THRESHOLDING TECHNIQUES

4.1.

Introduction

This chapter describes several standard thresholding techniques. Thresholding is one of the simplest and most widely used image segmentation techniques. The goal of thresholding is to segment an image into regions of interest and to remove all other regions deemed inessential. The simplest thresholding methods use a single threshold in order to isolate objects of interest. In many cases, however, no single threshold provides a good segmentation result over an entire image. In such cases variable and multilevel threshold techniques based on various statistical measures are used. The material presented in this chapter provides some insight into different strategies of threshold selection.

4.2.

Global Thresholding

The global thresholding technique is used to isolate objects of interest having values different from the background. Each pixel is classified as either belonging to an object of interest or to the background. This is accomplished by assigning to a pixel the value 1 if the source image value is within a given threshold range and 0 otherwise [1]. A sampling of algorithms used to determine threshold levels can be found in Sections 4.6, 4.7, and 4.8.

The global thresholding procedure is straightforward. Let ÿ be the source

image and be a given threshold range. The thresholded image

is given by

"!$#%'&µÿ()&* ,+-./1023!$4/

for all 556 .

Two special cases of this methodology are concerned with the isolation of uniformly high values or uniformly low values. In the first the thresholded image will

be given by

7

"!$#ˆÿ()8* ,+-./0923!:4/

while in the second case

;=<

>!$#ˆÿ(&? @+A-9./0923!B49/A

where denotes the suitable threshold value.

Image Algebra Formulation

Let ÿ'5; be the source image and C be a given threshold range. The

thresholded result imageDE11

can be computed using the characteristic function

138 CHAPTER 4. THRESHOLDING TECHNIQUES The characteristic functions RDSTVUW(XYZ\[

and

RDST]U^ X YZ\[

can be used to isolate object of high values and low values, respectively.

Comments and Observations

Global thresholding is effective in isolating objects of uniform value placed against a background of different values. Practical problems occur when the background is non- uniform or when the object and background assume a broad range of values. Note also thatRDST]UJ_`a

Xcb

YOZ\[TdYU W ` YeZ\[9[;fYOU ^ X YZ\[9[g

4.3.

Semithresholding

Semithresholding is a useful variation of global thresholding [1]. Pixels whose values lie within a given threshold range retain their original values. Pixels with values lying outside of the threshold range are set to 0. For a source image

Z'h5i;j

and a threshold range klmCno, the semithresholded image

Rhpiqj is given by RYr[T s ZtYr[vu$w l'x ZtYyr[ x*n z {|}~1€ u$ ~ for all r5h5‚ .

Regions of high values can be isolated using

RYr[;SGT s Z(Yr[vu$wZ(Yer[ƒ n z {|}~„€ u: ~

and regions of low values can be isolated using

RYr[;SGT s Z(Yr[vu$wZ(Yer[ x…n z {|}~„€ u: ~ g

Image Algebra Formulation

The image algebra formulation for the semithresholded image

Rhi;j

over the range of values klm†n‡o is R5SGTZfCU _G`a

X†b YOZ\[g

The images semithresholded over the unbounded sets kGn(m9ˆ

[

and

Y‰

ˆŠmCno are given by RY7r[SGT]Z‹f„UW(XYyZP[

and

RYr[;SGTZfCU ^ X YZ\[

m

4. 3 Semithresholding 139

Alternate Image Algebra Formulation

The semithresholded image can also be derived by restricting the source image to those points whose values lie in the threshold range, and then extending the restriction toŒ with value  . The image algebra formulation for this method of semithresholding is

Ž5G ‘O’J“”G•–—9˜7™š›\œ

If appropriate, instead of constructing a result over Œ , one can construct the subimage 

of

’

containing only those pixels lying in the threshold range, that is,



G’“ž”•–—c˜yœ

Comments and Observations

Figures 4.3.2and4.3.3below show the thresholded and semithresholded images of the original image of the Thunderbirds in Figure 4.3.1.

Figure 4.3.1. Image of Thunderbirds Ÿ .

140 CHAPTER 4. THRESHOLDING TECHNIQUES

Figure 4.3.3. Semithresholded image of Thunderbirds «¬­V®¯1°±²e³C´1µe®\¶.

4.4.

Multilevel Thresholding

Global thresholding and semithresholding techniques (Sections 4.2 and 4.3) seg- ment an image based on the assumption that the image contains only two types of regions. Certainly, an image may contain more than two types of regions. Multilevel thresholding is an extension of the two earlier thresholding techniques that allows for segmentation of pixels into multiple classes [1].

For example, if the image histogram contains three peaks, then it is possible to segment the image using two thresholds. These thresholds divide the value set into three nonoverlapping ranges, each of which can be associated with a unique value in the resulting image. Methods for determining such threshold values are discussed in Sections 4.6, 4.7, and 4.8.

Let®'·5¸;¹ be the source image, and letº ²E» ¯ž¯1¯» º¼ be threshold values satisfying º ²½ º¿¾ ½ ¯ž¯1¯ ½ º¼ . These values partition¸ intoÀÁ5 intervals which are associated with

values à ² » ¯1¯ž¯ » à ¼AÄ ² in the thresholded result image. A typical sequence of result values

might be  » ¼ÆÅ ² ¼ » ¯1¯ž¯ » ² ¼

»Ç . The thresholded image «]·p¸;¹ is defined by

«µ7ȶ­ÊÉ

ËÌ Ã² Í$Î]ºÆ²Ï…®(µOȶ Ã1Ð Í$ÎpºÐ;υ®tµÈ¶Ñ?ºÐÅ ² à ¼¿Ä ² Í$ÎÒ®(µÔÓ\¶Ñ?º ¼(Õ

Image Algebra Formulation