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Relative Pose

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---
title: Relative Pose
categories: session
---

**Reading** Ma 2004 Chapter 5

**Briefing** [Relative Pose Lecture]()

# Exercises

## Exercise 5.5

> Explain under what conditions the family of epipolar lines in at least
  one of the image planes will be parallel to each other.
  Where is the corresponding epipole 
  (in terms of is homogeneous co-ordinates)?

To make sense of the question, you have two draw the situation.

1.  Draw the two origins and the two image planes.
    Where are the epipoles?
2.  Draw two or three separate object points (3D) and the corresponding
    epipolar planes.  (Avoid object points in the same epipolar plane.)
    Where are the corresponding epipolar lines?
3.  If you have now identified different epipolar planes within one
    fixed camera configuration, you will have a *family of epipolar lines*
    in each image plane.
    Where do these lines intersect?
4.  Try to rotate one of the cameras.
    - Can you rotate it so that the epipolar lines become parallel?
    - How does this come about?
    - Where is the epipole when this happens?
5.  Review the original exercise text.  Does it make sense in terms
    of the thought experiment that you have just conducted?

## Exercise 5.8

> Given two images $x_1$, $x_2$ of a point $p$ together with the
  relative camera motion $(R,T)$, $\mathbf{X}_2=R\mathbf{X}_1+T$:

> 1. express the depth
> 1. express the depth
> 1. express the depth of $p$ with respect to the first image,
     i.e. $\lambda_1$ in terms of $x_1$, $x_2$, and $(R,T)$.
> 1. express the depth of $p$ with respect to the second image,
     i.e. $\lambda_2$ in terms of $x_1$, $x_2$, and $(R,T)$.


1.  Again, you need to draw the situation.

    - Draw the two image planes and origins.
    - Identify the points $p$, $x_1$, and $x_2$.
    - Identify the lengths $\lambda_1$ and $\lambda_2$
2.  Imagine that you move $p$ closer to camera 1, but keeping
    it on the same line so that the image point $x_1$ remains fixed.

    - what happens to $\lambda_1$?
    - what happens to $x_2$?


## Exercise 5.9


## Optional exercises

+ Exercise 5.3
+ Exercise 5.4
+ Exercise 5.2 (1)

# Debrief