Understanding Rigid Object Motion in Computer Vision
Defining the Motion Field
- The motion field describes the apparent movement of pixels in an image sequence.
- It's a 2D vector field, where each vector represents the velocity of a corresponding pixel.
- For rigid objects, all points move coherently, unlike non-rigid objects with independent movements.
Motion of a Rigid Object: Translation
- Pure translation involves only a change in position.
- Let `(x, y)` be the pixel coordinates and `(u, v)` be the displacement vector.
- Then, `u = t_x` and `v = t_y`, where `t_x` and `t_y` are the components of the translation vector.
Motion of a Rigid Object: Rotation
- Pure rotation around a point (e.g., the object's center) is described using a rotation matrix.
- The rotation matrix R is a 2x2 matrix dependent on the rotation angle θ.
- The rotation equations are: `x' = x cosθ - y sinθ` and `y' = x sinθ + y cosθ`. `x', y'` are the rotated coordinates.
Combining Translation and Rotation
- The general motion of a rigid object involves both translation and rotation.
- It's represented by a transformation matrix combining translation and rotation.
- This transformation matrix operates on homogeneous coordinates for efficient computation.
Optical Flow and Motion Field Estimation
- Optical flow is the apparent motion of brightness patterns in an image sequence.
- It's often used to estimate the motion field.
- Algorithms like Lucas-Kanade and Horn-Schunck are common approaches to estimate optical flow.
Limitations and Challenges
- Occlusion: Parts of an object disappearing from view can lead to errors.
- Aperture problem: Local ambiguity in motion estimation for edge points.
- Noise and image quality: Poor image quality affects accuracy.