest, 0, data.size(), numForEstimate, 0, arr);

       //compute the least squares estimate using the largest sub set
    for(int j=0; j<numDataObjects; j++) {
        if(bestVotes[j])
            leastSquaresEstimateData.push_back(&(data[j]));
    }
        // 对局内点再次用最小二乘法拟合出模型
    paramEstimator->leastSquaresEstimate(leastSquaresEstimateData,parameters);

    delete [] arr;
    delete [] bestVotes;
    delete [] curVotes;

    return (double)leastSquaresEstimateData.size()/(double)numDataObjects;
}