Various coordinate systems in computer vision _ image pixel coordinates

Introduction to 一、四个 coordinate system
I think first we have to understand the relationship between the four plane coordinate systems in the camera model: Pixel plane coordinate system (U,V), image physical coordinates (X,Y), Camera coordinate system (XC,YC,ZC) and World coordinate system (XW,YW,ZW).



second, the image coordinates: I want to talk to the world coordinates

In this blog, corn will be the second question in this dialogue about image coordinates and world coordinates: how to talk. How the world coordinates are transformed into the camera and projected into the image coordinates.

The corn made a simple diagram and made an outline here. The graph shows that the world coordinate system reaches the camera coordinate system through a rigid body transformation, and then the camera coordinate system reaches the image coordinate system through perspective projection transformation. It can be seen that the relationship between world coordinates and image coordinates is based on the transformation of rigid body and perspective projection. In order to reward the rigid body change and perspective projection transformation to communicate the "furthest distance in the world", corn on the map rewarded them two small red flowers. Ha ha

First, let's take a look at how the rigid body transformation links the world coordinate system to the image coordinate system. Here, first on the rigid body transformation to do an introduction:

Rigid Body Transformation (regidbody Motion): in three-dimensional space, when the object does not deform, the rotation of a geometric object, translational movement, called a rigid body transformation.

Because the world coordinate system and the camera coordinates are all right-handed coordinate systems, they do not deform. We want to convert the coordinates of the world coordinate system to the coordinates of the camera coordinates, as shown in the figure below, which can be transformed by a rigid body. A coordinate system in space can always be converted to another coordinate system by a rigid body transformation. Take a turn, walk, go to another coordinate system. may have been facing the sea before, after the translation of rotation, may end up facing the iceberg, haha

Let me take a look at the mathematical expression of the rigid body change between the two.

Where XC represents the camera coordinate system and x represents the world coordinate system. R represents rotation, and t represents panning. R, T has nothing to do with the camera, so the two parameters are called the camera's external parameters (extrinsic parameter) can be understood as the distance between the two coordinate origin, because it is controlled by the components of the x,y,z in three directions, so it has three degrees of freedom.

R is the sum of the effects of rotation around the XYZ tri-axis respectively. As shown in the following:

R=r1*r2*r3. It is controlled by θ in three directions, so it has three degrees of freedom.

Well, the rigid body transformation is finished. You should all understand that the transformation process between the world coordinate system and the camera coordinate system.

Next, let's look at how the coordinates of the camera's coordinates are projected into the image coordinate system and eventually become a pixel in the photo. This includes two processes: One is the perspective projection from camera coordinates to space image coordinates (X,Y), and the second is from "continuous image coordinates" to "discrete image coordinates" (u,v). The latter we have explained in the first article posting. So here, the main introduction of perspective projection.

Perspective Projection (Perspective projection): The Center projection method is used to project the shape onto the projection plane, thus obtaining a one-sided projection drawing which is close to the visual effect. A bit like shadow play. It accords with people's psychological habits, that is, the object near the point of view, far from the point of view objects, not parallel to the imaging plane will intersect in the hidden point of parallel lines (vanish points).

Wordy so much, in fact, we look at schematic, look at the formula, seconds to understand.

Taking B (XB,YB) point in the figure as an example, a small aperture imaging camera model (the most common model of geometrical analysis) is used. Here the F is the focal length of the camera, which belongs to the internal parameters of the camera (intrinsic parameter). The coordinates of the projection point B (XB,YB) on the imaging plane can be easily obtained by using a simple similarity-triangular proportional relation:

The above two expressions also illustrate the perspective projection relationship between camera coordinates and image coordinates.

Well, now corn has explained the three twists and turns of this dialogue between image coordinates and world coordinates. Namely: the rigid body transforms, the perspective projection, (x,y) Change (u,v) (Ps. This is mentioned in the previous blog post). Then the corn joins the three processes with a picture. Implements transitions from world coordinates (X,Y,Z) to (U,V). Let image coordinates directly with world coordinates.

The transformation relationship in the diagram below is expressed in homogeneous coordinates, and you will find that the expression is very neat.

In fact, this picture shows the process also has a name: Camera model (camera models). In fact, the camera's geometric model.

By multiplying the three, you can put the three processes together and write a matrix:

P is the direct contact of world coordinates to the image coordinates, and p represents a projection camera with the following formula:

Note that when you represent the homogeneous coordinates, you need to add a small cap to the symbol. Removing the homogeneous coordinate control bit p23,p has 11 degrees of freedom.

The camera model and the coordinate system involved are the basis of the 3D reconstruction geometry framework. They can be treated as basic operational relationships. Three basic coordinate systems and camera models are used for the derivation of the three-dimensional reconstructed geometric framework.