in real images. On the sides of the objects, the contour lines are closed and
generally convex curves, but they may have local concavities. We can assume
that each threshold gray level deﬁnes a single closed contour for each object.
Under these conditions we need to consider only the range of gray levels
corresponding to the sloping sides of the object, and the ways to establish
the maximum slope threshold can be summarized as follows:
1. Select T at a local minimum in the histogram. This is the easiest tech-
nique, and it minimizes the sensitivity of the area measurement to small
variations in T.
2. Select T corresponding to the inﬂection point in the h-equivalent CCS
proﬁle function. This is a simple computation, and it involves considerable
averaging for noise reduction.
3. Select T to maximize the average boundary gradient. This involves com-
puting the perimeter function but requires no approximation regarding
equivalent spot images.
4. Select T corresponding to the inﬂection point in the p-equivalent CCS
For large-scale studies, one may use one of these methods to characterize the
objects under study. Then a shortcut method can be implemented for efﬁciency.
If a proﬁle analysis shows, for example, that the optimal threshold gray level for
isolated cells in microscope images occurs midway between the peak and the
background gray level, then this simpliﬁed technique can be employed for
9.2.2 Morphological Processing
After thresholding, a given image is segmented into a binary image of object
(foreground) and background. If this initial segmentation is not satisfactory,
a set of morphological operations or the procedures based on these operations
and their variants can be utilized to improve the segmentation results. The
techniques of morphological processing provide versatile and powerful tools
for image segmentation. The design of particular algorithms involves using one’s
knowledge of what effect each of the primitive operations has on an image and
combining them appropriately to obtain the desired result. For a more thorough
discussion of morphological operations, see Chapter 8.
Many of the binary morphological operations can be implemented as 3 3
neighborhood operations. In a binary image, any pixel, together with its
eight neighbors (assuming 8-connectivity), represents 9 bits of information.
9.2 Region-Based Segmentation
Thus there are only 512 possible conﬁgurations for a 3 3 neighborhood in
a binary image. Convolution of a binary image with the 3 3 kernel
64 128 256
generates a 9-bit (512-gray-level) image, in which the gray level uniquely speci-
ﬁes the conﬁguration of the 3 3 binary neighborhood centered on that pixel.
Neighborhood operations thus can be implemented with a 512-entry lookup
table with 1-bit output. Whether the operation is implemented in software or in
specially designed hardware, it is often much more efﬁcient to use a lookup table
for fast ‘‘pipeline processing’’ [21–23] than other ways of implementation.
In the general case, morphological image processing operates by sliding
a structuring element over the image, manipulating a square of pixels at a time
similar to convolution (Fig. 9.5). Like the convolution kernel, the structuring
element can be of any size, and it can contain any complement of 1’s and 0’s.
At each position, a speciﬁed logical operation is performed between the struc-
turing element and the underlying binary image. The binary result of that logical
operation is stored in the output image at that pixel position. The effect created
depends on the size and content of the structuring element and on the nature of
the logical operation.
Binary erosion is the process of eliminating all the boundary points from
an object, leaving it smaller in area by one pixel all around the perimeter.
By deﬁnition, a boundary point is a pixel that is located inside the object but
that has at least one neighbor outside the object. If the object is circular, its
diameter decreases by two pixels with each erosion. If it narrows to less than
three pixels thick at any point, it will become disconnected (into two objects) at
FIGURE 9.5 Implementation of morphological image processing operation. (After .)
9 Image Segmentation