估计摄像机投影矩阵和基本矩阵实验报告

使用最小二乘法解空间3D点到图像2D点的投影矩阵
实验目的：

​ 老师题目中给了一个三维空间里面20个点的坐标，分别用两个角度拍摄这个场景，并给出两张照片中这20个点的二维坐标。要求求两张照片对应的投影矩阵。

​ 想要将三维空间点坐标投影到二维平面坐标需要求出对应的投影矩阵，通过投影矩阵与三维坐标矩阵相乘可以得到二维点坐标。而反之如果知道三维空间点坐标和投影后的二维平面点坐标就可以找到其对应的投影矩阵。

实验过程：

首先在MATLAB中读取图像的2D和3D坐标：

clear
close allformatSpec = '%f';
size2d_norm = [2 Inf];
size3d_norm = [3 Inf];file_2d_pic_b = fopen('E:\硕士\计算机视觉\Assignment2\data\pts2d-pic_a.txt','r');
file_3d = fopen('E:\硕士\计算机视觉\Assignment2\data\pts3d.txt','r');
Points_2D = fscanf(file_2d_pic_b,formatSpec,size2d_norm)';
Points_3D = fscanf(file_3d,formatSpec,size3d_norm)';

使用最小二乘法求三维坐标投影到二维的投影公式推导：
[ s u s v s ] = [ m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 ] [ X Y Z 1 ] \begin{bmatrix} su\\ sv\\ s \end{bmatrix}=\begin{bmatrix} m_{11}&m_{12}&m_{13}&m_{14}\\ m_{21}&m_{22}&m_{23}&m_{24}\\ m_{31}&m_{32}&m_{33}&m_{34} \end{bmatrix}\begin{bmatrix} X\\ Y\\ Z\\ 1 \end{bmatrix} ⎣⎡​susvs​⎦⎤​=⎣⎡​m11​m21​m31​​m12​m22​m32​​m13​m23​m33​​m14​m24​m34​​⎦⎤​⎣⎢⎢⎡​XYZ1​⎦⎥⎥⎤​

s u = m 11 X + m 12 Y + m 13 Z + m 14 s v = m 21 X + m 22 Y + m 23 Z + m 24 s = m 31 X + m 32 Y + m 33 Z + m 34 u = m 11 X + m 12 Y + m 13 Z + m 14 m 31 X + m 32 Y + m 33 Z + m 34 v = m 21 X + m 22 Y + m 23 Z + m 24 m 31 X + m 32 Y + m 33 Z + m 34 su=m_{11}X+m_{12}Y+m_{13}Z+m_{14}\\ sv=m_{21}X+m_{22}Y+m_{23}Z+m_{24}\\ s=m_{31}X+m_{32}Y+m_{33}Z+m_{34}\\ \\ u=\frac{m_{11}X+m_{12}Y+m_{13}Z+m_{14}}{m_{31}X+m_{32}Y+m_{33}Z+m_{34}}\\ v=\frac{m_{21}X+m_{22}Y+m_{23}Z+m_{24}}{m_{31}X+m_{32}Y+m_{33}Z+m_{34}}\\ \\ su=m11​X+m12​Y+m13​Z+m14​sv=m21​X+m22​Y+m23​Z+m24​s=m31​X+m32​Y+m33​Z+m34​u=m31​X+m32​Y+m33​Z+m34​m11​X+m12​Y+m13​Z+m14​​v=m31​X+m32​Y+m33​Z+m34​m21​X+m22​Y+m23​Z+m24​​
( m 31 X + m 32 Y + m 33 Z + m 34 ) u = m 11 X + m 12 Y + m 13 Z + m 14 ( m 31 X + m 32 Y + m 33 Z + m 34 ) v = m 21 X + m 22 Y + m 23 Z + m 24 m 31 u X + m 32 u Y + m 33 u Z + m 34 u = m 11 X + m 12 Y + m 13 Z + m 14 m 31 v X + m 32 v Y + m 33 v Z + m 34 v = m 21 X + m 22 Y + m 23 Z + m 24 0 = m 11 X + m 12 Y + m 13 Z + m 14 − m 31 u X − m 32 u Y − m 33 u Z − m 34 u 0 = m 21 X + m 22 Y + m 23 Z + m 24 − m 31 v X − m 32 v Y − m 33 v Z − m 34 v (m_{31}X+m_{32}Y+m_{33}Z+m_{34})u=m_{11}X+m_{12}Y+m_{13}Z+m_{14}\\ (m_{31}X+m_{32}Y+m_{33}Z+m_{34})v=m_{21}X+m_{22}Y+m_{23}Z+m_{24}\\ \\ m_{31}uX+m_{32}uY+m_{33}uZ+m_{34}u=m_{11}X+m_{12}Y+m_{13}Z+m_{14}\\ m_{31}vX+m_{32}vY+m_{33}vZ+m_{34}v=m_{21}X+m_{22}Y+m_{23}Z+m_{24}\\ \\ 0=m_{11}X+m_{12}Y+m_{13}Z+m_{14}-m_{31}uX-m_{32}uY-m_{33}uZ-m_{34}u\\ 0=m_{21}X+m_{22}Y+m_{23}Z+m_{24}-m_{31}vX-m_{32}vY-m_{33}vZ-m_{34}v (m31​X+m32​Y+m33​Z+m34​)u=m11​X+m12​Y+m13​Z+m14​(m31​X+m32​Y+m33​Z+m34​)v=m21​X+m22​Y+m23​Z+m24​m31​uX+m32​uY+m33​uZ+m34​u=m11​X+m12​Y+m13​Z+m14​m31​vX+m32​vY+m33​vZ+m34​v=m21​X+m22​Y+m23​Z+m24​0=m11​X+m12​Y+m13​Z+m14​−m31​uX−m32​uY−m33​uZ−m34​u0=m21​X+m22​Y+m23​Z+m24​−m31​vX−m32​vY−m33​vZ−m34​v

考虑系统为非齐次线性方程组，常数项不为0：
m 11 X + m 12 Y + m 13 Z + m 14 − m 31 u X − m 32 u Y − m 33 u Z = m 34 u m 21 X + m 22 Y + m 23 Z + m 24 − m 31 v X − m 32 v Y − m 33 v Z = m 34 v m_{11}X+m_{12}Y+m_{13}Z+m_{14}-m_{31}uX-m_{32}uY-m_{33}uZ=m_{34}u\\ m_{21}X+m_{22}Y+m_{23}Z+m_{24}-m_{31}vX-m_{32}vY-m_{33}vZ=m_{34}v m11​X+m12​Y+m13​Z+m14​−m31​uX−m32​uY−m33​uZ=m34​um21​X+m22​Y+m23​Z+m24​−m31​vX−m32​vY−m33​vZ=m34​v
等式右边的系数m34不影响最终结果，所以这里可以将m34的值设为1。将方程整理为矩阵形式：
[ X 1 Y 1 Z 1 1 0 0 0 0 − u 1 Y 1 − u 1 Y 1 − u 1 Z 1 0 0 0 0 X 1 Y 1 Z 1 1 − v 1 Y 1 − v 1 Y 1 − v 1 Z 1 . . . X n Y n Z n 1 0 0 0 0 − u n Y n − u n Y n − u n Z n 0 0 0 0 X n Y n Z n 1 − v n Y n − v n Y n − v n Z n ] [ m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 ] = [ u 1 v 1 . . . u n v n ] \begin{bmatrix} X_1&Y_1&Z_1&1&0&0&0&0&-u_1Y_1&-u_1Y_1&-u_1Z_1\\ 0&0&0&0&X_1&Y_1&Z_1&1&-v_1Y_1&-v_1Y_1&-v_1Z_1\\ &&&&&&.\\ &&&&&&.\\ &&&&&&.\\ X_n&Y_n&Z_n&1&0&0&0&0&-u_nY_n&-u_nY_n&-u_nZ_n\\ 0&0&0&0&X_n&Y_n&Z_n&1&-v_nY_n&-v_nY_n&-v_nZ_n \end{bmatrix}\begin{bmatrix} m_{11}\\ m_{12}\\ m_{13}\\ m_{14}\\ m_{21}\\ m_{22}\\ m_{23}\\ m_{24}\\ m_{31}\\ m_{32}\\ m_{33} \end{bmatrix}=\begin{bmatrix} u_1\\ v_1\\ .\\ .\\ .\\ u_n\\ v_n \end{bmatrix} ⎣⎢⎢⎢⎢⎢⎢⎢⎢⎡​X1​0Xn​0​Y1​0Yn​0​Z1​0Zn​0​1010​0X1​0Xn​​0Y1​0Yn​​0Z1​...0Zn​​0101​−u1​Y1​−v1​Y1​−un​Yn​−vn​Yn​​−u1​Y1​−v1​Y1​−un​Yn​−vn​Yn​​−u1​Z1​−v1​Z1​−un​Zn​−vn​Zn​​⎦⎥⎥⎥⎥⎥⎥⎥⎥⎤​⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡​m11​m12​m13​m14​m21​m22​m23​m24​m31​m32​m33​​⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤​=⎣⎢⎢⎢⎢⎢⎢⎢⎢⎡​u1​v1​...un​vn​​⎦⎥⎥⎥⎥⎥⎥⎥⎥⎤​
此时问题就是求解该非齐次线性方程组：
A x = b Ax=b Ax=b
在MATLAB的calcuate文件中求解方程组，并将求解的结果转化为矩阵形式，求得的M即为投影矩阵：

M = A\b;
M = [M;1];
M = reshape(M,[],3)';

得到的结果为：

M = [ − 2.33259098129747 − 0.109993113051944 0.337413915879523 736.673920205946 − 0.231050254183619 − 0.479506029384648 2.08717636190678 153.627755683397 − 0.00126379605778312 − 0.00206770917182951 0.000514635232581790 1 ] M=\begin{bmatrix} -2.33259098129747&-0.109993113051944&0.337413915879523&736.673920205946\\ -0.231050254183619&-0.479506029384648&2.08717636190678&153.627755683397\\ -0.00126379605778312&-0.00206770917182951&0.000514635232581790&1 \end{bmatrix} M=⎣⎡​−2.33259098129747−0.231050254183619−0.00126379605778312​−0.109993113051944−0.479506029384648−0.00206770917182951​0.3374139158795232.087176361906780.000514635232581790​736.673920205946153.6277556833971​⎦⎤​

实验结果：

图像a对应的投影矩阵M1为：
M 1 = [ − 2.33259098129747 − 0.109993113051944 0.337413915879523 736.673920205946 − 0.231050254183619 − 0.479506029384648 2.08717636190678 153.627755683397 − 0.00126379605778312 − 0.00206770917182951 0.000514635232581790 1 ] M_1=\begin{bmatrix} -2.33259098129747&-0.109993113051944&0.337413915879523&736.673920205946\\ -0.231050254183619&-0.479506029384648&2.08717636190678&153.627755683397\\ -0.00126379605778312&-0.00206770917182951&0.000514635232581790&1 \end{bmatrix} M1​=⎣⎡​−2.33259098129747−0.231050254183619−0.00126379605778312​−0.109993113051944−0.479506029384648−0.00206770917182951​0.3374139158795232.087176361906780.000514635232581790​736.673920205946153.6277556833971​⎦⎤​
图像b对应的投影矩阵M2为：
M 2 = [ − 2.04662531866668 1.18743051850851 0.388938200241278 243.732984539506 − 0.456886722410772 − 0.302017127919076 2.14721847547261 165.932475135729 − 0.00224678720499548 − 0.00109380145729559 0.000558547111046076 1 ] M_2=\begin{bmatrix} -2.04662531866668&1.18743051850851&0.388938200241278&243.732984539506\\ -0.456886722410772&-0.302017127919076&2.14721847547261&165.932475135729\\ -0.00224678720499548&-0.00109380145729559&0.000558547111046076&1 \end{bmatrix} M2​=⎣⎡​−2.04662531866668−0.456886722410772−0.00224678720499548​1.18743051850851−0.302017127919076−0.00109380145729559​0.3889382002412782.147218475472610.000558547111046076​243.732984539506165.9324751357291​⎦⎤​

结果分析：

为检验所求投影矩阵是否正确，可以使用投影矩阵与三维空间坐标点相乘得到二维平面矩阵 [ s u , s v , s ] [su,sv,s] [su,sv,s]。

首先检验图像a的投影矩阵M1是否正确。在MATLAB的verification中将3D坐标矩阵最右边加上全为1的一列并将值赋给变量D，然后用求得的投影矩阵M与矩阵D的转置相乘，得到的矩阵命名为E：

D = [312.747000000000	309.140000000000	30.0860000000000	1;
305.796000000000	311.649000000000	30.3560000000000	1;
307.694000000000	312.358000000000	30.4180000000000	1;
310.149000000000	307.186000000000	29.2980000000000	1;
311.937000000000	310.105000000000	29.2160000000000	1;
311.202000000000	307.572000000000	30.6820000000000	1;
307.106000000000	306.876000000000	28.6600000000000	1;
309.317000000000	312.490000000000	30.2300000000000	1;
307.435000000000	310.151000000000	29.3180000000000	1;
308.253000000000	306.300000000000	28.8810000000000	1;
306.650000000000	309.301000000000	28.9050000000000	1;
308.069000000000	306.831000000000	29.1890000000000	1;
309.671000000000	308.834000000000	29.0290000000000	1;
308.255000000000	309.955000000000	29.2670000000000	1;
307.546000000000	308.613000000000	28.9630000000000	1;
311.036000000000	309.206000000000	28.9130000000000	1;
307.518000000000	308.175000000000	29.0690000000000	1;
309.950000000000	311.262000000000	29.9900000000000	1;
312.160000000000	310.772000000000	29.0800000000000	1;
311.988000000000	312.709000000000	30.5140000000000	1];
E = M*D';

求得的矩阵E为：
E = [ − 16.6887473176187 − 0.659778369979904 − 5.14410150685183 − 10.6796315710174 − 15.1990420836781 − 12.7113263563845 − 3.76272942821424 − 9.00784957660574 − 4.66836094965765 − 6.30028677523524 − 2.88313483050308 − 5.82557289161616 − 9.83668607563857 − 6.57673501387308 − 4.87789108084496 − 13.1007302174069 − 4.72863567476850 − 10.4302874647503 − 15.8384635714726 − 15.1664630273000 − 4.07222406140946 − 3.10573675459039 − 3.75483497742637 − 4.17969569280536 − 6.16364010873583 − 1.71942885292421 − 4.65958141911187 − 4.58551349188365 − 4.93261715283837 − 4.18713461175354 − 5.20566641579476 − 3.75638874711973 − 5.42093004908442 − 5.13454117396678 − 4.96173106633606 − 6.15680234695733 − 4.52399732398634 − 4.64385722555516 − 6.81885082223766 − 4.71490245446180 − 0.0189767234554149 − 0.0152410078570877 − 0.0190737911931670 − 0.0120586101388139 − 0.0203961206517722 − 0.0134752679660926 − 0.00790022617009989 − 0.0214946212293559 − 0.0147491326278135 − 0.00804506537900285 − 0.0122120442774548 − 0.00875197282297702 − 0.0150005372159331 − 0.0154064207943787 − 0.0118919727915201 − 0.0175545043337072 − 0.0108963785499872 − 0.0198789697267456 − 0.0221270995818992 − 0.0251768924022721 ] E=\begin{bmatrix} -16.6887473176187& -0.659778369979904& -5.14410150685183& -10.6796315710174& -15.1990420836781& -12.7113263563845& -3.76272942821424& -9.00784957660574& -4.66836094965765& -6.30028677523524& -2.88313483050308& -5.82557289161616& -9.83668607563857& -6.57673501387308& -4.87789108084496& -13.1007302174069& -4.72863567476850& -10.4302874647503& -15.8384635714726& -15.1664630273000\\ -4.07222406140946& -3.10573675459039& -3.75483497742637& -4.17969569280536& -6.16364010873583& -1.71942885292421& -4.65958141911187& -4.58551349188365& -4.93261715283837& -4.18713461175354& -5.20566641579476& -3.75638874711973& -5.42093004908442& -5.13454117396678& -4.96173106633606& -6.15680234695733& -4.52399732398634& -4.64385722555516& -6.81885082223766& -4.71490245446180\\ -0.0189767234554149& -0.0152410078570877& -0.0190737911931670& -0.0120586101388139& -0.0203961206517722& -0.0134752679660926& -0.00790022617009989& -0.0214946212293559& -0.0147491326278135& -0.00804506537900285& -0.0122120442774548& -0.00875197282297702& -0.0150005372159331& -0.0154064207943787& -0.0118919727915201& -0.0175545043337072& -0.0108963785499872& -0.0198789697267456& -0.0221270995818992& -0.0251768924022721& \end{bmatrix} E=⎣⎡​−16.6887473176187−4.07222406140946−0.0189767234554149​−0.659778369979904−3.10573675459039−0.0152410078570877​−5.14410150685183−3.75483497742637−0.0190737911931670​−10.6796315710174−4.17969569280536−0.0120586101388139​−15.1990420836781−6.16364010873583−0.0203961206517722​−12.7113263563845−1.71942885292421−0.0134752679660926​−3.76272942821424−4.65958141911187−0.00790022617009989​−9.00784957660574−4.58551349188365−0.0214946212293559​−4.66836094965765−4.93261715283837−0.0147491326278135​−6.30028677523524−4.18713461175354−0.00804506537900285​−2.88313483050308−5.20566641579476−0.0122120442774548​−5.82557289161616−3.75638874711973−0.00875197282297702​−9.83668607563857−5.42093004908442−0.0150005372159331​−6.57673501387308−5.13454117396678−0.0154064207943787​−4.87789108084496−4.96173106633606−0.0118919727915201​−13.1007302174069−6.15680234695733−0.0175545043337072​−4.72863567476850−4.52399732398634−0.0108963785499872​−10.4302874647503−4.64385722555516−0.0198789697267456​−15.8384635714726−6.81885082223766−0.0221270995818992​−15.1664630273000−4.71490245446180−0.0251768924022721​​⎦⎤​
将矩阵中的数据转化为数组，并提取其中的数据放到Excel中：

1234567891011121314151617181920
su-16.6887473176187-0.659778369979904-5.14410150685183-10.6796315710174-15.1990420836781-12.7113263563845-3.76272942821424-9.00784957660574-4.66836094965765-6.30028677523524-2.88313483050308-5.82557289161616-9.83668607563857-6.57673501387308-4.87789108084496-13.1007302174069-4.72863567476850-10.4302874647503-15.8384635714726-15.1664630273000
sv-4.07222406140946-3.10573675459039-3.75483497742637-4.17969569280536-6.16364010873583-1.71942885292421-4.65958141911187-4.58551349188365-4.93261715283837-4.18713461175354-5.20566641579476-3.75638874711973-5.42093004908442-5.13454117396678-4.96173106633606-6.15680234695733-4.52399732398634-4.64385722555516-6.81885082223766-4.71490245446180
s-0.0189767234554149-0.0152410078570877-0.0190737911931670-0.0120586101388139-0.0203961206517722-0.0134752679660926-0.00790022617009989-0.0214946212293559-0.0147491326278135-0.00804506537900285-0.0122120442774548-0.00875197282297702-0.0150005372159331-0.0154064207943787-0.0118919727915201-0.0175545043337072-0.0108963785499872-0.0198789697267456-0.0221270995818992-0.0251768924022721

在Excel中将矩阵E的第一行数据su和第二行数据sv分别除以第三行s求得二维图像坐标 [ u , v ] [u,v] [u,v]：

1234567891011121314151617181920
u879.432543.28968269.6948885.6437745.1928943.3079476.2812419.0746316.5177783.1244236.0895665.6297655.7556426.8827410.1835746.2888433.964524.6895715.7948602.3961
v214.5905203.775196.8583346.615302.1967127.5989589.8035213.3331334.4344520.46426.2731429.2048361.3824333.2728417.2336350.7249415.1836233.6065308.1674187.271

而题中所给的图a中原二维平面坐标 [ u , v ] [u,v] [u,v]为：

1234567891011121314151617181920
u88043270886745943476419317783235665655427412746434525716602
v214203197347302128590214335521427429362333415351415234308187

将坐标与原二维平面坐标做对比，误差较小，可知计算出的投影矩阵M1正确，所使用方法无误且数据正确。

接下来使用同样的方法检验图像b的投影矩阵M2是否正确：

首先也是求得对应的矩阵E：
E = [ − 17.5590788133619 − 0.248709738306957 − 3.26720218709886 − 14.8686687704218 − 15.0938180890912 − 16.0271265816996 − 9.25703395821051 − 6.50525463249699 − 5.78561740417763 − 12.2025178351287 − 5.34896394645625 − 11.0756202054919 − 12.0381207494619 − 7.71642239532446 − 7.97713401310159 − 14.4351569877847 − 8.39869562205990 − 9.35227830399947 − 14.8110949745414 − 11.6012851262338 − 5.72183650990732 − 2.72402888800136 − 3.67220248535237 − 5.63671149580264 − 7.51128486685765 − 3.26264145582385 − 5.52270527720329 − 4.85727297009922 − 5.24925734595043 − 5.39825719904761 − 5.52068794049694 − 4.81321784660351 − 6.49364563920307 − 5.67421724350423 − 5.59773198819127 − 7.47872373003449 − 5.22505049953509 − 5.29093766641400 − 8.10663774294156 − 5.53394910963843 − 0.0240092921261503 − 0.0109854144005990 − 0.0159906918280182 − 0.0164749860435208 − 0.0237308488830045 − 0.0184900311312094 − 0.00965528718380537 − 0.0198866161109655 − 0.0136091559478236 − 0.0114788835569917 − 0.0111463767100629 − 0.0114742507748944 − 0.0173538557340275 − 0.0152656222729534 − 0.0123745669106787 − 0.0208924058758417 − 0.0117733658368726 − 0.0200996955288196 − 0.0250374104088368 − 0.0259686998751191 ] E=\begin{bmatrix} -17.5590788133619& -0.248709738306957& -3.26720218709886& -14.8686687704218& -15.0938180890912& -16.0271265816996& -9.25703395821051& -6.50525463249699& -5.78561740417763& -12.2025178351287& -5.34896394645625& -11.0756202054919& -12.0381207494619& -7.71642239532446& -7.97713401310159& -14.4351569877847& -8.39869562205990& -9.35227830399947& -14.8110949745414& -11.6012851262338\\ -5.72183650990732& -2.72402888800136& -3.67220248535237& -5.63671149580264& -7.51128486685765& -3.26264145582385& -5.52270527720329& -4.85727297009922& -5.24925734595043& -5.39825719904761& -5.52068794049694& -4.81321784660351& -6.49364563920307& -5.67421724350423& -5.59773198819127& -7.47872373003449& -5.22505049953509& -5.29093766641400& -8.10663774294156& -5.53394910963843\\ -0.0240092921261503& -0.0109854144005990& -0.0159906918280182& -0.0164749860435208& -0.0237308488830045 &-0.0184900311312094& -0.00965528718380537& -0.0198866161109655& -0.0136091559478236& -0.0114788835569917& -0.0111463767100629& -0.0114742507748944& -0.0173538557340275& -0.0152656222729534& -0.0123745669106787& -0.0208924058758417& -0.0117733658368726& -0.0200996955288196& -0.0250374104088368& -0.0259686998751191 \end{bmatrix} E=⎣⎡​−17.5590788133619−5.72183650990732−0.0240092921261503​−0.248709738306957−2.72402888800136−0.0109854144005990​−3.26720218709886−3.67220248535237−0.0159906918280182​−14.8686687704218−5.63671149580264−0.0164749860435208​−15.0938180890912−7.51128486685765−0.0237308488830045​−16.0271265816996−3.26264145582385−0.0184900311312094​−9.25703395821051−5.52270527720329−0.00965528718380537​−6.50525463249699−4.85727297009922−0.0198866161109655​−5.78561740417763−5.24925734595043−0.0136091559478236​−12.2025178351287−5.39825719904761−0.0114788835569917​−5.34896394645625−5.52068794049694−0.0111463767100629​−11.0756202054919−4.81321784660351−0.0114742507748944​−12.0381207494619−6.49364563920307−0.0173538557340275​−7.71642239532446−5.67421724350423−0.0152656222729534​−7.97713401310159−5.59773198819127−0.0123745669106787​−14.4351569877847−7.47872373003449−0.0208924058758417​−8.39869562205990−5.22505049953509−0.0117733658368726​−9.35227830399947−5.29093766641400−0.0200996955288196​−14.8110949745414−8.10663774294156−0.0250374104088368​−11.6012851262338−5.53394910963843−0.0259686998751191​⎦⎤​
将矩阵中的数据转化为数组，并提取其中的数据放到Excel中：

1234567891011121314151617181920
su-17.5590788133619-0.248709738306957-3.26720218709886-14.8686687704218-15.0938180890912-16.0271265816996-9.25703395821051-6.50525463249699-5.78561740417763-12.2025178351287-5.34896394645625-11.0756202054919-12.0381207494619-7.71642239532446-7.97713401310159-14.4351569877847-8.39869562205990-9.35227830399947-14.8110949745414-11.6012851262338
sv-5.72183650990732-2.72402888800136-3.67220248535237-5.63671149580264-7.51128486685765-3.26264145582385-5.52270527720329-4.85727297009922-5.24925734595043-5.39825719904761-5.52068794049694-4.81321784660351-6.49364563920307-5.67421724350423-5.59773198819127-7.47872373003449-5.22505049953509-5.29093766641400-8.10663774294156-5.53394910963843
s-0.0240092921261503-0.0109854144005990-0.0159906918280182-0.0164749860435208-0.0237308488830045-0.0184900311312094-0.00965528718380537-0.0198866161109655-0.0136091559478236-0.0114788835569917-0.0111463767100629-0.0114742507748944-0.0173538557340275-0.0152656222729534-0.0123745669106787-0.0208924058758417-0.0117733658368726-0.0200996955288196-0.0250374104088368-0.0259686998751191

在Excel中将矩阵E的第一行数据su和第二行数据sv分别除以第三行s求得图像b的二维图像坐标 [ u , v ] [u,v] [u,v]：

1234567891011121314151617181920
u731.345122.64204.319902.4996636.0421866.7982958.7528327.1172425.12681063.04479.8837965.2587693.6857505.4771644.6395690.9284713.364465.2945591.5586446.7411
v238.3176247.9678229.6463342.1376316.5199176.4541571.9877244.2483385.7151470.2772495.2899419.4799374.1904371.699452.3578357.9637443.8026263.2347323.781213.1007

而题中所给的图b中原二维平面坐标 [ u , v ] [u,v] [u,v]为：

1234567891011121314151617181920
u731222049036358679583284261064480964695505645692712465591447
v238248230342316177572244386470495419374372452359444263324213

将坐标与原二维平面坐标做对比，误差较小，可知计算出的投影矩阵M2正确，所使用方法无误且数据正确。