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(c) Figure 1412: This figure shows two frames of a sequence and the disparities of the matched feature points (shown magnified by a factor of 5) after applying the relaxation labeling algorithm
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CHAPTER
14 DYNAMIC
VISION
The probabilities are now refined with the iterative calculation: (1419) for constants A and B Constants A and B are selected to control the rate of convergence of the algorithm The updated probabilities must be normalized Usually, a good solution is obtained after only a few iterations To speed up the algorithm, matches with low probability are removed In Figure 1412, we show two frames of a sequence and disparities found using the above algorithm Interested readers should see [21] for an in-depth analysis of disparity calculations
Image Flow
Image flow is the distribution of velocity, relative to the observer, over the points of an image Image flow carries information which is valuable for analyzing dynamic scenes Several methods for dynamic-scene analysis have been proposed which assume that image flow information is available Unfortunately, however, although image flow has received a significant amount of attention from researchers, the techniques developed for computing image flow do not produce results of the quality which will allow the valuable information to be recovered Current methods for computing image flow, information which is critical in optical flow, and the recovery of such information are discussed in this section Definition 141 Image flow is the velocity field in the image plane due to the motion of the observer, the motion of objects in the scene, or apparent motion which is a change in the image intensity between frames that mimics object or observer motion
Computing
Image Flow
Image flow is determined by the velocity vector of each pixel in an image Several schemes have been devised for calculating image flow based on two or more frames of a sequence These schemes can be classified into two general categories: feature-based and gradient-based If a stationary camera is used, most of the points in an image frame will have zero velocity This is assuming
144 IMAGE FLOW
that a very small subset of the scene is in motion, which is usually true Thus, most applications for image flow involve a moving camera
Feature-Based
Methods
Feature-based methods for computing image flow first select some features in the image frames and then match these features and calculate the disparities between frames As discussed in an earlier section, the correspondence may be solved on a stereo image pair using relaxation The same approach may be used to solve the correspondence problem in dynamic scenes However, the problem of selecting features and establishing correspondence is not easy Moreover, this method only produces velocity vectors at sparse points This approach was discussed above as disparity analysis
Gradient-Based
Methods
spatial and to segment is given as will remain (1420)
Gradient-based methods exploit the relationship between the temporal gradients of intensity This relationship can be used images based on the velocity of points Suppose the image intensity at a point in the image plane E(x, y, t) Assuming small motion, the intensity at this point constant, so that dE 0 dt =
Using the chain rule for differentiation, we see that BE dx BE dy BE _ 0 Bx dt + By dt + Bt -
Using (1421)
U=and v=
dx dt
(1422)
dy (1423) dt' the relationship between the spatial and temporal gradients and the velocity components is: (1424)
CHAPTER 14 DYNAMIC
VISION
Figure 1413: If one sees a point using a tube such that only one point is visible, then motion of the point cannot be determined One can only get the sense of the motion, not the components of the motion vector This problem is commonly called the aperture problem In the above equation, Ex, Ey, and Et can be computed directly from the image Thus, at every point in an image, there are two unknowns, u and v, and only one equation Using information only at a point, image flow cannot be determined This can be explained using Figure 1413 This is known as the aperture problem The velocity components at a point cannot be determined using the information at only one point in the image without making further assumptions It can be assumed that the velocity field varies smoothly over an image Under this assumption, an iterative approach for computing image flow using two or more frames can be developed The following iterative equations are used for the computation of image flow These equations can be derived using the variational approach discussed below p
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