在多篇文章中提到玻尔兹曼熵来表征评价均匀性（还有各分区内的星数标准差）。

image-20220228111816120

image-20220301160342968

image-20220301211333236

多尺度像面分割《CN201510107562.0-用于星敏感器的筛选导航星的方法》

玻尔兹曼熵《Boltzmann entropy-based guide star selection algorithm for star tracker》2005

正交网格法《A general method of the automatical selection of guide star》

用到uniformity的计算

均匀性计算《Star trackers, star catalogs, and attitude determination-Probabilistic aspects of system design》1993

《基于螺旋基准点的导航星选取方法》

均匀性很好

image-20220302143041244

image-20220302151757196

clear all ;close all;
figure(1);
sphere(50);
alpha(0.1);
hold on ;

z=[];
theta=[];
fai=[];

n = 30;
k = 0:n-1;
z = 1-(2*k+1)/n;
theta=acos(z);
fai=sqrt(n*pi).*theta;
x = sin(theta).*cos(fai);
y = sin(theta).*sin(fai);
A = [x.', y.' ,z.'];
for i=1:length(A)-1 
    delta(i,:) = acosd(A(i,:)*A(i+1,:).');
end
mean = mean(delta);
newmatrix = 1/length(A).*(A.'*A)
eigen = eig(newmatrix)
eigen(1)*log(3*eigen(1)) + eigen(2)*log(3*eigen(2)) + eigen(3)*log(3*eigen(3))

scatter3(x,y,z,'filled');
plot3(x,y,z,'r-','LineWidth',2);
hold on;
quiver3(0,0,0,1,0,0,1,'k','filled','LineWidth',2);
text(1.1,0,0,'x','FontSize',14);
quiver3(0,0,0,0,1,0,1,'k','filled','LineWidth',2);
text(0,1.1,0,'y','FontSize',14);
quiver3(0,0,0,0,0,1,1,'k','filled','LineWidth',2);
text(0,0,1.1,'z','FontSize',14);
quiver3(0,0,0,A(2,1),A(2,2),A(2,3),1,'LineWidth',2);
hold on;
s=20;
xmin = cosd(s) * A(2,1) - sind(s) * sqrt(1-A(2,1)^2);
xmax = cosd(s) * A(2,1) + sind(s) * sqrt(1-A(2,1)^2);
ymin = cosd(s) * A(2,2) - sind(s) * sqrt(1-A(2,2)^2);
ymax = cosd(s) * A(2,2) + sind(s) * sqrt(1-A(2,2)^2);
zmin = cosd(s) * A(2,3) - sind(s) * sqrt(1-A(2,3)^2);
zmax = cosd(s) * A(2,3) + sind(s) * sqrt(1-A(2,3)^2);
                     
plotcube([xmax-xmin ymax-ymin zmax-zmin],[ xmin  ymin  zmin],.8,[1 0 0]);

view(-30,10)
axis equal

image-20220320155757356

后来我又看到 https://zhuanlan.zhihu.com/p/25988652?group_id=828963677192491008 菲波那契网格

clear all ;close all;
figure(1);
sphere(50);
axis equal;
alpha(0.1);
hold on ;

n = 30;
k = 1:n;
c = (sqrt(5)-1)/2;
z = (2*k-1)/n - 1;
x = sqrt(1-z.^2).*cos(2*pi.*k*c);
y = sqrt(1-z.^2).*sin(2*pi.*k*c);

scatter3(x,y,z);


A = [x.', y.' ,z.'];

newmatrix = 1/length(A).*(A.'*A)
eigen = eig(newmatrix)
eigen(1)*log(3*eigen(1)) + eigen(2)*log(3*eigen(2)) + eigen(3)*log(3*eigen(3))

image-20220320160017779

效果并不好

image-20220321101324772

阿基米德螺旋线

image-20220320105605922

clear all; close all;
a=0;%起始位置
b=3;%螺线间距参数
theta = 0:0.05*pi:10*pi;%θ的范围和步长，同时也可以控制螺线的旋转方向
coe=a+b*theta;%阿基米德螺线方程
x = (coe.*cos(theta));%因使用需要，获取直角坐标系下x轴的坐标并进行四舍五入
y = (coe.*sin(theta));%因使用需要，获取直角坐标系下y轴的坐标并进行四舍五入
r = pi*b;
for i = 1 :length(x)
    hold on;
    rectangle('Position',[x(i)-r,y(i)-r,2*r,2*r],'Curvature',[1,1],'linewidth',1);
end
plot(x,y,'c');%将获取的坐标打印在图纸上
xt=x';%将x轴数据转置成列，便于导出使用
yt=y';%将y周数据转置成列，便于导出使用
axis equal

image-20220307114759208

弧长公式 https://www.zhihu.com/question/27384632

《均匀快速的导航星选取方法》

在赤纬带平分点，矩形范围的代码见下。（两角和余弦公式）

image-20220302200145394

image-20220302211206988

《16mv精细导星星库构建与评价》

分区后取最亮几个。

国内研究构建星库的方法主要分为两类:第一类从星表划分角度出发,通过不同的划分方式进行导星筛选,现有的星表划分方法主要有赤纬带法、圆锥法、球矩形法、内接正方体法等;第二类从局部天区出发,主要包括回归筛星法、星等加权算法等。

评价标准：完备性 、均匀性和冗余性。（均匀性、速度、多于3个星的分区数概率）

image-20220301210751502

image-20220301210838500

《适用于小视场星敏感器的导航星表构建方法》

image-20220303214512056

在1993年的均匀性计算文章中提到这一计算方法

总体均匀性计算方法：

image-20220228225805510

image-20220228225835383

局部均匀性计算方法：

image-20220304094711632

以上一篇《基于星座聚类的星敏感器导航星优选算法研究》为例，

所有4908个点的全局均匀性为

% boltzmannEntropy.m
clear
clc
profile on
load("附件2  简易星表.mat");
A  =  [  cosd(star_data(:,3)).*cosd(star_data(:,2))   ,   cosd(star_data(:,3)).*sind(star_data(:,2))  ,  sind(star_data(:,3))  ];
newmatrix = 1/length(A).*(A.'*A)
eigen = eig(newmatrix)

eigen(1)*log(3*eigen(1)) + eigen(2)*log(3*eigen(2)) + eigen(3)*log(3*eigen(3))

newmatrix =

​ 0.2799 -0.0070 0.0483
-0.0070 0.3631 0.0007
​ 0.0483 0.0007 0.3570

eigen =

​ 0.2562
​ 0.3632
​ 0.3805

ans =

​ 0.0142

均匀化后 得到39个点 data=

clear
clc
profile on
load("附件2  简易星表.mat");
bandwidth = 0;
%% 加载数据
% : = 1:end;
data =  [  cosd(star_data(:,3)).*cosd(star_data(:,2))   ,   cosd(star_data(:,3)).*sind(star_data(:,2))  ,  sind(star_data(:,3))  ];

x=data';

tic
    while length(data)>150
    	bandwidth = bandwidth + 0.1;
        [clustCent,point2cluster,clustMembsCell] = MeanShiftCluster(data.',bandwidth);
        numClust = length(clustMembsCell);
        while (length(find(point2cluster==mode(point2cluster)))>1)
            [val,idx]=min(dist(clustCent(:,mode(point2cluster)).',data.'))   ;
            data(idx,:)=[];
            x(:,idx)=[];
            [clustCent,point2cluster,clustMembsCell] = MeanShiftCluster(data.',bandwidth);
        end
        numClust = length(clustMembsCell);
        
    end

figure(1),clf,hold on;axis equal
cVec = 'bgrcmykbgrcmykbgrcmykbgrcmykbgrcmykbgrcmykbgrcmykbgrcmykbgrcmykbgrcmykbgrcmykbgrcmyk';%, cVec = [cVec cVec];
for k = 1:min(numClust,length(cVec))
    myMembers = clustMembsCell{k};
    myClustCen = clustCent(:,k);
    plot(x(2,myMembers),x(1,myMembers),[cVec(k) '.']);
    plot(myClustCen(2),myClustCen(1),'o','MarkerEdgeColor','k','MarkerFaceColor',cVec(k), 'MarkerSize',10)
end

title(['no shifting, numClust:' int2str(numClust)])
hold on 
figure(2);
sphere(50);
alpha(0.1);
hold on ;
scatter3(data(:,1),data(:,2),data(:,3),'filled');
view(-30,10)
axis equal

A  =  data;
newmatrix = 1/length(A).*(A.'*A)
eigen = eig(newmatrix)

eigen(1)*log(3*eigen(1)) + eigen(2)*log(3*eigen(2)) + eigen(3)*log(3*eigen(3))
toc

0.721096202739022 0.143274305623194 -0.677858937938827
0.968970810392417 0.208978537391637 0.131998255735122
0.569826352357379 0.276879108656515 0.773715637265177
0.228610689351435 0.164381998103028 -0.959539322494872
0.680467391588214 0.574452415904999 0.454937963733565
0.755501580234282 0.652710692710075 0.0564456719814448
0.504277122250002 0.768039960865827 -0.394739410863010
0.154912237007631 0.405692531666614 0.900786194707282
0.210384928706513 0.600544339596624 -0.771417317637872
0.143977198200687 0.816855223274374 0.558585813109905
-0.229099414143074 0.950795223816936 -0.208571092930452
-0.282616508078908 0.919125613504918 0.274474071563222
-0.273028245935124 0.599816587126275 0.752114112837791
-0.384654835774312 0.742153343311259 -0.548861614915233
-0.256254169146301 0.406094432070341 -0.877166525259945
-0.739931959521445 0.571829210408285 -0.354276797719793
-0.806338802865188 0.579353885367924 0.119024411374415
-0.760075452992603 0.337837739926240 0.555113472399671
-0.487550601625308 0.150466743789691 0.860031493532724
-0.987416677336434 0.156220148544471 0.0245676719820876
-0.898581139351745 -0.0465144783456416 0.436335122704501
-0.374139568095630 -0.0287413085026819 -0.926926923101692
-0.963578273457490 -0.260891712855464 -0.0587573406820316
-0.799916216534544 -0.556325600463240 0.225024160467884
-0.732478694296601 -0.536144597346316 -0.419552062487963
-0.478334336301847 -0.740621679799566 0.471885356973005
-0.194005559456996 -0.435699798511738 0.878935451826020
-0.314232785230549 -0.886073947077770 -0.340779572445699
-0.00720050076528926 -0.0591052787344646 0.998225785488659
-0.0490019433066575 -0.732931424546231 -0.678535434936750
0.0801942976395903 -0.872038194549829 0.482823220104800
0.368597406332163 -0.929570349282381 0.00590912685900466
0.323982389195822 -0.553545295684837 0.767217711680429
0.514345456211916 -0.579655346834382 -0.632019327679517
0.827337679920563 -0.510494571429276 -0.234323827053289
0.448212896485379 -0.212583754752496 -0.868281835949317
0.964675223452813 -0.215965114222275 0.150866771342216
0.724877321462857 -0.104703930494290 0.680874405281827
0.927771336185871 -0.108671846203537 -0.356974477511487

image-20220301152248993

newmatrix =

​ 0.3527 -0.0057 -0.0221
-0.0057 0.3007 0.0039
-0.0221 0.0039 0.3466

eigen =

​ 0.3001
​ 0.3273
​ 0.3726

ans =

​ 0.0040

均匀性不错。

又做一次实验

newmatrix =
0.3364 -0.0062 -0.0127
-0.0062 0.3195 0.0007
-0.0127 0.0007 0.3441

eigen =
0.3169
0.3290
0.3541

ans =
0.0011

image-20220301165031830

随机选取39个点

clc;clear;close all;
load("附件2  简易星表.mat");
star_data_xyz(:,1) = cos(star_data(:,2) * pi / 180) .* cos(star_data(:,3) * pi / 180);
star_data_xyz(:,2) = sin(star_data(:,2) * pi / 180) .* cos(star_data(:,3) * pi / 180);
star_data_xyz(:,3) = sin(star_data(:,3) * pi / 180);
A = star_data_xyz(randperm(length(star_data_xyz), 39),:)
figure(1)
scatter3(star_data_xyz(:,1),star_data_xyz(:,2),star_data_xyz(:,3))

figure(2);
sphere(50);
alpha(0.1);
hold on ;
scatter3(A(:,1),A(:,2),A(:,3),'filled');
view(-30,10)
axis equal
newmatrix = 1/length(A).*(A.'*A)
eigen = eig(newmatrix)
eigen(1)*log(3*eigen(1)) + eigen(2)*log(3*eigen(2)) + eigen(3)*log(3*eigen(3))
axis equal

newmatrix =
0.3068 -0.0422 0.0801
-0.0422 0.3637 -0.0381
0.0801 -0.0381 0.3295

eigen =
0.2368
0.3226
0.4406

ans =
0.0314

image-20220301161540597

% 矩形范围
clear all ;close all;
figure(1);
sphere(50);
alpha(0.1);
hold on ;

z=[];
theta=[];
fai=[];

n = 30;
k = 0:n-1;
z = 1-(2*k+1)/n;
theta=acos(z);
fai=sqrt(n*pi).*theta;
x = sin(theta).*cos(fai);
y = sin(theta).*sin(fai);
A = [x.', y.' ,z.'];
for i=1:length(A)-1 
    delta(i,:) = acosd(A(i,:)*A(i+1,:).');
end
mean = mean(delta);
newmatrix = 1/length(A).*(A.'*A)
eigen = eig(newmatrix)
eigen(1)*log(3*eigen(1)) + eigen(2)*log(3*eigen(2)) + eigen(3)*log(3*eigen(3))

scatter3(x,y,z,'filled');
plot3(x,y,z,'r-','LineWidth',2);
hold on;
quiver3(0,0,0,1,0,0,1,'k','filled','LineWidth',2);
text(1.1,0,0,'x','FontSize',14);
quiver3(0,0,0,0,1,0,1,'k','filled','LineWidth',2);
text(0,1.1,0,'y','FontSize',14);
quiver3(0,0,0,0,0,1,1,'k','filled','LineWidth',2);
text(0,0,1.1,'z','FontSize',14);
quiver3(0,0,0,A(2,1),A(2,2),A(2,3),1,'LineWidth',2);
hold on;
s=20;
xmin = cosd(s) * A(2,1) - sind(s) * sqrt(1-A(2,1)^2);
xmax = cosd(s) * A(2,1) + sind(s) * sqrt(1-A(2,1)^2);
ymin = cosd(s) * A(2,2) - sind(s) * sqrt(1-A(2,2)^2);
ymax = cosd(s) * A(2,2) + sind(s) * sqrt(1-A(2,2)^2);
zmin = cosd(s) * A(2,3) - sind(s) * sqrt(1-A(2,3)^2);
zmax = cosd(s) * A(2,3) + sind(s) * sqrt(1-A(2,3)^2);

plotcube([xmax-xmin ymax-ymin zmax-zmin],[ xmin  ymin  zmin],.8,[1 0 0]);
view(-30,10)
axis equal