Ste*_*eng 4 matlab ellipse mathematical-optimization data-fitting
论坛
我有一组数据，显然在 3D 空间中形成了一个椭圆（不是椭圆体，而是 3D 中的曲线）。受到以下线程http://au.mathworks.com/matlabcentral/newsreader/view_thread/65773的启发 ，并在某人的帮助下，我设法运行优化代码并输出一组最佳参数x（向量）。然而，当我尝试使用这个 x 来复制椭圆时，结果是空间中的一条奇怪的直线。我已经为此好几天了。仍然不知道出了什么问题......非常沮丧......我希望有人能对此有所启发。椭圆的 Mathematica 公式与上面的线程相同，其中
3D 椭圆由以下公式给出：(x;y;z) = (z1;z2;z3) + R(alpha,beta,gamma)。(a cos(phi); b*sin(phi);0)
其中： * z 是平移向量。* R 是旋转矩阵（使用欧拉角，我们首先绕 x 轴旋转 alpha rad，然后绕 y 轴旋转 beta rad，最后再次绕 z 轴旋转 gamma rad）。* a 是椭圆的长轴 * b 是椭圆的短轴。
这是我的优化目标函数（ellipsefit.m）
function [merit]= ellipsefit(x, vmatrix) % x is the initial parameters, vmatrix stores the datapoints
load vmatrix.txt % In vmatrix, the data are stored: N rows x 3 columns
a = x(1);
b = x(2);c
alpha = x(3);
beta = x(4);
gamma = x(5);
z = [x(6),x(7),x(8)];
t = z'
[dim1, dim2]=size(vmatrix);
% construction of rotation matrix R(alpha,beta,gamma)
R1 = [cos(alpha), sin(alpha), 0; -sin(alpha), cos(alpha), 0; 0, 0, 1];v
R2 = [1, 0, 0; 0, cos(beta), sin(beta); 0, -sin(beta), cos(beta)];
R3 = [cos(gamma), sin(gamma), 0; -sin(gamma), cos(gamma), 0; 0, 0, 1];
R = R3*R2*R1;
% first compute vector phi (in the length of the data) by minimizing for every
% point in the data set the distance of this point to the ellipse
% (with its initial parameters a,b,alpha,beta,gamma, z1,z2,z3 held fixed) with respect to phi.
for i=1:dim1
point=vmatrix(i,:)';
dist=@(phi)sum((R*[a*cos(phi); b*sin(phi); 0]+t-point)).^2;
phi(i)=fminbnd(dist,0,2*pi);
end
v = [a*cos(phi); b*sin(phi); zeros(size(phi))];
P = R*v;
%The targetfunction is: g = (xi1,xi2,xi3)' -(z1,z2,z3)'-R(alpha,beta,gamma)(a cos(phi), b sin(phi),0)'
% Construction of distance function
merit = [vmatrix(:,1)-z(1)-P(1),vmatrix(:,2)-z(2)-P(2),vmatrix(:,3)-z(3)-P(3)];
merit = sqrt(sum(sum(merit.^2)))
end
这是参数初始化和调用 opts 的主要函数（xfit.m）
function [y] = xfit (x)
x= [1 1 1 1 1 1 1 1] % initial parameters
[x] = fminsearch(@ellipsefit,x) % set the distance minimum as the target function
y=x
end
在散点中重建椭圆的代码（ellipsescatter.txt）
x= [0.655,0.876,1.449,2.248,1.024,0.201,-0.11,0.002] % values obtained according to above routines
a = x(1);
b = x(2);
alpha = x(3);
beta = x(4);
gamma = x(5);
z = [x(6),x(7),x(8)];
R1 = [cos(alpha), sin(alpha), 0; -sin(alpha), cos(alpha), 0; 0, 0, 1];
R2 = [1, 0, 0; 0, cos(beta), sin(beta); 0, -sin(beta), cos(beta)];
R3 = [cos(gamma), sin(gamma), 0; -sin(gamma), cos(gamma), 0; 0, 0, 1];
R = R3*R2*R1;
phi=linspace(0,2*pi,100)
v = [a*cos(phi); b*sin(phi); zeros(size(phi))];
P = R*v;
u = P'
最后是数据点（vmatrix）
0.002037965 0.004225765 0.002020202
0.005766671 0.007269193 0.004040404
0.010004802 0.00995638 0.006060606
0.014444336 0.012502725 0.008080808
0.019083408 0.014909533 0.01010101
0.023967745 0.017144827 0.012121212
0.03019849 0.01969697 0.014591289
0.038857283 0.022727273 0.017839321
0.045443501 0.024730475 0.02020202
0.051213405 0.026346492 0.022222222
0.061038174 0.028787879 0.02555121
0.069408829 0.030575164 0.028282828
0.075785861 0.031818182 0.030321465
0.088818543 0.033954681 0.034343434
0.095538223 0.03490652 0.036363636
0.109421234 0.036499949 0.04040404
0.123800737 0.037746182 0.044444444
0.131206601 0.038218171 0.046464646
0.146438211 0.038868525 0.050505051
0.162243245 0.039117883 0.054545455
0.178662839 0.03893748 0.058585859
0.195740664 0.038296774 0.062626263
0.204545539 0.037790433 0.064646465
0.222781268 0.036340005 0.068686869
0.23715887 0.034848485 0.071748051
0.251787024 0.033009003 0.074747475
0.26196429 0.031542949 0.076767677
0.278510276 0.028787879 0.079919236
0.294365342 0.025757576 0.082799669
0.306221108 0.023197784 0.084848485
0.31843759 0.020305704 0.086868687
0.331291367 0.016967964 0.088888889
0.342989936 0.013636364 0.090622484
0.352806191 0.010606061 0.091993214
0.36201461 0.007575758 0.093211986
0.376385537 0.002386324 0.094949495
0.386214665 -0.001515152 0.096012
0.396173756 -0.005800677 0.096969697
0.406365393 -0.010606061 0.097799682
0.417897899 -0.016666667 0.098516141
0.428059375 -0.022727273 0.098889844
0.436894505 -0.028787879 0.09893196
0.444444123 -0.034848485 0.098652697
0.45074522 -0.040909091 0.098061305
0.455830971 -0.046969697 0.097166076
0.457867157 -0.05 0.096591789
0.46096663 -0.056060606 0.095199991
0.461974832 -0.059090909 0.094368708
0.462821268 -0.063709158 0.092929293
0.46279206 -0.068181818 0.091323015
0.462224312 -0.071212121 0.090097745
0.461247257 -0.074242424 0.088770148
0.459194871 -0.07812596 0.086868687
0.456406121 -0.0818267 0.084848485
0.45309565 -0.085162601 0.082828283
0.449335762 -0.088184223 0.080808081
0.445185841 -0.090933095 0.078787879
0.440695103 -0.093443633 0.076767677
0.435904796 -0.095744683 0.074747475
0.429042582 -0.098484848 0.072052312
0.419877272 -0.101489369 0.068686869
0.41402731 -0.103049401 0.066666667
0.407719192 -0.104545455 0.064554798
0.395265308 -0.106881864 0.060606061
0.388611992 -0.107880111 0.058585859
0.374697979 -0.10945186 0.054545455
0.360058411 -0.11051623 0.050505051
0.352443612 -0.11084211 0.048484848
0.336646801 -0.111097219 0.044444444
0.320085063 -0.110817414 0.04040404
0.31150078 -0.110465333 0.038383838
0.293673303 -0.109300395 0.034343434
0.275417637 -0.107575758 0.030396076
0.265228963 -0.106361993 0.028282828
0.251914589 -0.104545455 0.025603647
0.234385536 -0.101745907 0.022222222
0.223443994 -0.099745394 0.02020202
0.212154519 -0.097501571 0.018181818
0.20046153 -0.09497557 0.016161616
0.188298809 -0.092121085 0.014141414
0.17558878 -0.088883868 0.012121212
0.162241674 -0.085201142 0.01010101
0.148154337 -0.081000773 0.008080808
0.136529019 -0.077272727 0.006507255
0.127611912 -0.074242424 0.005361311
0.116762238 -0.070350086 0.004040404
0.103195122 -0.065151515 0.002507114
0.095734279 -0.062121212 0.001725236
0.081719652 -0.056060606 0.000388246
0 0 0
这个答案不是直接拟合 3D 数据，而是首先涉及数据旋转，使点平面与 xy 平面重合，然后拟合 2D 数据。
% input: data, a N x 3 array with one set of Cartesian coords per row
% remove the center of mass
CM = mean(data);
datap = data - ones(size(data,1),1)*CM;
% now rotate all points into the xy plane ...
% start by finding the plane:
[u s v]=svd(datap);
% rotate the data into the principal axes frame:
datap = datap*v;
% fit the equation for an ellipse to the rotated points
x= [0.25 0.07 0.037 0 0]'; % initial parameters
options=1;
xopt = fmins(@fellipse,x,options,[],datap) % set the distance minimum as the target function
fellipse这是基于提供的函数的函数：
function [merit]= fellipse(x,data) % x is the initial parameters, data stores the datapoints
a = x(1);
b = x(2);
alpha = x(3);
z = x(4:5);
R = [cos(alpha), sin(alpha), 0; -sin(alpha), cos(alpha), 0; 0, 0, 1];
data = data*R;
merit = 0;
[dim1, dim2]=size(data);
for i=1:dim1
dist=@(phi)sum( ( [a*cos(phi);b*sin(phi)] + z - data(i,1:2)').^2 );
phi=fminbnd(dist,0,2*pi);
merit = merit+dist(phi);
end
end
另外，请再次注意，这可以直接转化为 3D 拟合，但如果您可以假设数据点大约位于 3D 位置，那么这个答案同样好。在 2D 平面中。当前的解决方案可能比具有附加参数的 3D 解决方案更有效。
希望代码是不言自明的。我建议查看OP中包含的链接，它解释了循环的目的phi。
这是检查拟合结果的方法：
a = xopt(1);
b = xopt(2);
alpha = xopt(3);
z = [xopt(4:5) ; 0]';
phi = linspace(0,2*pi,100)';
simdat = [a*cos(phi) b*sin(phi) zeros(size(phi))];
R = [cos(alpha), -sin(alpha), 0; sin(alpha), cos(alpha), 0; 0, 0, 1];
simdat = simdat*R + ones(size(simdat,1), 1)*z ;
figure, hold on
plot3(datap(:,1),datap(:,2),datap(:,3),'o')
plot3(simdat(:,1),simdat(:,2),zeros(size(simdat,1),1),'r-')
以下是 3D 方法。它看起来不是很鲁棒，因为它对起始参数的选择非常敏感。可能需要进行一些改进。
CM = mean(data);
datap = data - ones(size(data,1),1)*CM;
xopt = [ 0.07 0.25 1 -0.408976 0.610120 0 0 0]';
options=1;
xopt = fmins(@fellipse3d,xopt,options,[],datap) % set the distance minimum as the target function
函数fellipse3d是
function [merit]= fellipse3d(x,data) % x is the initial parameters, data stores the datapoints
a = abs(x(1));
b = abs(x(2));
alpha = x(3);
beta = x(4);
gamma = x(5);
z = x(6:8)';
[dim1, dim2]=size(data);
R1 = [cos(alpha), sin(alpha), 0; -sin(alpha), cos(alpha), 0; 0, 0, 1];
R2 = [1, 0, 0; 0, cos(beta), sin(beta); 0, -sin(beta), cos(beta)];
R3 = [cos(gamma), sin(gamma), 0; -sin(gamma), cos(gamma), 0; 0, 0, 1];
R = R3*R2*R1;
data = (data - z(ones(dim1,1),:))*R;
merit = 0;
for i=1:dim1
dist=@(phi)sum( ([a*cos(phi);b*sin(phi);0] - data(i,:)').^2 );
phi=fminbnd(dist,0,2*pi);
merit = merit+dist(phi);
end
end
您可以使用以下命令可视化结果
a = xopt(1);
b = xopt(2);
alpha = -xopt(3);
beta = -xopt(4);
gamma = -xopt(5);
z = xopt(6:8)' + CM;
dim1 = 100;
phi = linspace(0,2*pi,dim1)';
simdat = [a*cos(phi) b*sin(phi) zeros(size(phi))];
R1 = [cos(alpha), sin(alpha), 0; ...
-sin(alpha), cos(alpha), 0; ...
0, 0, 1];
R2 = [1, 0, 0; ...
0, cos(beta), sin(beta); ...
0, -sin(beta), cos(beta)];
R3 = [cos(gamma), sin(gamma), 0; ...
-sin(gamma), cos(gamma), 0; ...
0, 0, 1];
R = R1*R2*R3;
simdat = simdat*R + z(ones(dim1,1),:);
figure, hold on
plot3(data(:,1),data(:,2),data(:,3),'o')
plot3(simdat(:,1),simdat(:,2),simdat(:,3),'r-')