RANSAC在2D线性回归（可靠线配合） [英] RANSAC linear regression in 2D (robust line fit)
本文介绍了RANSAC在2D线性回归（可靠线配合）的处理方法，对大家解决问题具有一定的参考价值，需要的朋友们下面随着小编来一起学习吧！
问题描述
该文件包含C ++ $ C $下RANSAC线性回归与使用OpenCV的单元测试。一个典型的用一条线拟合的是
This file includes C++ code for RANSAC Linear regression with a unit test that uses OpenCV. A typical use of a line fit is
float x[N], float y[N];
bool inliers[N];
float RMSE = RANSAC_line(x, y, npoints, param, NITERATIONS, MAX_ER, inliers);
下面是文件
/*
* RANSAC.h
*
* Created on: Mar 4, 2013
* Author: vivanchenko
*
* Description: line is represented by equation
* a*x+b*y=d, where a=cos(alpha), b=sin(alpha), alpha - the angle
* between a horizontal axis and a line normal, and d is the distance
* from the origin to the line;
*
* Representing a line in such a way is well suited for fitting error
* evaluation and allows to avoid some marginal cases that arise from
* a traditional line representation y=ax+b
*/
#ifndef RANSAC_H_
#define RANSAC_H_
#include <string.h>
#include <assert.h>
#include <stdio.h> // NULL symbol
#include "cv.h"
#include "highgui.h"
using namespace std;
using namespace cv;
#define SQR(a) ((a)*(a))
#define MAX_FLOAT (std::numeric_limits<float>::digits)
#define LARGE_NUMBER (MAX_FLOAT/2)
#define SMALL_NUMBER (1.0F/LARGE_NUMBER)
#define SIGN(x) ((x)>0?1:(-1))
// unit test params
const int NDATA_UNIT_TEST = 100;
const float TRUE_PARAM_UNIT_TEST[2] = {0.9f, -27.0f};
const float NOISE_LEVEL_UNIT_TEST = 1.0f;
const float OUTLIER_PROPORTION_UNIT_TEST = 0.3f;
const float SHIFT_OUTLIERS_UNIT_TEST = 10*NOISE_LEVEL_UNIT_TEST; // shift introduced by outliers
const int UNIT_TEST_IMG_WIDTH = 800;
const int UNIT_TEST_IMG_HEIGHT = 800;
// conversion to string for many types
template <class T>
string toString(T a) {
stringstream ss (stringstream::in | stringstream::out);
ss << a;
return(ss.str());
}
// outputs redctangle that includes the data
cv::Rect dataRange(float* x, float* y, const int N) {
float minx = x[0];
float miny = y[0];
float maxx = x[0];
float maxy = y[0];
for (int i=1; i<N; i++) {
if (minx > x[i])
minx = x[i];
if (miny > y[i])
miny = y[i];
if (maxx < x[i])
maxx = x[i];
if (maxy < y[i])
maxy = y[i];
}
Rect_<float> rect(minx, miny, maxx-minx+1, maxy-miny+1);
return rect;
}
// simple linear transformation
inline float linear(float src, float mult_val, float plus_val) {
return(src*mult_val+plus_val);
}
// shows points, marks inliers, draws a fit line (Y points up);
void showPoints(Mat& image, float* x, float* y, const int N,
bool* inlrs, float* param) {
bool* inliers = inlrs;
if (inlrs==NULL) {
inliers = new bool[N];
std::fill_n(inliers, N, true);
}
// image dimensions
int h = image.rows;
int w = image.cols;
const int pad = 10; // border padding
// range of data
Rect_<float> range = dataRange(x, y, N);
string str = "x: " + toString(range.x) + ".." + toString(range.x+range.width) +
"; y: " + toString(range.y) + ".." + toString(range.y+range.height) +
"; data: y = " + toString(TRUE_PARAM_UNIT_TEST[0]) + "x + " +
toString(TRUE_PARAM_UNIT_TEST[1]) + " + noise";
putText(image, str, Point(20, 20), FONT_HERSHEY_PLAIN, 1, Scalar(255) );
// maping data to image
float dataToImg = min((h-2*pad)/range.height,
(w-2*pad)/range.width);
// inverse y direction
int x0 = pad;
int y0 = h-pad;
// points
for (int i=0; i<N; i++) {
// adjust for origin
float datax = x[i]-range.x;
float datay = y[i]-range.y;
// transform to image coordiantes
int imgx = linear(datax, dataToImg, x0) + 0.5f;
int imgy = linear(datay, -dataToImg, y0) + 0.5f;
int thickness = inliers[i]?2:1;
circle(image, Point(imgx, imgy), 2, Scalar(255), thickness);
}
// line points
float datax1 = range.x;
float datax2 = range.x+range.width-1;
float datay1 = linear(datax1, param[0], param[1]);
float datay2 = linear(datax2, param[0], param[1]);
// adjust for origin
datax1-=range.x;
datax2-=range.x;
datay1-=range.y;
datay2-=range.y;
// transform to image coordiantes
int x1 = linear(datax1, dataToImg, x0) + 0.5f;
int x2 = linear(datax2, dataToImg, x0) + 0.5f;
int y1 = linear(datay1, -dataToImg, y0) + 0.5f; // Y points up
int y2 = linear(datay2, -dataToImg, y0) + 0.5f;
cv::line(image, Point(x1, y1), Point(x2, y2), Scalar(255), 1);
if (inlrs==NULL)
delete inliers;
}
// Root mean sqaure error (RMSE) of line fit y=param[0]*x+param[1]
float errorLine(float* x, float* y, int N, float* param,
const bool* inliers = NULL) {
if (N<=0 || x==NULL || y==NULL || param==NULL)
return MAX_FLOAT;
int ninliers = (inliers==NULL?N:0);
double RMSE = 0;
// error accumulation loop
for (int i = 0; i < N; i++) {
if (inliers!=NULL) {
if (inliers[i])
ninliers++;
else
continue;
}
float y_predicted = param[0] * x[i] + param[1];
RMSE += SQR(y[i] - y_predicted);
}
if (ninliers==0)
return LARGE_NUMBER;
else
return sqrt(RMSE/ninliers);
}
// simple line fit using sum of squared differences
// inliers are used to specify a subset of points for a fit
float fitLine(float* x, float* y, int N, float* param,
const bool* inliers = NULL, const bool debug = false) {
if (N<=0 || x==NULL || y==NULL || param==NULL) {
if (debug)
cout<<"ERROR fit line quadratic: N<=0 || x==NULL || y==NULL || param==NULL"<<endl;
return MAX_FLOAT;
}
int ninliers = (inliers==NULL?N:0);
double sum_x = 0;
double sum_y = 0;
double sum_xy = 0;
double sum_x2 = 0;
// create specific sums of x, y, xy, x^2
for (int i = 0; i < N; i++) {
// use inliers only
if (inliers!=NULL) {
if (!inliers[i])
continue;
else
ninliers++;
}
sum_x += x[i];
sum_y += y[i];
sum_xy += x[i] * y[i];
sum_x2 += x[i] * x[i];
}
if(ninliers < 2) {
if (debug)
cout<<"ERROR fit line quadratic: less than 2 data points"<<endl;
return MAX_FLOAT;
}
// means
double mean_x = sum_x / ninliers;
double mean_y = sum_y / ninliers;
float varx = sum_x2 - sum_x * mean_x;
float cov = sum_xy - sum_x * mean_y;
// eliminate bias: variance is e a bit underestimated since df=N-1, see
// http://davidmlane.com/hyperstat/B16616.html)
// if (ninliers>1)
// varx *= (float)ninliers/(ninliers-1);
// quadratic fit
if (abs(varx) < SMALL_NUMBER) {
if (debug)
cout<<"ERROR fit line quadratic: zero variance" <<endl;
return MAX_FLOAT;
}
// see http://easycalculation.com/statistics/learn-regression.php
param[0] = cov / varx;
param[1] = mean_y - param[0] * mean_x;
return errorLine(x, y, N, param, inliers);
}
// RANSAC line fit (number of inliers has priority over RMSE).
float RANSAC_line(float* x, float* y, const int N, float* param,
const int niter = 10, const float maxError = 1.0f,
bool* inlrs = NULL, bool debug = false) {
if (x==NULL || y==NULL || N<2) {
if (debug)
cout<<"x==NULL || y==NULL || N<2" <<endl;
return MAX_FLOAT;
}
srand (time(NULL));
// internal stopping criterions
const float RMSE_OK = 0.1f;
const float INLIERS_RATIO_OK = 0.7f;
int ninliers, best_ninliers = 0;
float inliers_ratio = 0;
float RMSE, bestRMSE = MAX_FLOAT;
float cur_param[2];
bool* inliers = inlrs;
if (inlrs==NULL) {
inliers = new bool[N];
std::fill_n(inliers, N, true);
}
// iterations
int iter;
for (iter = 0; iter<niter; iter++) {
// 1. select a random set of 2 inliers
int i1 = rand() % N; // [0, N[
int i2 = i1;
while (i2==i1)
i2 = rand() % N;
// 2. select minimum number of points (2)
float x1 = x[i1];
float x2 = x[i2];
float y1 = y[i1];
float y2 = y[i2];
// TODO: we may parameterize the line differently (alpha, d)
if (abs(x1-x2) < SMALL_NUMBER)
cur_param[0] = LARGE_NUMBER * SIGN(cur_param[0]);
else
cur_param[0] = (y1-y2)/(x1-x2);
cur_param[1] = y1-cur_param[0]*x1;
if (abs(cur_param[0]) < SMALL_NUMBER) // flat line?
cur_param[0] = SMALL_NUMBER * SIGN(cur_param[0]);
// 3. determine inliers using the whole set
for (int i=0; i<N; i++) {
float y_fit = cur_param[0]*x[i] + cur_param[1];
float x_fit = (y[i] - cur_param[1])/cur_param[0];
if (max(abs(y[i]-y_fit), abs(x[i]-x_fit)) < maxError) { // block distance
inliers[i] = true;
} else {
inliers[i] = false;
}
}
// 4. re-calculate params via quadratic fit on all inliers
RMSE = fitLine(x, y, N, cur_param, (const bool*)inliers);
if (abs(cur_param[0]) < SMALL_NUMBER) // flat line?
cur_param[0] = SMALL_NUMBER * SIGN(cur_param[0]);
// 5. re-calculate inliers
ninliers = 0;
for (int i=0; i<N; i++) {
float y_fit = cur_param[0]*x[i] + cur_param[1];
float x_fit = (y[i] - cur_param[1])/cur_param[0];
if (max(abs(y[i]-y_fit), abs(x[i]-x_fit)) < maxError) {
inliers[i] = true;
ninliers++;
} else {
inliers[i] = false;
}
}
// 6. calculate the error
RMSE = errorLine(x, y, N, cur_param, (const bool*)inliers);
// found a better solution?
if (best_ninliers < ninliers) {
best_ninliers = ninliers;
param[0] = cur_param[0];
param[1] = cur_param[1];
bestRMSE = RMSE;
}
// 7. check exit condition
inliers_ratio = (float)best_ninliers/N;
if (RMSE < RMSE_OK && inliers_ratio > INLIERS_RATIO_OK) {
if (debug)
cout<<"Breaking early after "<< iter+1<<" iterations"<<endl;
break;
}
} // iterations
if (debug)
cout<<"inliers ratio = "<< inliers_ratio <<endl;
// 8. recreate inliers for the best parameters
for (int i=0; i<N; i++) {
float y_fit = param[0]*x[i] + param[1];
float x_fit = (y[i] - param[1])/param[0];
if (max(abs(y[i]-y_fit), abs(x[i]-x_fit)) < maxError) { // block distance
inliers[i] = true;
} else {
inliers[i] = false;
}
}
if (inlrs==NULL)
delete inliers;
return bestRMSE;
}
// generates the data give the random seed
void generateData(float* x, float* y, unsigned int seed = 0, bool hasOutl = false) {
// initialize pseudo-random generator
srand (seed);
for (int i=0; i<NDATA_UNIT_TEST; i++) {
// uniform noise (negative and positive -50..50)
float noise = (float)(rand() % 100-50) *
NOISE_LEVEL_UNIT_TEST / 100.0f;
//cout<<noise<<"; ";
// shift
int noutlisers = (float)NDATA_UNIT_TEST * OUTLIER_PROPORTION_UNIT_TEST;
int start = NDATA_UNIT_TEST/2-noutlisers/2; // put outliers in the middle
bool outlier = (i > start ) && (i < start + noutlisers);
if (hasOutl && outlier)
noise += SHIFT_OUTLIERS_UNIT_TEST ;
x[i] = i-NDATA_UNIT_TEST/2; // center x data around 0
y[i] = TRUE_PARAM_UNIT_TEST[0]*x[i]+TRUE_PARAM_UNIT_TEST[1]+noise;
//cout<<"x, y = "<<x[i]<<"; "<<y[i]<<endl;
}
}
// generates the data from pasted numbers
void generateDataPaste(float* x, float* y) {
const int n = 63;
// compiler will generate an error if n is inaccurate
float data[n][2] = {
{1, 1},
{2, 1},
{3, 1},
{4, 1},
{5, 1},
{6, 1},
{7, 1},
{8, 1},
{9, 1},
{2, 2},
{3, 2},
{4, 2},
{5, 2},
{6, 2},
{7, 2},
{8, 2},
{9, 2},
{10, 2},
{11, 2},
{12, 2},
{13, 2},
{14, 2},
{15, 2},
{16, 2},
{17, 2},
{8, 3},
{10, 3},
{12, 3},
{13, 3},
{14, 3},
{15, 3},
{16, 3},
{17, 3},
{18, 3},
{9, 4},
{10, 4},
{11, 4},
{12, 4},
{13, 4},
{14, 4},
{15, 4},
{16, 4},
{17, 4},
{18, 4},
{19, 4},
{20, 4},
{21, 4},
{22, 4},
{23, 4},
{24, 4},
{25, 4},
{17, 5},
{18, 5},
{19, 5},
{20, 5},
{21, 5},
{22, 5},
{23, 5},
{24, 5},
{25, 5},
{26, 5},
{27, 5},
{28, 5}};
// crop or repeate data to get a required sample size
for (int i=0; i<NDATA_UNIT_TEST; i++) {
x[i] = data[i%n][0];
y[i] = data[i%n][1];
}
}
// unit test of quadratic line fit
bool uniTest_fitLine_QUADRATIC(int col = 0, int row = 0) {
const bool hasOutliers = false; // cannot handle outliers well
const bool dataFromParam = false; // either data from param or pasted ones
string str = hasOutliers?" with outliers":" with no outliers";
cout<<"unit-test_fitLine()"<<str<<endl;
bool res = false;
// opencv window
Mat img(UNIT_TEST_IMG_HEIGHT, UNIT_TEST_IMG_WIDTH, CV_8U);
cv::namedWindow("QUADRATIC", CV_WINDOW_AUTOSIZE);
cv::moveWindow("QUADRATIC", col*(img.cols+50), row*(img.rows+50));
// create data
float x[NDATA_UNIT_TEST], y[NDATA_UNIT_TEST];
if (dataFromParam)
generateData(x, y, time(NULL), hasOutliers);
else
generateDataPaste(x, y);
// fit the line
float param[2];
float er = fitLine(x, y, NDATA_UNIT_TEST, param) ;
if (er < NOISE_LEVEL_UNIT_TEST || !dataFromParam)
res = true;
// print the result
cout<<"a, b = "<<param[0]<<"; "<<param[1]<<"; RMSE = "<<er<<endl;
if (dataFromParam) {
cout<<"True a = "<<TRUE_PARAM_UNIT_TEST[0]<<"; b = "<<
TRUE_PARAM_UNIT_TEST[1]<<endl;
cout<<"combined param error = "<<
abs(TRUE_PARAM_UNIT_TEST[0]-param[0])+
abs(TRUE_PARAM_UNIT_TEST[1]-param[1])<<endl;
if (res)
cout<<"===passed"<<endl;
else
cout<<"===FAIL!"<<endl;
cout<<endl;
}
// visualize
showPoints(img, x, y, NDATA_UNIT_TEST, NULL, param);
imshow("QUADRATIC", img);
cv::waitKey(10);
return res;
}
// unit test of RANSAC line fit
bool uniTest_fitLine_RANSAC(int col = 0, int row = 0) {
const bool hasOutliers = true; // handles outliers gracefully
const bool dataFromParam = false; // either data from param or pasted ones
string str = hasOutliers?" with outliers":" with no outliers";
cout<<"unit-test_RANSAC()"<<str<<endl;
bool res = false;
// opencv window
Mat img(UNIT_TEST_IMG_HEIGHT, UNIT_TEST_IMG_WIDTH, CV_8U);
cv::namedWindow("RANSAC", CV_WINDOW_AUTOSIZE);
cv::moveWindow("RANSAC", col*(img.cols+50), row*(img.rows+50));
// create data
float x[NDATA_UNIT_TEST], y[NDATA_UNIT_TEST];
if (dataFromParam)
generateData(x, y, time(NULL), hasOutliers);
else
generateDataPaste(x, y);
// parameters
float param[2];
int niter = 20;
float maxError = 1.0f;
bool inliers[NDATA_UNIT_TEST];
bool debug = true;
float er;
// function call
er = RANSAC_line(x, y, NDATA_UNIT_TEST, param, niter, maxError,
inliers, debug);
if (er < NOISE_LEVEL_UNIT_TEST || !dataFromParam)
res = true;
// print the result
cout<<"outliers: ";
int noutliers = 0;
for (int i=0; i<NDATA_UNIT_TEST; i++) {
if (!inliers[i]) {
noutliers++;
cout<<i<<"; ";
if (noutliers % 30==0)
cout<<endl;
}
}
cout<<" overall "<<noutliers<<endl;
cout<<"a, b = "<<param[0]<<"; "<<param[1]<<"; RMSE = "<<er<<endl;
if (dataFromParam) {
cout<<"True a = "<<TRUE_PARAM_UNIT_TEST[0]<<"; b = "<<
TRUE_PARAM_UNIT_TEST[1]<<endl;
cout<<"combined param error = "<<
abs(TRUE_PARAM_UNIT_TEST[0]-param[0])+
abs(TRUE_PARAM_UNIT_TEST[1]-param[1])<<endl;
if (res)
cout<<"===passed"<<endl;
else
cout<<"===FAIL!"<<endl;
cout<<endl;
}
// visualize
showPoints(img, x, y, NDATA_UNIT_TEST, inliers, param);
imshow("RANSAC", img);
cv::waitKey(10);
return res;
}
#endif /* RANSAC_H_ */
在code提供耐受高达90％的异常值的同时迅速找到解决办法的能力。
The code provide the ability to tolerate up to 90% of outliers while quickly finding a solution.
推荐答案
使用代替：
RMSE + = SQR（Y [I] - Y_ predicted）; 的点来计算的线
的距离
您应该只是使用点的实际距离与
行
ABS（y_i-cur_m * x_i-cur_b）/开方（cur_m * cur_m + 1）或
ABS（pt.y-cur_m * pt.x-cur_b）* POW（cur_m * cur_m + 1， - 5）（少师）
或至少接近：
ABS（y_i -cur_m * x_i-cur_b）/ cur_m
Instead of using:
RMSE += SQR(y[i] - y_predicted); for the distance of the point to the calculated line
you should rather use the actual distance of the point to the line with
abs(y_i-cur_m*x_i-cur_b)/sqrt(cur_m*cur_m+1) or abs(pt.y-cur_m*pt.x-cur_b)* pow(cur_m*cur_m+1,-.5) (less divisions)
or at least approximately: abs(y_i-cur_m*x_i-cur_b)/cur_m
这篇关于RANSAC在2D线性回归（可靠线配合）的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！
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