当前位置:网站首页>Understanding rotation matrix R from the perspective of base transformation
Understanding rotation matrix R from the perspective of base transformation
2022-07-05 08:54:00 【Li Yingsong~】
In understanding the camera coordinate system , We will definitely touch the camera's external parameter matrix R, It converts coordinates from the world coordinate system to the camera coordinate system :
P c = R ∗ P w + t P_c=R*P_w+t Pc=R∗Pw+t
This is actually a transformation between two coordinate systems , We know R R R A matrix is an orthogonal matrix , So it's 3 Row ( Column ) The vector is 3 A set of orthonormal bases in a vector space , And a set of orthonormal bases can be used as three basis vectors of a coordinate system . So our R R R How does a matrix relate to the basis vectors of two coordinate systems ?
Let's draw two coordinate systems first X w Y w Z w X_wY_wZ_w XwYwZw and X c Y c Z c X_cY_cZ_c XcYcZc:
We're going to talk about how to put a certain point P P P Coordinates in the world coordinate system are converted into coordinates in the camera coordinate system .
Let's not consider the translation between the two coordinate systems , So move the origin of the camera coordinate system to the origin of the world coordinate system , like this :
We can mark the set of basis vectors of two coordinate systems e w ( e ⃗ w x , e ⃗ w y , e ⃗ w z ) e_w(\vec{e}_{wx},\vec{e}_{wy},\vec{e}_{wz}) ew(ewx,ewy,ewz) and e c ( e ⃗ c x , e ⃗ c y , e ⃗ c z ) e_c(\vec{e}_{cx},\vec{e}_{cy},\vec{e}_{cz}) ec(ecx,ecy,ecz). They're all in the world coordinate system .
Next , And how to put a point in the world coordinate system P ( X w , Y w , Z w ) P(X_w,Y_w,Z_w) P(Xw,Yw,Zw) Convert to camera coordinates
P ( X w , Y w , Z w ) → P ( X c , Y w , Z w ) P(X_w,Y_w,Z_w)→P(X_c,Y_w,Z_w) P(Xw,Yw,Zw)→P(Xc,Yw,Zw)
In the world coordinate system , Basis vector set e w ( e ⃗ w x , e ⃗ w y , e ⃗ w z ) e_w(\vec{e}_{wx},\vec{e}_{wy},\vec{e}_{wz}) ew(ewx,ewy,ewz) Is the unit matrix , That is to say

among e ⃗ w x = ( 1 , 0 , 0 ) T \vec{e}_{wx}=(1,0,0)^T ewx=(1,0,0)T, e ⃗ w y = ( 0 , 1 , 0 ) T \vec{e}_{wy}=(0,1,0)^T ewy=(0,1,0)T, e ⃗ w z = ( 0 , 0 , 1 ) T \vec{e}_{wz}=(0,0,1)^T ewz=(0,0,1)T.
We know P P P The coordinates in the world coordinate system are actually a linear combination of the above three sets of basis vectors , namely P w = X w ∗ e ⃗ w x + Y w ∗ e ⃗ w x + Z w ∗ e ⃗ w x P_w=X_w*\vec{e}_{wx}+Y_w*\vec{e}_{wx}+Z_w*\vec{e}_{wx} Pw=Xw∗ewx+Yw∗ewx+Zw∗ewx

This is the base vector representation of coordinates .
So let's take P P P The coordinates of the points are transformed into a set of basis vectors e c ( e ⃗ c x , e ⃗ c y , e ⃗ c z ) e_c(\vec{e}_{cx},\vec{e}_{cy},\vec{e}_{cz}) ec(ecx,ecy,ecz) Then we get the transformation in the camera coordinate system . let me put it another way , We're going to calculate P P P The point is in the base vector set e c ( e ⃗ c x , e ⃗ c y , e ⃗ c z ) e_c(\vec{e}_{cx},\vec{e}_{cy},\vec{e}_{cz}) ec(ecx,ecy,ecz) The coordinates under P c = X c ∗ e ⃗ c x + Y c ∗ e ⃗ c x + Z c ∗ e ⃗ c x P_c=X_c*\vec{e}_{cx}+Y_c*\vec{e}_{cx}+Z_c*\vec{e}_{cx} Pc=Xc∗ecx+Yc∗ecx+Zc∗ecx.
From the perspective of the rotation matrix , The formula is :
P c = R P w P_c=RP_w Pc=RPw
Let's forget for a moment P P P, Let's think about the set of basis vectors e c ( e ⃗ c x , e ⃗ c y , e ⃗ c z ) e_c(\vec{e}_{cx},\vec{e}_{cy},\vec{e}_{cz}) ec(ecx,ecy,ecz) adopt R R R What does a matrix look like in camera coordinates ?
The answer is obvious , It's a unit array E E E.
That is to say, by left multiplying the rotation matrix R R R, We can set the basis vectors e c ( e ⃗ c x , e ⃗ c y , e ⃗ c z ) e_c(\vec{e}_{cx},\vec{e}_{cy},\vec{e}_{cz}) ec(ecx,ecy,ecz) Into a unit matrix E E E, The expression is as follows :
R ( e ⃗ c x , e ⃗ c y , e ⃗ c z ) = E R(\vec{e}_{cx},\vec{e}_{cy},\vec{e}_{cz})=E R(ecx,ecy,ecz)=E
So we know
( e ⃗ c x , e ⃗ c y , e ⃗ c z ) = R − 1 = R T (\vec{e}_{cx},\vec{e}_{cy},\vec{e}_{cz})=R^{-1}=R^T (ecx,ecy,ecz)=R−1=RT
This is our rotation matrix R R R Understanding from the perspective of basis transformation , R R R The inverse matrix ( Or transpose matrix ) Three column vectors of , It is the coordinates of the three base vectors of the camera coordinate system in the world coordinate system .
边栏推荐
猜你喜欢
Summary of "reversal" problem in challenge Programming Competition
Guess riddles (4)
Guess riddles (3)
Halcon: check of blob analysis_ Blister capsule detection
Pytorch entry record
Business modeling | process of software model
Halcon clolor_ pieces. Hedv: classifier_ Color recognition
My experience from technology to product manager
Solution to the problems of the 17th Zhejiang University City College Program Design Competition (synchronized competition)
Halcon affine transformations to regions
随机推荐
notepad++
c#比较两张图像的差异
Codeforces Round #648 (Div. 2) D. Solve The Maze
[daiy4] copy of JZ35 complex linked list
Programming implementation of ROS learning 6 -service node
Shift operation of complement
Solution to the problem of the 10th Programming Competition (synchronized competition) of Harbin University of technology "Colin Minglun Cup"
RT-Thread内核快速入门,内核实现与应用开发学习随笔记
Search data in geo database
Guess riddles (2)
资源变现小程序添加折扣充值和折扣影票插件
Use arm neon operation to improve memory copy speed
C [essential skills] use of configurationmanager class (use of file app.config)
皮尔森相关系数
Halcon snap, get the area and position of coins
Warning: retrying occurs during PIP installation
交通运输部、教育部:广泛开展水上交通安全宣传和防溺水安全提醒
Blue Bridge Cup provincial match simulation question 9 (MST)
The first week of summer vacation
Multiple linear regression (sklearn method)