Motion field

Introduction

A camera model maps each point $(x_{1}, x_{2}, x_{3})$ in 3D space to a 2D image point $(y_{1}, y_{2})$ according to some mapping functions $m_{1}, m_{2}$ :

(\begin{matrix} y_{1} \\ y_{2} \end{matrix}) = (\begin{matrix} m_{1} (x_{1}, x_{2}, x_{3}) \\ m_{2} (x_{1}, x_{2}, x_{3}) \end{matrix})

Assuming that the scene depicted by the camera is dynamic; it consists of objects moving relative each other, objects which deform, and possibly also the camera is moving relative to the scene, a fixed point in 3D space is mapped to varying points in the image. Differentiating the previous expression with respect to time gives

(\begin{matrix} \frac{d y_{1}}{d t} \\ \frac{d y_{2}}{d t} \end{matrix}) = (\begin{matrix} \frac{d m_{1} (x_{1}, x_{2}, x_{3})}{d t} \\ \frac{d m_{2} (x_{1}, x_{2}, x_{3})}{d t} \end{matrix}) = (\begin{matrix} \frac{d m_{1}}{d x_{1}} & \frac{d m_{1}}{d x_{2}} & \frac{d m_{1}}{d x_{3}} \\ \frac{d m_{2}}{d x_{1}} & \frac{d m_{2}}{d x_{2}} & \frac{d m_{2}}{d x_{3}} \end{matrix}) (\begin{matrix} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \\ \frac{d x_{3}}{d t} \end{matrix})

Here

u = (\begin{matrix} \frac{d y_{1}}{d t} \\ \frac{d y_{2}}{d t} \end{matrix})

is the motion field and the vector u is dependent both on the image position $(y_{1}, y_{2})$ as well as on the time t. Similarly,

x^{'} = (\begin{matrix} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \\ \frac{d x_{3}}{d t} \end{matrix})

is the motion of the corresponding 3D point and its relation to the motion field is given by

u = M x^{'}

where $M$ is the image position dependent $2 \times 3$ matrix

M = (\begin{matrix} \frac{d m_{1}}{d x_{1}} & \frac{d m_{1}}{d x_{2}} & \frac{d m_{1}}{d x_{3}} \\ \frac{d m_{2}}{d x_{1}} & \frac{d m_{2}}{d x_{2}} & \frac{d m_{2}}{d x_{3}} \end{matrix})

This relation implies that the motion field, at a specific image point, is invariant to 3D motions which lies in the null space of $M$ . For example, in the case of a pinhole camera all 3D motion components which are directed to or from the camera focal point cannot be detected in the motion field.

2 more chapters are free with a Lably account.

Introduction

Special cases

Relation to optical flow

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