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

proﬁle function.

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

routine use.

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

169

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

124

81632

64 128 256

2

4

3

5

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

Structuring

Element

Input

Image

Output

Image

Output

Pixel

Row

Row

Column

Logical

Operation

Column

FIGURE 9.5 Implementation of morphological image processing operation. (After [2].)

9 Image Segmentation

170

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