Summary | three coordinate systems in machine vision and their relationships

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author ： Lin Qingchun

link ：https://zhuanlan.zhihu.com/p/59289459

This article reprints self-knowledge , The author has authorized , Do not reprint without permission .

 

One 、 Introduction to machine vision geometric coordinates

  Machine vision system has three coordinate systems , Namely ：1、 World coordinate system ,2、 Camera coordinate system ,3、 Images （ Pixels ） Coordinate system .

1、 World coordinate system

World coordinate system （Xw,Yw,Zw） Is the reference frame of the position of the target object , Set the dot position freely according to the convenience of operation , It can be located on the base of the robot hand or on the actuator at the front of the robot hand .

Its main function is

(1) The three-dimensional coordinates of the object ;

(2) During calibration, the position of the calibration object is determined according to the origin ;

(3) Give the positions of the two cameras relative to the world coordinate system , So as to find out the coordinate relationship between two or more cameras ;

2、 Camera coordinate system

Camera coordinate system （Xc,Yc,Zc） It is the coordinate system of the camera at its own angle , The origin is on the optical center of the camera ,Z The axis is parallel to the optical axis of the camera , That is, the shooting direction of the camera .

3、 Images （ Pixels ） Coordinate system

3.1、 Image coordinate system

Image coordinate system （x,y） Unit: meter or millimeter , Are continuous image coordinates or spatial coordinates , The coordinate system established with the intersection of diagonal lines of the picture as the datum origin .

3.2、 Pixel coordinate system

Pixel coordinate system （u,v） The unit scale is one pixel, Is discrete image coordinates or pixel coordinates , The origin is in the upper left corner of the picture .

4、 The relationship between coordinate systems

When we determine the position of an object in the picture , How to let the robot hand go to the actual position in space to grasp ？ This requires the transformation of coordinates . The conversion from pixel to spatial point is opposite to the conversion from spatial point to pixel , Let's start with the derivation of the latter .

4.1、 Images Coordinate system and Pixels Coordinate system

The relationship between image coordinate system and pixel coordinate system is ：

dx Represents the width of a pixel （x Direction ）, And x Same unit ,x/dx Express x How many pixels are there on the axis , Empathy y/dy Express y Number of pixels on the axis ,(u0,v0) Is the center of the image plane .

Convert the above relationship into matrix form ：

4.2、 The camera Coordinate system and Images Coordinate system

From the camera coordinate system to the image coordinate system, it is a three-dimensional coordinate to a two-dimensional coordinate （3D->2D） The process of , It is called perspective projection transformation . In order to solve the relationship between them , Put ordinary image coordinates （x,y） Expand to homogeneous coordinates （x,y,1）. A point in space , The point projected on the image plane is in line with the optical center of the camera . Establish the camera coordinate system with the optical center as the origin ：

According to the similar triangle relationship, we can get the following ：

f Focus the camera （ The distance from the optical center of the camera to the imaging plane ）
Expressed in matrix form as ：

Unified use of points on the imaging plane (u,v) Express ：

The relationship between image points and spatial points is ：

4.3、 The world Coordinate system and The camera Coordinate system

World coordinates （Xw,Yw,Zw） And camera coordinates （Xc,Yc,Zc） Both are three-dimensional coordinates （ Right hand system , The three axes are perpendicular to each other ）, The relationship between the two coordinate systems is rigid body transformation （ Rigid body transformation ： When an object does not deform , Rotate a geometric object , Translational motion ）. You can imagine it out of thin air , There are two coordinate systems A And B, How to integrate A The coordinates in the coordinate system are converted to B Coordinate system representation , First of all, will A The coordinate system rotates arbitrarily based on the origin , Make it x Axis ,y Axis ,z Shaft with B The coordinate axes are parallel and in the same direction , Then pan AB The linear distance of the origin of the coordinate system , It can be A The coordinates in the coordinate system are converted to B Coordinate system , This rotation Rotation And translation Transport It is the relationship between the two three-dimensional coordinates that needs to be obtained .

Use the following equation to express the relationship between the two coordinate systems ：

Where the rotation matrix R It can be regarded as space coordinates along X,Y,Z The result of dot multiplication of three rotation matrices of the axis .

When around Z Shaft rotation θ angle , The relationship between the old and new coordinates is ：

It is represented by a matrix as ：

Empathy , Around the X Axis ,Y Shaft rotation φ and ω angle , You can get ：

therefore , Get the rotation matrix R = R1*R2*R3, Dimension for 3X3,T For the translation matrix , Dimension for 3X1.

Expand to secondary coordinates ：

4.4、 From world coordinates to pixel coordinates

Synthesize the process deduced above , World coordinates （Xw,Yw,Zw）—> Camera coordinates （Xc,Yc,Zc）—> Image coordinates （x,y）—> Pixel coordinates （u,v）, The above sequence is represented by a matrix as continuous left multiplication of the next step , namely ：

The product of the first two matrices on the right of the equation is called camera internal parameter , The third matrix is called camera external parameter , Calibration of the rear monocular camera , It is to solve the internal and external parameters of the camera .

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