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CamCal 005 Translation and Rotation

CamCal 005 Translation and Rotation

Highlights: In this post we will talk more about the process of going from World to Camera frame, and it is really important that you understand what we were talking in the last few posts.

Tutorial Overview:

  1. Translation and Rotation
  2. Going from World to Camera

1. Translation and Rotation

In order to go from World to Camera frame we need to do translation and rotation. So to review, a translation and rotation from frame \(A\) to \(B\), first we express in the non-homogeneous or regular coordinates. We take the location of some point \(p \) in the \(A \) frame, we rotate it and then we translate it.

translation-and-rotation

In Homogeneous coordinates, we write this as a single matrix multiplication. This matrix consists of the rotation matrix in it in the top left and the translation vector located in the right-hand column.

single-matrix-multiplication

And the key is that homogeneous coordinates allow us to write this coordinate transforms as a single matrix. So, now finally, we can talk about going from World to Camera frame.

2. Going from World to Camera

Here is our equation to go from World to Camera, using nonhomogeneous (regular) coordinates.

Nonhomogeneous coordinates:

from-real-world-to-camera

Where the idea is, if we have some point p in the world, so it’s a point location in the world frame, we have to rotate it, orient it with, to know which way it would be oriented in the camera frame, and then we have the translation from the World to Camera frame. So, we have this sort of ugly equation that would get us from a point in the world to a point in the camera so that \(\overrightarrow{_{ }^{C}\textrm{p}} \) frame, that’s now the point in the camera frame.

Inhomogeneous coordinates, it’s just expressed like this.

Homogeneous coordinates:

homogenous-coordinates

The top left \(3\times 3 \) is the rotation, the right-hand column is the translation. And that whole \(4\times 4 \) is referred to as the extrinsic parameter matrix. This is the matrix that transforms a point in the world to a point in the camera frame. By the way, that bottom row is not so important unless we are doing inverses. What is important is that bottom row is what makes this equation invertible.

Summary:

So that ends the lesson on extrinsics. This is not really a calibration, because we are going to do the calibration part later. So, this was about extrinsic geometry. Later, we are going to do extrinsic calibration where we figure out how a camera is oriented in the world. We are going to have to revisit this whole thing when we talk about mapping from world points to a location on an image plane. But before we can do that, we are going to have to talk about, once we have the location of a point in a camera frame, where does that point end up in the image? And that is the intrinsic and we are going to do that at the next post.

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