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Lecture: Distorted Space
Distorted Space
What is a distorted space?
- Consider a pixmap image with pixels \(1\times2\) mm. What is the distance from origo to the points \((0,10)\), \((10,0)\), and \((\sqrt{50},\sqrt{50})\approx(7,7)\)?
- \(\psi: \mathbb{R}^3 \to \mathbb{R}^3\), \(\psi: \mathbb{X}\mapsto \mathbb{X}' = K\mathbb{X}\)
- 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 \(S=K^{-T}K^{-1}\)
- Norm \(||u||_S=\sqrt{\langle u,u\rangle}\)
- This gives rise to a distorted space
- angles are different
- norms are different
3D Motion in Distorted Space
- Movement in canonical space: \(X = RX_0+T\)
- Co-ordinates in uncalibrated camera frame
- before: \(X_0' = KX_0\)
- after: \(X' = KX = KRX_0 + KT = KRK^{-1}X_0' + T'\)
- where \(T'=KT\)
- Thus the movement in distorted (uncalibrated) space is \((R',T') = (KRK^{-1},KT)\)
Conjugate Matrix Group
- The set of all Euclidean motions: \(\mathsf{SE}(3)=\{(R,T)|R\in\mathsf{SO}(3), T\in\mathbb{R}^3\}\)
- Conjugate of \(\mathsf{SE}(3)\) \[G' = \bigg\{ g' = \begin{bmatrix} KRK^{-1} & T'\\0&1\end{bmatrix} \bigg|R\in\mathsf{SO}(3), T\in\mathbb{R}^3\bigg\}\]
- Note commutative diagram in Fig 6.3 in the textbook
Image Formation
- Calibrated (5.1) \(\lambda x = \Pi_0X\)
- 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)
- \(\lambda x' = KRX_0 + KT\)
- abuse of notation! we switch between homogeneous and non-homogeneous co-ordinates
- \(\lambda x' = KRK^{-1}KX_0 + KT\)
- 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.