Coordinate System

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• World Coordinate System

 

A coordinate system based on a point in space where we live, arbitrarily coordinate, for example,

you can point one corner of the room to the origin, and you can set one wall to the X-axis, the other to the Y-axis, and the sky to the Z-axis.

 

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• Camera Coordinate System

 

A coordinate system based on a camera.

 

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• Image Coordinate System(Pixel Image Coordinate System)

 

A coordinate system for the images we see with our real eyes.

 

 

 

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• image plane

 

The image obtained through a camera.

The plane determined by the x-axis and y-axis on the Pixel Image Coordinate System.

 

A point in the 3D space is projected through the camera(or the focus of the lens) and onto a point in the image plane.

Therefore from the 3D point P, the can be determined, but on the contrary, it is impossible to obtain P from the image pixel without additional information.

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• Normalized Image Coordinate System

 

Image coordinate system that removes the influence of the camera's internal parameters.

Virtual image plane with 1 distance from camera focus.

In other words, an image plane in which the original image plane was moved in parallel to a point where the distance from the camera focal point was 1.

 

The reason why using Virtual image plane is because geometric interpretation is easier to do on a Normalized Image Coordinate System with camera parameters removed.

 

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Transformation between pixel coordinates and normal coordinates.

 

 

$\left.\left[\begin{array}{c}x\\ y\\ 1\end{array}\right.\right]=\left[\begin{array}{ccc}{f}_x& 0& c_x\\ 0& f_y& c_y\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}u\\ v\\ 1\end{array}\right]$

 

- : focal length

- : principal point

- 3 * 3 matrix : camera matrix

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• Homogeneous Coordinates

 

 

To represent a 2D point as a 3D vector, 1 is added to the bottom.

An affine transformation or a projective transformation can be expressed as a single matrix using with homogeneous coordinate systems.

 

-  2D to Homogeneous Coordinates

(x, y) -> (x, y, 1)

(x, y) -> (wx, wy, w) 
w : nonzero constant

 

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- 3D to Homogeneous Coordinates

(X, Y, Z) -> (X, Y, Z, 1) 
Or
(X, Y, Z) -> (wX, wY, wZ, w)

 

p' = (u, v) -> p' = (u, v, 1)

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-  Homogeneous Coordinates to 2D

 

Change the end to 1 and eliminate.

(x, y, α) = (x/α, y/α, 1) -> (x/α, y/α)

 

https://velog.io/@richpin/Computer-Vision-07-2D-Transformations

https://darkpgmr.tistory.com/77

https://darkpgmr.tistory.com/78