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title: Lecture: Distorted Space
categories: lecture
---
# Distorted Space
## What is a distorted space?
1. Consider a pixmap image with pixels $1\times2$ mm.
What is the distance from origo to the points $(0,10)$, $(10,0)$,
and $(\sqrt{50},\sqrt{50})\approx(7,7)$?
2. $\psi: \mathbb{R}^3 \to \mathbb{R}^3$,
$\psi: \mathbb{X}\mapsto \mathbb{X}' = K\mathbb{X}$
2. Redifining the Inner Product
- $\langle\psi^{-1}(u),\psi^{-1}(v)\rangle
= u^TK^{-T}K^{-1}v
=\langle u,v\rangle_{K^{-T}K^{-1}}
=\langle u,v\rangle_{S}$
- where $S=K^{-T}K^{-1}$
4. Norm $||u||_S=\sqrt{\langle u,u\rangle}$
4. This gives rise to a **distorted space**
- angles are different
- norms are different
## 3D Motion in Distorted Space
1. Movement in canonical space: $X = RX_0+T$
2. Co-ordinates in uncalibrated camera frame
- before: $X_0' = KX_0$
- after: $X' = KX = KRX_0 + KT = KRK^{-1}X_0' + T'$
- where $T'=KT$
3. Thus the movement in distorted (uncalibrated) space is
$(R',T') = (KRK^{-1},KT)$
## Conjugate Matrix Group
1. The set of all Euclidean motions:
$\mathsf{SE}(3)=\{(R,T)|R\in\mathsf{SO}(3), T\in\mathbb{R}^3\}$
2. Conjugate of $\mathsf{SE}(3)$
$$G' = \bigg\{ g' = \begin{bmatrix} KRK^{-1} & T'\\0&1\end{bmatrix}
\bigg|R\in\mathsf{SO}(3), T\in\mathbb{R}^3\bigg\}$$
3. *Note commutative diagram in Fig 6.3 in the textbook*
## Image Formation
1. Calibrated (5.1) $\lambda x = \Pi_0X$
1. Uncalibrated (6.1) $\lambda x' = K\Pi_0gX_0$
- $g$ is camera pose
- $K$ is camera calibration matrix
- $\Pi_0$ is the projection (as before)
2. $\lambda x' = KRX_0 + KT$
- **abuse of notation!** we switch between homogeneous
and non-homogeneous co-ordinates
4. $\lambda x' = KRK^{-1}KX_0 + KT$
5. Rewriting in uncalibrated co-ordinates:
- $\lambda x'=KRK^{-1}X'_0 + T' = \Pi_0g'X_0'$
# Uncalibrated Epipolar Geometry
Two views by the same camera.
This gives one and the same calibration matrix $K$ for both views.