Have you ever wondered how motion in a three-dimensional scene translates into the images we see? It’s a fascinating concept to explore. In this article, we’ll delve into the concepts of motion field and optical flow, and understand the intricate relationship between the two.
Contents
What is Motion Field and Optical Flow?
Picture a point in a three-dimensional scene, moving in a certain direction. This motion projects onto the image plane, creating a motion on the image itself. This motion on the image plane is known as the motion field corresponding to that particular point. However, measuring this motion field is not always feasible.
What we can measure, instead, is the motion of brightness patterns in the image. This is referred to as the optical flow. The question then arises: When does the optical flow align with the motion field?
Let’s consider a scenario. Suppose we have a point, denoted as P₀, in the three-dimensional scene. Its location on the image plane is described by the vector rᵢ. Now, if the point P₀ has a velocity, denoted as V₀, in the three-dimensional scene, it will move to a new location given by the vector r₀ + Δr₀ over a period of time Δt.
The velocity Vₛ₀ of the scene point P₀ is the rate of change of the vector r₀ with respect to time. Our goal is to relate the image velocity Vᵢ to the scene velocity V₀. To do this, we utilize perspective projection. By applying perspective projection, we can find the derivative of the vector rᵢ in the perspective projection equation with respect to time, giving us the image velocity Vᵢ.
Optical Flow vs Motion Field
Ideally, optical flow should be equal to the motion field, accurately representing the motion of points within a scene. However, there are instances where this correspondence breaks down.
Consider two images captured by a moving camera. The motion of points between these two images depends on the depth of the point. Ideally, we would want to determine the motion field, i.e., the movement of each point in the scene. However, what we can measure is the motion of brightness patterns in the scene. By developing algorithms to track the movement of brightness patterns between images, we can obtain the optical flow at each pixel. The optical flow vector provides information about the speed and direction of motion on the image plane.
Nevertheless, optical flow does not always align with the motion field. Let’s explore a few scenarios:
Case 1: Motion Field, No Optical Flow
Imagine a sphere spinning about a vertical axis. Although the sphere’s motion creates a motion field in the three-dimensional scene, capturing images using a stationary camera results in no visible optical flow. The images taken by the camera appear identical.
Case 2: Optical Flow, No Motion Field
Now, consider a stationary sphere with changing lighting conditions. As the light source moves, the shading on the sphere changes, resulting in brightness patterns moving in the images. While there is optical flow, indicating motion of brightness patterns, there is no actual physical motion in the scene.
Case 3: Optical Flow and Motion Field, but No Correspondence
The barber pole illusion is an interesting example where both optical flow and motion field exist, but they do not correspond to each other. When a cylinder spins about a vertical axis, all points on the cylinder move in the horizontal direction. However, the pattern on the cylinder appears to move vertically in the images, resulting in a mismatch between optical flow and motion field.
Motion Illusions and Optical Flow Perception
These examples of incorrect motion perception also apply to human vision. Motion illusions, such as the dungaree wave illusion and the old chip pattern, demonstrate that optical flow can be perceived even when there is no motion field present. Our visual system measures optical flow, but our perception of motion may not always align with the actual motion in the scene.
In conclusion, understanding the relationship between motion field and optical flow provides valuable insights into how we perceive and analyze motion in images and the limitations of optical flow. These concepts play a crucial role in various fields, including computer vision, robotics, and augmented reality.
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FAQs
Q: Can we always measure the motion field?
A: Unfortunately, measuring the motion field is not always possible. We can only measure the motion of brightness patterns in the image, which is referred to as optical flow.
Q: How does optical flow differ from the motion field?
A: Optical flow represents the motion of brightness patterns in an image, while the motion field corresponds to the actual movement of points in a three-dimensional scene.
Q: Do optical flow and motion field always align?
A: No, optical flow and motion field do not always correspond to each other. There are cases where optical flow exists without a motion field and vice versa.
Conclusion
Motion field and optical flow are fascinating concepts that shed light on how motion in a three-dimensional scene translates into the images we see. While optical flow provides valuable information about the motion of brightness patterns, it may not always align with the actual motion field. Understanding the relationship between these two concepts enhances our ability to analyze and interpret motion in various fields.
