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98 Chapter 4. Basic Motion Estimation Techniques

or as simple as a translational 2-parameter model (which is used in block-
matching) [80]. Note that with parametric models, the constraint to regularize
the ill-posed motion estimation problem is implicitly included in the motion
model.
In nonparametric models, however, an explicit constraint (e.g., the smooth-
ness of the motion eld) is introduced to regularize the ill-posed problem of
motion estimation.

4.2.6 Region of Support

An important parameter in motion estimation is the region of support. This is
the set of pels to which the motion model applies. A region of support can
be as large as a frame or as small as a single pel, it can be of xed size or
of variable size, and it can have a regular shape or an arbitrary shape.
Large regions of support result in a small motion overhead but may su er
from the accuracy problem. This means that pels within the region belong
to di erent objects moving in di erent directions. Thus, the estimated motion
parameters will not be accurate for some or all of the pels within the region.
The accuracy problem can be overcome by using small regions of support.
This is, however, at the expense of an increase in motion overhead. Small
support regions may also su er from the ambiguity problem. This means that
several patterns similar to the region may appear at multiple locations within
the reference frame. This may lead to incorrect motion parameters.

4.3 Di-erentialMethods

Di erential methods are among the early approaches for estimating the motion
of objects in video sequences. They are based on the relationship between the
spatial and the temporal changes of intensity.
Di erential methods were rst proposed by Limb and Murphy in 1975 [81].
In their method, they use the magnitude of the temporal frame di erence,
FD, over a moving area, A, to measure the speed of this area. To remove
dependence on the area size, this measure is normalized by the horizontal,
HD, or vertical, VD, spatial pel di erences. Thus the estimated motion vector
is given by

 FD(s)sign(HD(s)) 
s∈A
ˆ

d x  s∈A |HD(s)| 
ˆ
d = =  ; (4.4)

ˆ
FD(s)sign(VD(s))
d y  s∈A 
|VD(s)|
s∈A