Image Segmentation - Bezier and Splines in Image Processing and Machine Vision - page 41

[0 ,s ]andin F 02 ( x, y ), it is limited to ( s, L 1]. From the standpoint of

object/background thresholding, F 01 ( x, y ) can be viewed as the object while
F 02 ( x, y ) is the background, without loss of generality.
To check the feasibility of global approximation of the subimages so ob-
tained, we approximate, first of all, F 01 ( x, y ) by a polynomial of order p

−

q
( q is a predefined upper limit on the order of polynomials) satisfying a crite-
rion C. It should be noted that F 01 ( x, y ) may consist of a number of isolated
regions or patches, say, Ω 1 ,Ω 2 ,

≤

Ω r . If the approximation satisfies the
criterion C, we accept the subimage F 01 ( x, y ). Otherwise, even when a poly-
nomial surface of order q cannot approximate the subimage subject to C, we
compute the variance in each of the regions. Next, we fit a global surface
of order q over the entire subimage and a local surface of order less than q
over the residual errors (defined with respect to surface of order q )ofthe
most dispersed region. This may give rise to one of the following four different
situations:
(1) The criterion C is satisfied for the most dispersed region (with respect to
global and local surface fitting) and also for rest of the regions (with respect
to global fitting).
(2) C is satisfied for the most dispersed region but not for rest of the regions.
(3) C is not satisfied for the most dispersed region but satisfied for rest of the
regions.
(4) C is not satisfied for both the most dispersed region and rest of the regions.
In situation in (1), both local and global fits are satisfied. Hence, it implies
that all segmented regions or surface patches are homogeneous and we accept
the subimage.
In situation (2), additionally fit a local surface of order less than q over the
residual errors (defined with respect to surface of order q ) of the second most
dispersed region. The process may continue for all regions in the subimage,
only in case of failure for the global surface approximation. But if the local
surface fit fails to satisfy the criterion C at any stage (cases 3 and 4), it indi-
cates the need for further decomposition and hence, we seek a new threshold
for the subimage F 01 ( x, y ). We accept the partition, F 01 when both local and
global fits satisfy the criterion C.
A new threshold s 1 divides the image F 01 into F 011 ( x, y )and F 012 ( x, y ).
The graylevels in F 011 ( x, y ) extend from zero to s 1 while in F 012 ( x, y ), they
extend from s 1 +1 to s . In other words, the graylevel bands are [0 ,s 1 ]and
( s 1 ,s ] respectively for F 011 ( x, y )and F 012 ( x, y ). The image F 02 ( x, y )may
likewise be examined and segmented if needed. The segmentation, therefore,
follows a binary tree structure as shown in Figure 2.2.
The criterion C plays a crucial role in the determination of polynomial
orders. If the segmented regions are more or less uniform, then low order
polynomials will fit the data reasonably well. However, if the approximation
criterion C is very strict and if the spatial distribution of gray values over a
region deviates from uniformity, higher order polynomial will be required to
justify the fit. This will result in better reconstruction of the image at the

···