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. \(\psi: \mathbb{R}^3 \to \mathbb{R}^3\), \(\psi: \mathbb{X}\mapsto \mathbb{X}' = K\mathbb{X}\)
3. 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 \(||u||_S=\sqrt{\langle u,u\rangle}\)
5. 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) \(\lambda x = \Pi_0X\)
2. 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)
3. \(\lambda x' = KRX_0 + KT\)
   • 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.