Computer Vision basic 2--Camera Imaging Geometrical Description _ Computer vision

Imagine a lot of tourists shooting Eiffel Tower at different angles at the same time, how to describe the process in a mathematical way. The first problem to be solved is positioning, or the choice of coordinates, the Eiffel Tower has only one, if according to Warp, latitude to depict, its coordinates is only certain, but the visitor obviously does not relate to this point, he (she) only according to own preference chooses the angle and the position, therefore, the object (scenery) has the object the coordinate system, The camera has a camera's coordinate system, even if the same camera, when adjusting parameters, in the same position, the same angle, can also get different images. To unify the description, it is necessary to introduce world coordinates (or object coordinates), camera coordinates and image plane coordinates.

The world coordinates use UVW to remember.

The camera coordinates are in XYZ notation. Middle School physics tells us that objects are like inverted relationships, but as a mathematical analysis, we use illusion. Like a plane with Xoy in mind.

And the digital Image uses (U,V) to express, does not confuse the picture plane and the digital image these two concepts, the same like through the translation, the stretching and so on, may obtain the different mathematical image (U,v).

Overall, it's

We need to describe this process in a mathematical language. First look at the middle part.

The red box is marked with a 3D object to the 2D image plane of perspective projection (if you do not understand the concept of perspective projection, you need to repair the higher geometry)

Obviously, the image of any point on the OP is P (x,y), and in order to describe this relationship, you need to introduce homogeneous coordinates.

By convention, we specify this given (x ', y ', z ') we can recover the 2D point (x,y) as

x=\frac{x^ '}{z^ '}x=\frac{x^ '}{z^ '} y=\frac{y^ '}{z^ '}y=\frac{y^ '}{z^ '}

Note: (x,y) = (x,y,1) = (2x, 2y, 2) = (k X, KY, k)

About the homogeneous coordinates, the more detailed introduction can refer to the higher geometry.

The process of the above perspective projection can be described as

As the beginning says, different visitors will choose different positions and angles to shoot the same object, so the relationship between the object and the camera is not the same, this is the object to the camera coordinate transformation problem.

The Red box section describes the process from the coordinates of the object (called World coordinates) to the coordinate transformation of the camera, which is a rigid body movement, which can be described by translation and rotation.

The above illustration represents the transformation from world coordinates to camera coordinates: PC=R (pw−c) pc=r (Pw−c), written in matrix form

The translation is easy to understand, and we discuss the simpler scenario of assuming that the origin of the world coordinate system and the camera coordinate system is coincident, and that the transformation is left to rotate.

The elements of the rotation matrix are also easy to determine. Imagine (u,v,w) = (1,0,0), and its coordinates in the camera coordinate system (X,Y,Z) = (a,b,c) (different coordinates of the same physical point) are:

So there are:

Since the rotation is a rigid body movement, it is an orthogonal transformation that satisfies the r−1=rtr−1=rt, so there are:

It is not difficult to draw:

Look at an example:

Due to the transformation of the coordinates of the objects to the camera coordinates, the external parameters (External Parameters) are often written in relation to the internal parameters of the camera (r and T), i.e.

Summarize

This section describes how to transform a point in the 3D world coordinate system into a camera coordinate system and then pivot projection to a point (x,y) on a 2D image plane.

Summary: This is harder than the first verse to understand, design to some matrix transformation, three-dimensional to two-dimensional information transformation.