利用Matlab实现单像空间后方交会

空间后方交会的定义为：根据影像覆盖范围内一定数量的分布合理的地面控制点（已知其像点和地面点的坐标），利用共线条件方程，求解影像外方位元素。

• 其原理可以简单地解释为：以单幅影像为基础，从该影像所覆盖地面范围内的若干控制点的已知地面坐标和相应点的像坐标量测值出发，根据共线条件方程，运用最小二乘间接平差方法，求解该影像在航空摄影时刻的外方位元素。
  

- 已知的四对影像点及其对应的地面点

由于可以将摄影瞬间近似看作垂直摄影情况，可以假设内方位元素已知：

摄影机主距fK=153.24mm；

相片比例尺分母m=50000；

像主点坐标x0=y0=0。

未知数的初始值可以表示为：

p=4;
q=5;
%定义cell矩阵，存储文件内容
BasicData=cell(p,q);
%以只读方式打开文件
fid=fopen('C:UsersDesktopBasicData.txt','r');
for i=1:p
    for j=1:q
        %以字符方式读取每个值，遇空格完成每个值的读取
        BasicData{i,j}=fscanf(fid,'%s',[1,1]);
    end
end
fclose(fid);

for i=1:p
    for j=1:q
        %将文本格式转为数字格式
        BasicData{i,j}=str2double(BasicData{i,j});
    end
end
for i=1:4
    for j=1:2
        BasicData{i,j}=0.001*BasicData{i,j};
    end
end
BasicData=cell2mat(BasicData);

%已知的内方位元素%
f=153.24*10^-3;
m=50000;
x0=0;
y0=0;

%确定未知数的近似值
Xs=(BasicData(1,3)+BasicData(2,3)+BasicData(3,3)+BasicData(4,3))/p;
Ys=(BasicData(1,4)+BasicData(2,4)+BasicData(3,4)+BasicData(4,4))/p;
Zs=m*f;
phi=0;
omega=0;
kappa=0;

while(1)
    %计算旋转矩阵
    R={cos(phi)*cos(kappa)-sin(phi)*sin(omega)*sin(kappa),-cos(phi)*sin(kappa)-sin(phi)*sin(omega)*cos(kappa),-sin(phi)*cos(omega);
        cos(omega)*sin(kappa),cos(omega)*cos(kappa),-sin(omega);
        sin(phi)*cos(kappa)+cos(phi)*sin(omega)*sin(kappa),-sin(phi)*sin(kappa)+cos(phi)*sin(omega)*cos(kappa),cos(phi)*cos(omega)};
    R=cell2mat(R);
    
%计算像点坐标的近似值
    x1(1) = x0-f*(R(1,1)*(BasicData(1,3)-Xs)+ R(2,1)*(BasicData(1,4)-Ys)+ R(3,1)*(BasicData(1,5)-Zs))/(R(1,3)*(BasicData(1,3)-Xs)+ R(2,3)*(BasicData(1,4)-Ys)+ R(3,3)*(BasicData(1,5)-Zs));
    y1(1) = y0-f*(R(1,2)*(BasicData(1,3)-Xs)+ R(2,2)*(BasicData(1,4)-Ys)+ R(3,2)*(BasicData(1,5)-Zs))/(R(1,3)*(BasicData(1,3)-Xs)+ R(2,3)*(BasicData(1,4)-Ys)+ R(3,3)*(BasicData(1,5)-Zs));
    x1(2) = x0-f*(R(1,1)*(BasicData(2,3)-Xs)+ R(2,1)*(BasicData(2,4)-Ys)+ R(3,1)*(BasicData(2,5)-Zs))/(R(1,3)*(BasicData(2,3)-Xs)+ R(2,3)*(BasicData(2,4)-Ys)+ R(3,3)*(BasicData(2,5)-Zs));
    y1(2) = y0-f*(R(1,2)*(BasicData(2,3)-Xs)+ R(2,2)*(BasicData(2,4)-Ys)+ R(3,2)*(BasicData(2,5)-Zs))/(R(1,3)*(BasicData(2,3)-Xs)+ R(2,3)*(BasicData(2,4)-Ys)+ R(3,3)*(BasicData(2,5)-Zs));
    x1(3) = x0-f*(R(1,1)*(BasicData(3,3)-Xs)+ R(2,1)*(BasicData(3,4)-Ys)+ R(3,1)*(BasicData(3,5)-Zs))/(R(1,3)*(BasicData(3,3)-Xs)+ R(2,3)*(BasicData(3,4)-Ys)+ R(3,3)*(BasicData(3,5)-Zs));
    y1(3) = y0-f*(R(1,2)*(BasicData(3,3)-Xs)+ R(2,2)*(BasicData(3,4)-Ys)+ R(3,2)*(BasicData(3,5)-Zs))/(R(1,3)*(BasicData(3,3)-Xs)+ R(2,3)*(BasicData(3,4)-Ys)+ R(3,3)*(BasicData(3,5)-Zs));
    x1(4) = x0-f*(R(1,1)*(BasicData(4,3)-Xs)+ R(2,1)*(BasicData(4,4)-Ys)+ R(3,1)*(BasicData(4,5)-Zs))/(R(1,3)*(BasicData(4,3)-Xs)+ R(2,3)*(BasicData(4,4)-Ys)+ R(3,3)*(BasicData(4,5)-Zs));
    y1(4) = y0-f*(R(1,2)*(BasicData(4,3)-Xs)+ R(2,2)*(BasicData(4,4)-Ys)+ R(3,2)*(BasicData(4,5)-Zs))/(R(1,3)*(BasicData(4,3)-Xs)+ R(2,3)*(BasicData(4,4)-Ys)+ R(3,3)*(BasicData(4,5)-Zs));
    
    %计算常数项
    for i=1:4
        L(2*i-1,1)= BasicData(i,1)-x1(i);
        L(2*i,1)=BasicData(i,2)-y1(i);
    end

%计算系数矩阵
    for i=1:4
        A(2*i-1,1)=(R(1,1)*f+R(1,3)*(BasicData(i,1)-x0))/(R(1,3)*(BasicData(i,3)-Xs)+R(2,3)*(BasicData(i,4)-Ys)+R(3,3)*(BasicData(i,5)-Zs));
        A(2*i-1,2)=(R(2,1)*f+R(2,3)*(BasicData(i,1)-x0))/(R(1,3)*(BasicData(i,3)-Xs)+R(2,3)*(BasicData(i,4)-Ys)+R(3,3)*(BasicData(i,5)-Zs));
        A(2*i-1,3)=(R(3,1)*f+R(3,3)*(BasicData(i,1)-x0))/(R(1,3)*(BasicData(i,3)-Xs)+R(2,3)*(BasicData(i,4)-Ys)+R(3,3)*(BasicData(i,5)-Zs));
        A(2*i-1,4)=(BasicData(i,2)-y0)*sin(omega)-cos(omega)*((BasicData(i,1)-x0)*((BasicData(i,1)-x0)*cos(kappa)-(BasicData(i,2)-y0)*sin(kappa))/f+f*cos(kappa));
        A(2*i-1,5)=-f*sin(kappa)-(BasicData(i,1)-x0)*((BasicData(i,1)-x0)*sin(kappa)+(BasicData(i,2)-y0)*cos(kappa))/f;
        A(2*i-1,6)=(BasicData(i,2)-y0);
        A(2*i,1)=(R(1,2)*f+R(1,3)*(BasicData(i,2)-y0))/(R(1,3)*(BasicData(i,3)-Xs)+R(2,3)*(BasicData(i,4)-Ys)+R(3,3)*(BasicData(i,5)-Zs));
        A(2*i,2)=(R(2,2)*f+R(2,3)*(BasicData(i,2)-y0))/(R(1,3)*(BasicData(i,3)-Xs)+R(2,3)*(BasicData(i,4)-Ys)+R(3,3)*(BasicData(i,5)-Zs));
        A(2*i,3)=(R(3,2)*f+R(3,3)*(BasicData(i,2)-y0))/(R(1,3)*(BasicData(i,3)-Xs)+R(2,3)*(BasicData(i,4)-Ys)+R(3,3)*(BasicData(i,5)-Zs));
        A(2*i,4)=-(BasicData(i,1)-x0)*sin(omega)-cos(omega)*((BasicData(i,2)-y0)*((BasicData(i,1)-x0)*cos(kappa)-(BasicData(i,2)-y0)*sin(kappa))/f-f*sin(kappa));
        A(2*i,5)=-f*cos(kappa)-(BasicData(i,2)-y0)*((BasicData(i,1)-x0)*sin(kappa)+(BasicData(i,2)-y0)*cos(kappa))/f;
        A(2*i,6)=-(BasicData(i,1)-x0);
    end

%计算法方程
    XX=inv(A'*A)*A'*L;
    XN=[Xs;Ys;Zs;phi;omega;kappa]+XX;
    Xs=XN(1,1);
    Ys=XN(2,1);
    Zs=XN(3,1);
    phi=XN(4,1);
    omega=XN(5,1);
    kappa=XN(6,1);
    if(abs(XX(4,1))<0.1*pi/10800 && abs(XX(5,1))<0.1*pi/10800 && abs(XX(6,1))<0.1*pi/10800 )
        break;
    end
end

Xs=vpa(XN(1,1),7)
Ys=vpa(XN(2,1),7)
Zs=vpa(XN(3,1),6)
phi=vpa(degrees2dms(rad2deg(XN(4,1))),2)
omega=vpa(degrees2dms(rad2deg(XN(5,1))),2)
kappa=vpa(degrees2dms(rad2deg(XN(6,1))),2)
R

原始数据，BasicData.txt文件内容

最后

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