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title: Lecture - Stratified Reconstruction
categories: lecture
---
# The Camera Projection
$$\lambda x = K\Pi gX$$
1. Euclidean transformation $g$ from world frame to camera frame
1. Projection $\Pi$ from 3D to 2D
1. Camera intrinsic transformation $K$
## Ambiguities
There are three ambiguities
$$\lambda x = (KR^{-1})(R\Pi H^{-1}) (H gg_w^{-1})g_wX$$
Due to the arbitrary choice of frame, $R$ and $g_w$ are
inconsequential
# Equivalence classes of the calibration matrix
- $K$ can be chosen to be upper triangular without loss of generality
- decomposition $K=QR$ where
- $R$ is a rotation
- $Q$ is upper triangular
- The special linear group $\mathsf{SL}(3)$
- invertible matrices with determinant $+1$
- The group of rotations $\mathsf{SO}(3) < \mathsf{SL}(3)$ (subgroup)
- self-orthogonal $R^{-1}=R^T$
- Equivalence classes $$\frac{\mathsf{SL}(3)}{\mathsf{SO}(3)}$$
- equivalent because they induce the same iner product $\langle\rangle_S$
- $S=K^{-T}K^{-1}$
- $(KR)^{-T}(KR)^{-1}=K^{-T}R^{-T}R^{-1}K^{-1}=K^{-T}RR^{-1}K^{-1}=K^{-T}K^{-1}=S$
- one-to-finite correspondence between $S$ and upper triangular $K$
- usually only one $K$ is a valid camera calibration matrix
- We cannot distinguish $K$ from $KR_0^T$
- and $g=[R,T]$ from $\tilde g=[R_0R,R_0T]$
# Intrinsic and Extrinsic
- intrinsic $K$
- extrinsic $g$
$$\Pi X = (\Pi H^{-1})(H X) = \tilde \Pi \tilde X$$
- $X$ in the true world
- $\tilde X$ in the distorted world
# Stratified Reconstruction
- Projective $\to$ Affine $\to$ Euclidean
- Decompose
$$H^{-1}=
\begin{bmatrix}
K^{-1} & 0 \\ v^T & v_4
\end{bmatrix}
=
\begin{bmatrix}
K^{-1} & 0 \\ 0 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
I & 0 \\ v^T & v_4
\end{bmatrix}
$$
- The first factor is an affine transformation
- The second factor is a projective transformation
- Equation 6.32
$$F\mapsto ([I,0],[\hat\tilde T^TF,\tilde T])$$