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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:

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

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

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

Next: Transferring Disparity Values Up: Hybrid rigid and non-rigid Previous: Computing new fist/elbow/shoulder positions

Trevor Darrell
10/29/1998