算法思想:

算法通过最小化约束条件4ac-b^2 = 1,最小化距离误差。利用最小二乘法进行求解,首先引入拉格朗日乘子算法获得等式组,然后求解等式组得到最优的拟合椭圆。

算法的优点:

  a、椭圆的特异性,在任何噪声或者遮挡的情况下都会给出一个有用的结果;

  b、不变性,对数据的Euclidean变换具有不变性,即数据进行一系列的Euclidean变换也不会导致拟合结果的不同;

  c、对噪声具有很高的鲁棒性;

  d、计算高效性。

算法原理:

代码实现(Matlab):

 %
function a = fitellipse(X,Y) % FITELLIPSE Least-squares fit of ellipse to 2D points.
% A = FITELLIPSE(X,Y) returns the parameters of the best-fit
% ellipse to 2D points (X,Y).
% The returned vector A contains the center, radii, and orientation
% of the ellipse, stored as (Cx, Cy, Rx, Ry, theta_radians)
%
% Authors: Andrew Fitzgibbon, Maurizio Pilu, Bob Fisher
% Reference: "Direct Least Squares Fitting of Ellipses", IEEE T-PAMI,
%
% @Article{Fitzgibbon99,
% author = "Fitzgibbon, A.~W.and Pilu, M. and Fisher, R.~B.",
% title = "Direct least-squares fitting of ellipses",
% journal = pami,
% year = 1999,
% volume = 21,
% number = 5,
% month = may,
% pages = "476--480"
% }
%
% This is a more bulletproof version than that in the paper, incorporating
% scaling to reduce roundoff error, correction of behaviour when the input
% data are on a perfect hyperbola, and returns the geometric parameters
% of the ellipse, rather than the coefficients of the quadratic form.
%
% Example: Run fitellipse without any arguments to get a demo
if nargin ==
% Create an ellipse
t = linspace(,); Rx = ;
Ry = ;
Cx = ;
Cy = ;
Rotation = .; % Radians NoiseLevel = .; % Will add Gaussian noise of this std.dev. to points x = Rx * cos(t);
y = Ry * sin(t);
nx = x*cos(Rotation)-y*sin(Rotation) + Cx + randn(size(t))*NoiseLevel;
ny = x*sin(Rotation)+y*cos(Rotation) + Cy + randn(size(t))*NoiseLevel; % Clear figure
clf
% Draw it
plot(nx,ny,'o');
% Show the window
figure(gcf)
% Fit it
params = fitellipse(nx,ny);
% Note it may return (Rotation - pi/) and swapped radii, this is fine.
Given = round([Cx Cy Rx Ry Rotation*])
Returned = round(params.*[ ]) % Draw the returned ellipse
t = linspace(,pi*);
x = params() * cos(t);
y = params() * sin(t);
nx = x*cos(params())-y*sin(params()) + params();
ny = x*sin(params())+y*cos(params()) + params();
hold on
plot(nx,ny,'r-') return
end % normalize data
mx = mean(X);
my = mean(Y);
sx = (max(X)-min(X))/;
sy = (max(Y)-min(Y))/; x = (X-mx)/sx;
y = (Y-my)/sy; % Force to column vectors
x = x(:);
y = y(:); % Build design matrix
D = [ x.*x x.*y y.*y x y ones(size(x)) ]; % Build scatter matrix
S = D'*D; % Build 6x6 constraint matrix
C(,) = ; C(,) = -; C(,) = ; C(,) = -; % Solve eigensystem
if
% Old way, numerically unstable if not implemented in matlab
[gevec, geval] = eig(S,C); % Find the negative eigenvalue
I = find(real(diag(geval)) < 1e-8 & ~isinf(diag(geval))); % Extract eigenvector corresponding to negative eigenvalue
A = real(gevec(:,I));
else
% New way, numerically stabler in C [gevec, geval] = eig(S,C); % Break into blocks
tmpA = S(:,:);
tmpB = S(:,:);
tmpC = S(:,:);
tmpD = C(:,:);
tmpE = inv(tmpC)*tmpB';
[evec_x, eval_x] = eig(inv(tmpD) * (tmpA - tmpB*tmpE)); % Find the positive (as det(tmpD) < ) eigenvalue
I = find(real(diag(eval_x)) < 1e-8 & ~isinf(diag(eval_x))); % Extract eigenvector corresponding to negative eigenvalue
A = real(evec_x(:,I)); % Recover the bottom half...
evec_y = -tmpE * A;
A = [A; evec_y];
end % unnormalize
par = [
A()*sy*sy, ...
A()*sx*sy, ...
A()*sx*sx, ...
-*A()*sy*sy*mx - A()*sx*sy*my + A()*sx*sy*sy, ...
-A()*sx*sy*mx - *A()*sx*sx*my + A()*sx*sx*sy, ...
A()*sy*sy*mx*mx + A()*sx*sy*mx*my + A()*sx*sx*my*my ...
- A()*sx*sy*sy*mx - A()*sx*sx*sy*my ...
+ A()*sx*sx*sy*sy ...
]'; % Convert to geometric radii, and centers thetarad = 0.5*atan2(par(),par() - par());
cost = cos(thetarad);
sint = sin(thetarad);
sin_squared = sint.*sint;
cos_squared = cost.*cost;
cos_sin = sint .* cost; Ao = par();
Au = par() .* cost + par() .* sint;
Av = - par() .* sint + par() .* cost;
Auu = par() .* cos_squared + par() .* sin_squared + par() .* cos_sin;
Avv = par() .* sin_squared + par() .* cos_squared - par() .* cos_sin; % ROTATED = [Ao Au Av Auu Avv] tuCentre = - Au./(.*Auu);
tvCentre = - Av./(.*Avv);
wCentre = Ao - Auu.*tuCentre.*tuCentre - Avv.*tvCentre.*tvCentre; uCentre = tuCentre .* cost - tvCentre .* sint;
vCentre = tuCentre .* sint + tvCentre .* cost; Ru = -wCentre./Auu;
Rv = -wCentre./Avv; Ru = sqrt(abs(Ru)).*sign(Ru);
Rv = sqrt(abs(Rv)).*sign(Rv); a = [uCentre, vCentre, Ru, Rv, thetarad];

实验效果:

a、同等噪声条件下,不同长度的样本点,导致的拟合结果,如下所示:

b、相同长度的样本点下,不同噪声的样本点,导致的拟合结果,如下所示:

c、少样本点下,拟合结果如下:

源码下载:

      地址: FitEllipse

参考文献:

[1]. Andrew W. Fitzgibbon, Maurizio Pilu and Robert B. Fisher. Direct Least Squares Fitting of Ellipses. 1996.

[2]. http://research.microsoft.com/en-us/um/people/awf/ellipse/

基于直接最小二乘的椭圆拟合(Direct Least Squares Fitting of Ellipses)的更多相关文章

  1. 基于MATLAB的多项式数据拟合方法研究-毕业论文

    摘要:本论文先介绍了多项式数据拟合的相关背景,以及对整个课题做了一个完整的认识.接下来对拟合模型,多项式数学原理进行了详细的讲解,通过对文献的阅读以及自己的知识积累对原理有了一个系统的认识.介绍多项式 ...

  2. opencv学习之路(27)、轮廓查找与绘制(六)——外接圆、椭圆拟合、逼近多边形曲线、计算轮廓面积及长度、提取不规则轮廓

    一.最小外接圆 #include "opencv2/opencv.hpp" #include<iostream> using namespace std; using ...

  3. 【Matlab&Mathematica】对三维空间上的点进行椭圆拟合

    问题是这样:比如有一个地心惯性系的轨道,然后从轨道上取了几个点,问能不能根据这几个点把轨道还原了? 当然,如果知道轨道这几个点的速度的情况下,根据轨道六根数也是能计算轨道的,不过真近点角是随时间变动的 ...

  4. 基于EM的多直线拟合

    作者:桂. 时间:2017-03-22  06:13:50 链接:http://www.cnblogs.com/xingshansi/p/6597796.html 声明:欢迎被转载,不过记得注明出处哦 ...

  5. 基于EM的多直线拟合实现及思考

    作者:桂. 时间:2017-03-22  06:13:50 链接:http://www.cnblogs.com/xingshansi/p/6597796.html 声明:欢迎被转载,不过记得注明出处哦 ...

  6. Dlib Opencv cv2.fitEllipse用于人眼轮廓椭圆拟合

    dlib库的安装以及人脸特征点的识别分布分别在前两篇博文里面 Dlib Python 检测人脸特征点 Face Landmark Detection Mac OSX下安装dlib (Python) 这 ...

  7. 6、基于highcharts实现的线性拟合,计算部分在java中实现,画的是正态概率图

    1.坐标点类 package cn.test.domain; public class Point { double x; double y; public Point(){ } public Poi ...

  8. C# + Matlab 实现计件工时基于三层BP神经网络的拟合--真实项目

    工序工时由该工序的工艺参数决定,有了工时后乘以固定因子就是计件工资.一般参考本地小时工资以及同类小时工资并考虑作业的风险等因素给出固定因子 采用的VS2010 , Matlab2015a 64,  开 ...

  9. 【翻译】拟合与高斯分布 [Curve fitting and the Gaussian distribution]

    参考与前言 英文原版 Original English Version:https://fabiandablander.com/r/Curve-Fitting-Gaussian.html 如何通俗易懂 ...

随机推荐

  1. 查看sql语句加锁信息

    问题: 最近使用quartz集群,总是报deadlock问题,所以需要查看一下执行的sql导致的加锁冲突. 步骤: 1.在要测试的库中创建指定表innodb_lock_monitor create t ...

  2. WCF - Autofac IOC

    /// <summary> /// IOC实例提供者,基于AutoFac /// /// </summary> public class IocInstanceProvider ...

  3. Python_oldboy_自动化运维之路_全栈考试(七)

    1. 计算100-300之间所有能被3和7整除的所有数之和 # -*- coding: UTF-8 -*- #blog:http://www.cnblogs.com/linux-chenyang/ c ...

  4. show engine innodb status 详细介绍

    Contents Header1 SEMAPHORES. 1 LATEST DETECTED DEADLOCK. 3 TRANSACTIONS. 5 什么是purge操作... 5 FILE I/O. ...

  5. ***四种参数传递的形式——URL,超链接,js,form表单

    什么时候用GET,  查,删 什么时候用POST,增,改  (特列:登陆用Post,因为不能让用户名和密码显示在URL上) 4种get传参方式 <html xmlns="http:// ...

  6. Kubernetes监控:部署Heapster、InfluxDB和Grafana

    本节内容: Kubernetes 监控方案 Heapster.InfluxDB和Grafana介绍 安装配置Heapster.InfluxDB和Grafana 访问 grafana 访问 influx ...

  7. 一步一步学习IdentityServer3 (8)

    IdentityServer3结合Hangfire及Cookies中间件实现授权 Idr3数据库Token过期管理 GlobalConfiguration.Configuration.UseSqlSe ...

  8. Django实战(10):单元测试

    尽早进行单元测试(UnitTest)是比较好的做法,极端的情况甚至强调“测试先行”.现在我们已经有了第一个model类和Form类,是时候开始写测试代码了. Django支持python的单元测试(u ...

  9. ifdown eth0或service network restart

    错误提示信息如下: Shutting down interface eth0:  Error: Device 'eth0' (/org/freedesktop/NetworkManager/Devic ...

  10. Ionic入门五:表单

    一.输入框 list 类同样可以用于 input 元素.item-input 和 item 类指定了文本框及其标签. 1.输入框属性:placeholder 以下实例中,默认为100%宽度(左右两侧没 ...