The study of motion in computer vision enables us to extract visual information from the spatial and temporal changes occurring in an *image sequence*. An image sequence is defined to be a series of *n* images, or frames, acquired at discrete times intervals *t _{k}* =

Consider an image of random dots, generated by assigning to each pixel a random grey level. Consider a second image obtained by shifting a squared, central region of the first image by a few pixels, say, to the right, and filling the gap thus created with more random dots. Displaying the 2 images in sequence, one after the other at a sufficiently fast rate, will induce the viewer to see a square moving sideways back and forth against a steady background. When the 2 images are, instead, viewed one placed next to the other, such that each eye can see only one image, the impression of a square floating against the background is produced. The latter case can be attributed to depth perception due to the disparity between parts of the 2 images.

**Example 2: Computing Time-to-Collision**

Consider a planar version of the pinhole camera model, and a vertical bar perpendicular to the optical axis, travelling towards the camera with constant velocity as shown in Fig. 1. It is possible to compute the time, *T*, taken by the bar to reach the camera only from image information, without knowing either the real size of the bar or its velocity in 3D space.

Let *L* denote the real size of the bar, *V* its constant velocity, and *f* the focal length of the camera. The origin of the reference frame is the projection centre. If the position of the bar on the optical axis is *D*(0) at time *t* = 0, its position at a later time *t* is given by *D* = *D*(0) - *V* * *t*. From Fig. 1 we see that the apparent size of the bar at time *t* on the image plane is given by

If we compute the time derivative of

Taking the ratio

Substituting the expressions for

The motion analysis problem can be divided into 2 subproblems: that of *correspondence*, and that of *reconstruction*. The correspondence problem states: which elements of a frame correspond to which elements of the next frame of the sequence? The reconstruction problem states: given a number of corresponding elements, and possibly knowledge of the camera's intrinsic parameters, what can be said about the 3D motion and structure of the observed world? These problems are often solved using optical flow. Before we proceed to discuss more of the motion analysis problem we make the following simplifying assumption:

**Assumption**

*There is only one, rigid, relative motion between the camera and the observed scene, and the illumination conditions do not vary.*

The above stated assumption implies that the 3D objects observed cannot move with different motions. This assumption is violated, for example, by sequences of football matches, motorway traffic, or busy streets, but is satisfied by, say, the sequence of a building viewed by a moving observer. This assumption also excludes flexible objects, and nonrigid motions such as stretching and shearing; for example, deformable objects such as clothes moving are ruled out.

The *motion field* is the 2D vector field of velocities of the image points, induced by the relative motion between the viewing camera and the observed scene. The motion field can be thought of as the projection of the 3D velocity field on the image plane.

Let ** P** = [

Since the third coordinate of

where

Taking the time derivative on both sides of equation 1, we obtain the 2D velocity of point

Since the component of the motion field along the optical axis is always equal to 0, we write

Notice that the motion field is the sum of 2 components, one of which depends on translation only, the other on rotation only. In particular, the translation components of the motion are

and the rotational components are

This is the case in which the relative motion between the viewing camera and the scene has no rotational component. Since ** w** = 0, the 2D velocity components reduce to

The 2D velocity components can now be written as

The 2 equations obtained above indicate that the motion field of a pure translation is

Notice that the numerators in the above 2 equations, namely,

Motion parallax occurs when 2 3D points project to the same 2D location. Let ** P**, and

Likewise, the motion field

If, at some time instant

Notice that the relative motion field does not depend on the rotational component of motion. Other factors being equal,

where [