Simulating Object Motion with Camera Motion

  The image-based rendering problem of simulating camera movement for a fixed, Lambertian scene with depth/disparity/correspondence has been solved by several authors, such as [15,22,21,13]. The motion of a single object with a fixed camera is exactly equivalent to the opposite camera motion around a fixed object. Therefore, we can use these IBR techniques as a black box for simulating 3D object motion. (Dally et al. [6] describe an efficient image-based object representation.)

Specifically, we assume that an IBR algorithm takes an image, a rotation (specified by a rotation matrix R') and a translation (specified by a vector $\vec{t}'$). The algorithm produces a new view of a scene, with the translation and rotation applied to the 3D camera position and orientation.

We want to use such an algorithm to simulate the appearance of an object rotating by R about some pivot point x, and translated by $\vec{t}$. The problem is: given R, x, and $\vec{t}$, find R' and $\vec{t}'$.

The desired translation is composed of four translations and rotations:

1.

Translate the camera center from the origin to x.

2.

Rotate the camera by R^-1.

3.

Translate the camera center back to the origin.

4.

Translate the camera center by $-\vec{t}'$.

(Recall that a object translation or rotation corresponds to the opposite camera transformation.) This gives a linear system:

$$\begin{array} {rcl} R' T(\vec{t}') &=& T(-\vec{t}) T(-\vec{... ... \vec{x} - \vec{t}} \\ & 1 \end{array} \right ]$} \end{array}$$

where $T(\vec{t})$ is a translation by $\vec{t}$, and $\vec{x}$ is the vector from the origin to x. Solving, we get

$$\begin{array} {rcl} R' &=& R^{-1} \\ \vec{t}' &=& R^{-1} \vec{t} + R^{-1} \vec{x} - \vec{x} \end{array}$$