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# VO : The Problem

Modified 2019-01-07 by pravishsainath

The VO problem is modelled as a sequence of transformations Figure 3.1 VO Problem Model

Modified 2019-01-07 by pravishsainath

Image Sequence

• Monocular :

$$I_{0:n} = \{ I_0, I_1, \dots , I_n \}$$

• Stereo :

$$I_{l, 0:n} = \{ I_{l,0}, I_{l,1}, \dots , I_{l,n} \}$$ $$I_{r, 0:n} = \{ I_{r,0}, I_{r,1}, \dots , I_{r,n} \}$$

Modified 2019-01-07 by pravishsainath

Camera Poses

Poses of the camera w.r.t. The initial frame at k = 0

$$C_{0:n} = \{C_0, C_1, \dots , C_n\}$$

Modelled by relative transformations $T_{1:n}$

$$T_{1:n} = \{ T_{1,0}, T_{2,1}, \dots , T_{n,(n-1)}\}$$

For simplicity, denote $T_{k, k-1}$ as $T_k$,

$$T_{1:n} = \{ T_{1}, T_{2}, \dots , T_{n}\}$$

The transformation matrix at each transformation $T_k$ is viewed as :

$$T_{k, k-1} = T_{k} = \begin{bmatrix} \mathbf{R_{k, k-1}} & \mathbf{t_{k, k-1}} \\ 0 & 1 \end{bmatrix}$$

$\mathbf{R_{k, k-1}}$ = 3D Rotational Matrix

$\mathbf{t_{k, k-1}}$ = 3D Translation Vector

Concatenate to recover full trajectory :
$$C_n = C_{n-1} T_n$$

• In short : Estimate “relative motion” from image sequence
• Usually, the initial camera pose $C_0$ is considered to be the origin and these relative transformations are applied from there.
• This transformation matrix corresponds to 6-DoF pose parameters $=$ $\{t_x, t_y, t_z, \varphi, \theta, \psi \}$ where $t_x, t_y, t_z$ are the translation parameters along the 3 axes and $\varphi, \theta, \psi$ are the Euler angles indicating the rotation about the 3 axes (pitch, roll and yaw).

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