Motion and Gray Value Changes

Intuitively we associate motion with changes. Thus we start our discus- sion on motion analysis by observing the diff erences between two images of a sequence. Figure 14.1a and b shows an image pair of a construction area at Heidelberg University. There are diff erences between the left and right images which are not evident from direct comparison. However, if we subtract one image from the other, the diff erences immediately become visible (Fig. 14.3a). In the lower left of the image a truck has moved, while the car just behind it is obviously parked. In the center of the image we discover the outline of a pedestrian which is barely visible

14.2 Basics                                                                                  377

 

A                                                                   b

C                                                                   d

Figure 14.2: a to d Two pairs of images from an indoor lab scene. What changes can be seen between the left and right images?

 

in the original images. The bright spots in a row at the top of the image turn out to be bikers moving along a cycle lane. From the displacement of the double contours we can estimate that they move faster than the pedestrian. Even from this qualitative description, it is obvious that mo- tion analysis helps us considerably in understanding such a scene. It would be much harder to detect the cycle lane without observing the moving bikers.

Figure 14.1c and d show the same scene. Now we might even recog- nize the change in the original images. If we observe the image edges, we notice that the images have shifted slightly in a horizontal direction. What has happened? Obviously, the camera has been panned. In the diff erence image Fig. 14.3b all the edges of the objects appear as bright lines. However, the image is dark where the spatial gray value changes are small. Consequently, we can detect motion only in the parts of an image that show gray value changes. This simple observation points out the central role of spatial gray value changes for motion determination. So far we can sum up our experience with the statement that motion might result in temporal gray value changes. Unfortunately, the reverse

378                                                                                                                  14 Motion

 

A                                                                   b

Figure 14.3: Magnitude of the diff erence between a images a and b in Fig. 14.1;

b images c and d in Fig. 14.1.

 

A                                                                   b

Figure 14.4: Diff erence between a images a and b in Fig. 14.2; b images c and

d in Fig. 14.2.

 

 

conclusion that all temporal gray value changes are due to motion is not correct. At fi rst glance, the pair of images in Fig. 14.2a and b look iden- tical. Yet, the diff erence image Fig. 14.4a reveals that some parts in the upper image are brighter than the lower. Obviously the illumination has changed. Actually, a lamp outside the image sector shown was switched off before the image in Fig. 14.2b was taken. Can we infer where this lamp is located? In the diff erence image we notice that not all surfaces are equally bright. Surfaces which are oriented towards the camera show about the same brightness in both images, while surfaces facing the left hand side are considerably brighter. Therefore we can conclude that the lamp is located to the left outside of the image sector.

Another pair of images (Fig. 14.2c and d) shows a much more complex scene, although we did not change the illumination. We just closed the door of the lab. Of course, we see strong gray value diff erences where the door is located. The gray value changes, however, extend to the

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a                            b

Figure 14.5: Illustration of the aperture problem in motion analysis: a ambiguity of displacement vectors at an edge; b unambiguity of the displacement vector at a corner.

 

fl oor close to the door and to the objects located to the left of the door (Fig. 14.4b). As we close the door, we also change the illumination in the proximity of the door, especially below the door because less light is refl ected into this area.

 

The Aperture Problem

So far we have learned that estimating motion is closely related to spatial and temporal gray value changes. Both quantities can easily be derived with local operators that compute the spatial and temporal derivatives. Such an operator only “sees” a small sector — equal to the size of its mask — of the observed object. We may illustrate this eff ect by putting a mask or aperture onto the image.

Figure 14.5a shows an edge that moved from the position of the solid line in the fi rst image to the position of the dotted line in the second image. The motion from image one to two can be described by a dis- placement vector, or briefl y, DV. In this case, we cannot determine the displacement unambiguously. The displacement vector might connect one point of the edge in the fi rst image with any other point of the edge in the second image (Fig. 14.5a). We can only determine the component of the DV normal to the edge, while the component parallel to the edge remains unknown. This ambiguity is known as the aperture problem.

An unambiguous determination of the DV is only possible if a corner of an object is within the mask of our operator (Fig. 14.5b). This em- phasizes that we can only gain sparse information on motion from local operators.

 

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