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Pattern Recognition 38 (2005) 2363 2372www.elsevier.com/locate/patcog

Image thresholding using type II fuzzy sets

Hamid R. Tizhoosh

Pattern Analysis and Machine Intelligence Laboratory, Systems Design Engineering, University of Waterloo, 200 University Avenue West,

ON, Canada N2L 3G1

Received 5 September 2003; received in revised form 15 November 2004; accepted 18 February 2005

Abstract

Image thresholding is a necessary task in some image processing applications. However, due to disturbing factors, e.g.

non-uniform illumination, or inherent image vagueness, the result of image thresholding is not always satisfactory. In recent

years, various researchers have introduced new thresholding techniques based on fuzzy set theory to overcome this problem.

Regarding images as fuzzy sets (or subsets), different fuzzy thresholding techniques have been developed to remove the

grayness ambiguity/vagueness during the task of threshold selection. In this paper, a new thresholding technique is introduced

which processes thresholds as type II fuzzy sets. A new measure of ultrafuzziness is also introduced and experimental results

using laser cladding images are provided.

2005 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved.

Keywords: Image thresholding; Fuzzy sets; Type II fuzzy sets; Measures of fuzziness; Ultrafuzziness

1. Introduction

In some image processing applications, we often have

to threshold gray-level images into binary images. In these

cases, the image contains a background and one or more

objects. The generation of binary images mainly serves for

feature extraction and object recognition. Image threshold-

ing can be regarded as the simplest form of segmentation,

or more general, as a two-class clustering procedure. Exten-

sive research has been already conducted to introduce newand more robust thresholding techniques [14]. Sankura and

Sezginb list over 40 different thresholding techniques [5].

Fuzzy techniques are suitable for the development of new

algorithms because they are, as nonlinear knowledge-based

methods, able to remove grayness ambiguities in a robust

way [6]. In this paper, a new thresholding technique will

be introduced which processes thresholds as type II fuzzy

E-mail address: [email protected].

0031-3203/$30.00 2005 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.patcog.2005.02.014

sets (also called ultrafuzzy sets). The concept of ultrafuzzi-

ness aims at capturing/eliminating the uncertainties within

fuzzy systems using regular (type I) fuzzy sets (ultrafuzzy

sets should not only remove the vagueness/imprecision in

the data but also the uncertainty in assigning membership

values to the data). A measure of ultrafuzziness is also in-

troduced. Experimental results using laser cladding images

are provided in order to demonstrate the usefulness of the

proposed approach.

This paper is organized as follows: In Section 2, a briefreview of fuzzy thresholding techniques is provided. In Sec-

tion 3, the fuzzy information theoretical approach to im-

age thresholding is discussed. Section 4 describes briefly

the type II fuzzy sets. Here, a new measure for ultrafuzzi-

ness is introduced. Section 5 introduces image thresholding

using type II fuzzy sets and by means of the measure of

ultrafuzziness. In Section 6 laser cladding images are used

to demonstrate the advantage of the proposed technique.

Finally, the paper is summarized with some conclusions

in Section 7.
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2364 H.R. Tizhoosh / Pattern Recognition 38 (2005) 23632372

2. Fuzzy thresholding techniques

Under the title fuzzy thresholding, one should distinguish

between different pixel classification techniques that, on the

one hand, are all based on the same idea (namely the use

of fuzzy sets [7]), but on the other hand are very different

because they use different aspects and tools of fuzzy set

theory. Generally, regarding existing fuzzy algorithms in

the literature, one can distinguish between following fuzzy

approaches to image thresholding:

Fuzzy clustering considers the thresholding as a two-class

clustering problem. There are some algorithms such as

fuzzy c-means (FCM), possibilistic c-means (PCM), etc.

that can be applied to image thresholding [811].

Rule-based approach uses fuzzy ifthen rules to find the

suitable threshold. This method is suitable if there exists

explicit expert knowledge about the image (e.g. in medical

applications) [12].Fuzzy-geometrical approach optimizes geometrical mea-

sures such as compactness, index of area coverage, etc.

This approach uses, in contrast to other fuzzy techniques,

spatial image information [6,1317].

Information-theoretical approach minimizes or maximizes

measures of fuzziness and image information such as in-

dex of fuzziness or crispness, fuzzy entropy, fuzzy diver-

gence, etc. Because of its simplicity and high speed, this

approach is the most used fuzzy technique in the litera-

ture [6,1724].

In this work, we focus on the last approach because it is the

most common fuzzy approach to image thresholding. How-

ever, all other approaches could be reviewed to verify how

extension of type I to type II could eventually be imple-

mented. The main purpose of this work is to demonstrate

that algorithms based on type II fuzzy sets are (can be) su-

perior to their counterparts which use ordinary fuzzy sets.

3. Information-theoretical approach

If we are to understand images as fuzzy sets (or subsets),

the question arises how fuzzy is a fuzzy set? For instance,

if the membership function is flat, then it is very fuzzy,

and if it is steep, then it is rather crisp. A flat membership

function (high fuzziness) indicates the high image data

vagueness, and hence a difficult thresholding. Measures of

fuzziness give a quantitative answer to this issue. The most

common measure of fuzziness is the linear index of fuzzi-

ness [6,22,2427]. For an M N image subset A X with

L gray levels g [0, L 1], the histogram h(g) and the

membership function X(g), the linear index of fuzziness

l can be defined as follows:

l (A) =2

MN

L1

g=0

h(g) min[A(g), 1 A(g)]. (1)

For the spatial case, the fuzziness can be calculated as

follows:

l (A) =2

MN

M1i=1

N1j=1

min[A(gij), 1 A(gij)]. (2)

To measure the global or local image fuzziness, a suitablemembership function A(g) should be defined. Different

functions are already used in the literature, such as the stan-

dard S-function [28,27] and the Huang and Wang function

[20]. Tizhoosh [17] defined the suitable threshold as an LR-

type fuzzy number (Fig. 1), which was defined as follows:

(g)=

0, ggmin or ggmax,

L(g) =

g gmin

T gmin

, gmingT ,

R(g) =

gmax g

gmin T

, Tggmax,

(3)where g is the gray level, gmin and gmax are the mini-

mum and maximum gray levels and T [0, L 1] is a

suitable constant. The linguistic hedges and (0, )

can be determined with respect to the statistical properties

of the image histogram. However, the proper selection of

parameters is not easy and can add more complexity to the

algorithm at hand. Using a fuzzy number seems to be more

natural since we usually try to segment the image by means

of a preferably single number (a unique threshold for the

entire image). Only if this fails, which occurs in many ap-

plications, advanced techniques for adaptive thresholding

are employed. A single threshold, globally determined foran entire image or locally calculated for an image region,

remains uncertain. Therefore, removing the uncertainty

around a crisp number by considering/representing it as a

fuzzy number seems to be beneficial.

The general algorithm for image thresholding based on

measures of fuzziness can be formulated as follows (Fig. 2):

(1) Select the shape of the membership function.

(2) Select a suitable measure of fuzziness (e.g. Eq. (1)).

(3) Calculate the image histogram.

(4) Initialize the position of the membership function.

(5) Shift the membership function along the gray-level

range (Fig. 2) and calculate in each position the amountof fuzziness, for instance using Eq. (1).

(6) Locate the position gopt with minimum/maximum

fuzziness.

(7) Threshold the image with T = gopt.

Fig. 3 shows an example of thresholding with measures

of fuzziness with different membership functions. It should

be noted that it is not possible to say which membership

function is the best one (Murthy and Pal [28] make some

considerations on the choice of an appropriate membership

function). One can always find images for which a certain

technique delivers good or bad results. This is one of the
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H.R. Tizhoosh / Pattern Recognition 38 (2005) 2363 2372 2365

gray levelsA B C

0

1

0.5

gray levels0 L-1

0

1

0.5

threshold T

object background

gray levels0 L-1

0

1

0.5

L(g)R(g)

T

Fig. 1. Different membership functions for image thresholding. From left to right: S-function used by Pal and Rosenfeld [14], function used

by Huang and Wang [20], and threshold as a fuzzy number used by Tizhoosh [17].

T

0

1

histogram

moving the fuzzy number

m,h

Fig. 2. The membership function is shifted over the gray-level

range to calculate the amount of fuzziness in each position. The

maximum fuzziness indicates the optimal threshold (how and what

we shift may differ for other membership functions).

major motivations of this work to remove the uncertainty of

membership values by using type II fuzzy sets (see the next

section).

4. Type II fuzzy sets

The main problem with fuzzy sets type I, regardless of

which shape we use and what algorithm is applied, is that

the assignment of a membership degree to an element/pixel

is not certain. Membership functions are usually defined by

the expert and are based on his intuition/knowledge. The

fact that different fuzzy techniques differ mainly in the way

that they define the membership function is for the most part

due to this dilemma. To find a more robust solution, type II

fuzzy sets should be introduced.

There are different sources of uncertainties in type I fuzzy

sets (see [29]): the meanings of the words that are used,

measurements may be noisy, the data that are used to tune

the parameters of type I fuzzy sets may also be noisy. Type

I fuzzy sets are not able to directly model such uncertainties

because their membership functions are totally crisp. On the

other hand, type-2 fuzzy sets are able to model such uncer-tainties because their membership functions are themselves

fuzzy (Mendel and Bob John [29]). The term footprint of

uncertainty (FOU) is used in the literature to verbalize the

shape of type II fuzzy sets (shaded area in Fig. 4) [29,30].

The FOU implies that there is a distribution that sits on top

of that shaded area. When they all equal one, the resulting

type II fuzzy sets are called interval type II fuzzy sets. Fuzzy

sets of type II are, therefore, fuzzy sets for which the mem-

bership function does not deliver a single value for every

element but an interval.

Definition. A type II fuzzy set A is defined by a type IImembership function

A(x,u), where x X and u Jx

[0, 1], i.e. [30],

A = {((x, u),A

(x, u))| x X, u Jx [0, 1]},

(4)

in which 0A

(x,u)1. A can also be expressed in the

usual notation of fuzzy sets as

A =

xX

uJx

A

(x,u)

(x,u), Jx [0, 1], (5)

where the double integral denotes the union over all x and u.

In order to define a type II fuzzy set, one can define a type

I fuzzy set and assign upper and lower membership degrees

to each element to (re)construct the footprint of uncertainty

(Fig. 4). A more practical definition for a type II fuzzy set

can be given as follows:

A = {(x, U(x),L(x))| x X,

L(x)(x)U(x), [0, 1]}. (6)

The upper and lower membership degrees U and L of

initial (skeleton) membership function can be defined by
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Fig. 3. From left to right: original image, thresholded using S-function (T = 51), using the Huang and Wang function (T = 39), and using

a fuzzy number as in Fig. 1 (T = 19).

type I fuzzy set type II fuzzy set

upper limit

lower limit

membership

1 1

00

Fig. 4. A possible way to construct type II fuzzy sets. The interval between lower and upper membership values (shaded region) should

capture the footprint of uncertainty (FOU).

means of linguistic hedges like dilation and concentration:

U(x) = [(x)]0.5, (7)

L(x) = [(x)]2. (8)

Of course, other linguistic hedges such as deaccentuation

and accentuation can also be employed:

U(x) = [(x)]0.75, (9)

L(x) = [(x)]1.25. (10)

Hedges are generally available as pairs, which represent di-

agonally different modifications of a basic term. It seems,

therefore, practical to use a linguistic hedge and its re-ciprocal value to draw the footprint of uncertainty. Hence,

the upper and lower membership values can be defined as

follows:

U(x) = [(x)]1/, (11)

L(x) = [(x)], (12)

where (1, ). In the conducted experiments, (1, 2]

has been used because ?2 is usually not meaningful for

image data.

4.1. A measure of ultrafuzziness

If we interpret images or thresholds as type II fuzzy sets,then the question arises as to how ultrafuzzy is a fuzzy set.

We have to answer this question to extend the aforemen-

tioned fuzzy thresholding to type II fuzzy sets. If the degrees

of membership can be defined without any uncertainty (ordi-

nary or type I fuzzy sets), then obviously the ultrafuzziness

should be 0. For the case that individual membership values

can only be indicated as an interval, the amount of ultra-

fuzziness will increase. The extreme case of maximal ultra-

fuzziness (=1) is comparable to total ignorance in measure

theory, whereas we absolutely do not know anything about

the nature of membership degrees of the problem at hand.

With respect to these thoughts and the way we define a typeII fuzzy set, a measure of ultrafuzziness for an M N

image subset A X with L gray levels g [0, L 1],

histogram h(g) and the membership function A

(g) can be

defined as follows:

(A) =1

MN

L1g=0

h(g) [U(g) L(g)], (13)

where

U(g) = [A(g)]1/, (14)

L(g) = [A(g)]

, (1, 2]. (15)

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For the spatial case, the ultrafuzziness can be calculated as

follows:

(A) =1

MN

M1i=1

N1j=1

[U(gij) L(gij)]. (16)

This basic definition relies on the assumption that the sin-gletons sitting on the FOU are all equal in height (which is

the reason why the interval-based type II is used). Hence, it

can only measure the variation in the length of the FOU.

Kaufmann [26] introduced first an index of fuzziness to

measure the imprecision/vagueness of a fuzzy set. He also

established four conditions that every measure of fuzziness

shouldsatisfy. Analogously, we can demandthat themeasure

of ultrafuzziness should satisfy the following conditions:

(1) Minimum ultrafuzziness: (A)=0 ifA

is a type I fuzzy

set (g X U(g) = L(g)).

(2) Equal ultrafuzziness: (A) = ( A).Proof:1 Let A be a type II fuzzy set: A = {(x, U,L)|

L = ,U =

1/}. Then the complement set A can

be defined as follows:A = {(x, U,L)|L = 1

1/,U = 1 }.

The ultrafuzziness for the complement set A can be

calculated as follows:

( A) =1

MN

L1g=0

h(g) [(1 L(g)) (1 U(g))],

=

1

MN

L1

g=0

h(g) [(1 L(g)) 1 + U(g)],

=1

MN

L1g=0

h(g) [U(g) L(g)],

= (A). (17)

(3) Reduced ultrafuzziness: (A) (A) ifA is an inten-

sified (crisper) version ofA (A has a shorter/narrower

FOU than A).

(4) Maximum ultrafuzziness: (A) = 1 ifg X U(g)

L(g) = 1.

5. Thresholding with fuzzy sets of type II

The general algorithm for image thresholding based on

type II fuzzy sets and measures of ultrafuzziness can be

formulated as follows:

(1) Select the shape of skeleton membership function (g)

and initialize .

1 We are considering the special case with dilation and con-

centration modifiers as means for constructing the FOU. The proof

of the general case will remain a subject for future works.

(2) Calculate the image histogram.

(3) Initialize the position of the membership function.

(4) Shift the membership function along the gray-level

range.

(5) Calculate in each position the upper and lower mem-

bership values U(g) and L(g).

(6) Calculate in each position the amount of ultrafuzziness

(Eq. (13)).

(7) Find out the position gopt with maximum ultrafuzzi-

ness.

(8) Threshold the image with T = gopt.

Using the fuzzy number in Eq. (3) the thresholding based

on this scheme can be formulated as solving the following

equation:

j

jT(A) =

j

jT

1

MN

L1g=0

h(g)

[U( g , T ) L( g , T )] = 0. (18)

6. Experimental results

In this section two sets of test images will be used to

investigate the effect of type II fuzzy sets on the results

of image thresholding. The purpose of these experiments is

mainly to compare the type I fuzzy thresholding with its

type II counterpart. However, results by other techniques are

also presented to have non-fuzzy references.

6.1. Experiments with laser cladding images

In order to test the performance of the proposed tech-

nique, images from laser cladding are used. Laser cladding

by powder injection is an advanced material processing

with applications in manufacturing, part repairing, metal-

lic rapid prototyping and coating [31,32]. A laser beam

melts powder and a thin layer of the substrate to cre-

ate a layer on the substrate. Having a reliable feedback

system for the closed loop control is crucial to this pro-

cess. For this purpose and beside other sensors, a CCD

camera is used to feed the required data to a controller.

From the captured images, the laser height hL should be

measured (Fig. 5) and sent to the controller. The measure-ment accuracy of laser height plays here a pivotal role.

Fig. 5. Calculation of laser height for control.
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Table 1

Results of thresholding for laser images

Test Type Otsuimage I fuzzy sets algorithm Type II fuzzy sets

The result of the proposed approach based on type II

fuzzy sets was compared to its counterpart with fuzzy

sets type I. The interval-based type II fuzzy set was

defined with = 2 (Eqs. (11), (12)). Also the Otsu

algorithm was considered. Results for different laser

cladding images are illustrated in Table 1. Based on sub-

jective determination of the optimal height hLopt, the

average difference d from the optimal height was cal-

culated for every algorithm. The results are presented in

Table 2.

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Fig. 6. Test images and the corresponding (manually generated) ground-truth images. From top left to bottom right: blocks, zimba, gearwheel,

shadow, stones, rice, potatoes, text, ultrasonic, and newspaper.

Table 2

Average difference d (in pixel) from optimal (manual) measurement

(smaller d means that the laser height measurement is closer to

the expert measurement: ideally d = 0)

Using type I Using Otsu Using type IIfuzzy sets algorithm fuzzy sets

d 6.3 6.8 2.1

6.2. Experiments with other images

The effect of thresholding with type II fuzzy sets was

also tested using 9 different images. These images contained

small and large objects, text, objects with clear or fuzzy

boundaries, and were noisy or smooth. In order to verify the

performance of the thresholding, the optimal thresholded

image was created manually and used as a gold standard

(ground-truth image). A measure of performance was used

to compare the individual gold images with the binary result

delivered by type I and type II thresholding. Based on the

misclassification error [5,33], the performance measure was

defined as

= 100 |BO BT| + |FO FT|

|BO | + |FO |, (19)

where BO and FO denote the background and foreground

of the original (ground-truth) image, BT and FT denote the

background and foreground area pixels in the result image,

and |.| is the cardinality of the set.

The Otsu technique, as a well-established algorithm, and

the clustering-based Kittler method were employed as well

to have non-fuzzy references (according to Sankur and

Sezgin, the Kittler algorithm is the best thresholding tech-

nique available [5]). The test images with corresponding
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Fig. 7. Results of four algorithms for thresholding of images in Fig. 6. From left to right: result of type I algorithm, Otsu method, type II

algorithm, and Kittler clustering.

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Table 3

Performance of individual methods based on comparison of their

results with the ground-truth images (see Figs. 6 and 7, and

Eq. (19))

Image Type I Otsu Type II Kittler

Blocks 71.21 94.32 98.98 98.35Zimba 86.31 97.87 99.52 98.85

Gearwheel 64.47 98.13 98.21 92.24

Shadow 75.75 90.64 94.39 78.33

Stones 39.96 95.96 96.99 81.10

Rice 99.98 94.34 99.65 93.44

Potatoes 98.96 98.01 99.77 99.21

Text 36.37 77.28 93.44 90.02

Ultrasonic 92.63 96.25 97.56 96.81

Newspaper 93.68 99.00 98.17 96.31

m 75.93 94.18 97.67 92.47

23.15 6.44 2.19 7.38

ground-truth images are illustrated in Fig. 6, and the results

of the four techniques are presented in Fig. 7. The perfor-

mance measure for every algorithm is listed in Table 3.

As is apparent from Table 3, type II thresholding has the

highest average performance of 97.67%with the lowest stan-

dard deviation of 2.19%. In contrast, the type I algorithm

with 75.93% average performance and 23.15% standard de-

viation is clearly inferior to the type II algorithm.

7. Concluding remarks

Image thresholding is a difficult task in image process-

ing. Probably, we will never find a super algorithm that can

be successfully applied to all kinds of images. Therefore, it

is appropriate to look for new techniques. Fuzzy set theory

provides us with knowledge-based and robust tools for de-

veloping new thresholding techniques. They, however, usu-

ally suffer from the problem that the optimal membership

function cannot be easily determined. Thecentral idea of this

work was to introduce the application of type II fuzzy sets

into fuzzy thresholding in order to overcome this dilemma.

For this purpose, a new measure of ultrafuzziness is intro-

duced to quantify the vagueness of a type II fuzzy set. A new

thresholding algorithm based on fuzzy numbers and type IIfuzzy sets was then introduced. A practical example from

laser cladding demonstrated the usefulness of the proposed

approach and its superiority to the same algorithm incorpo-

rating type I (ordinary) fuzzy sets. Additional experiments

with different test images reinforced this conclusion. In fu-

ture works, the effect of extension to type II fuzzy sets for

other algorithms, comparisons with non-fuzzy techniques,

and an adaptive version of the proposed technique will be

the subject of investigations. More extensive investigations

on other measures of ultrafuzziness and the effect of param-

eters influencing the width/length of FOU should certainly

be conducted.

Acknowledgements

The author wants to thank Dr. E. Toyserkani and Dr. A.

Khajepour (Mechanical Engineering, University of Water-

loo, Canada) for providing the test images and necessary

descriptions.

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About the AuthorHAMID R. TIZHOOSH received the M.S. degree in electrical engineering from University of Technology, Aachen,

Germany, in 1995. From 1993 to 1996, he worked at Management of Intelligent Technologies Ltd. (MIT GmbH), Aachen, Germany, in the

area of industrial image processing. Dr. Tizhoosh received his Ph.D. degree from University of Magdeburg, Germany, in 2000 in the field

of computer vision.

He was active as a scientist in the engineering department of Image Processing Systems Inc. (IPS), Markham, Canada, until 2001. For 6

months, he visited the Knowledge/Intelligence Systems Laboratory, University of Toronto, Canada.

Since September 2001, he is a faculty member at the Department of Systems Design Engineering, University of Waterloo, Canada. His

research encompasses machine intelligence, computer vision and soft computing. Currently he is a member of the Pattern Analysis and

Machine Intelligence Group at the University of Waterloo.

10.1.1.98.214

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