---
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
  $$F\mapsto ([I,0],[\hat\tilde T^TF,\tilde T])$$