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Image medium similarity measure and its applications

Ningning Zhou

, Long Hong, Shaobai Zhang

Computer School, Nanjing University of Posts and Telecommunications, Nanjing, 210003, PR China

article info

Article history:

Received 3 June 2013

Received in revised form

1 February 2014

Accepted 11 March 2014

Communicated by: Long Cheng

Available online 8 April 2014

Keywords:

Similarity measure

Measure of medium truth degree

Image edge detection

Image matching

Image ﬁdelity measure

abstract

Similarity measure plays an important role in many image processing ﬁelds. This paper introduces the

medium mathematic systems, and establishes a novel image medium similarity measure between two

pixels and that of between two image sets based on the measure of medium truth degree. Moreover, an

image edge detection algorithm, an image matching algorithm and an image medium ﬁdelity measure

method governed by the medium similarity measure are discussed in this paper. Experimental results

show that the proposed similarity measure is effective.

1. Introduction

Similarity measure, which is usually deﬁned as a certain cost

function or distance function, plays an important role in many

image processing ﬁelds, such as image matching, image edge

detection and image evaluation, etc. Correction function, covar-

iance function, Euclidean distance, Mahalanobis distance, Cheby-

chev distance, Minkovsky distance, Hausdorff distance are the

common similarity measures used in image processing. Gray,

color, texture, edge, shape and spatial relationship are the features

commonly used to measure the similarity degree of images. Gray

and color can be simply and directly obtained from the images,

and the similarity measure based on these two features can make

the most use of the information of images which also leads to high

computational-complexity. Similarity measure based on texture,

edge, shape, content, spatial relationship and other features has

lower computational-complexity, but the accuracy depends on the

extraction of the features. As a result, most of the research on

similarity measure of image focuses on distance deﬁnition and

feature extraction. Others studies the solution methods of simi-

larity measure, such as absolute balance searching (ABS), sequen-

tial similarity detection algorithm (SSDA), hierarchical searching,

Alpha–Beta searching and so on. As traditional similarity measures

are well-established and easy for computation, they are used

widely in image processing. Nevertheless, to some complicatedly

and vague images, they cannot get the satisﬁed result. As a result,

some new mathematical and computational methods such as

neural network [1,2], wavelet transformation [3,4], rough sets [5]

and other mathematical and computational methods [6,7] are

introduced into similarity measure. Nevertheless, due to their

complexities and poor portabilities, effective similarity measure

methods are yet to be seen.

Because the complexity of image information and the strong

relations among image pixels are evident, problems with uncer-

tainty and inaccuracy will appear in the image processing. As a

result, some scholars introduced fuzzy mathematics into image

similarity measure [8–10], which have yielded excellent results.

But the result of fuzzy methods is highly dependent on the

membership function which is decided by subjective experience.

The medium mathematics system is another mathematical tool

which deals with fuzzy and uncertain problem. This paper intro-

duces the medium mathematics system into image similarity

measure and establishes a new medium similarity measure that

is governed by the measure of medium truth degree. Some

properties of the proposed similarity measure are presented.

Finally we present some applications of the medium similarity

measure in image processing. The experimental results in these

algorithms show that the proposed measure is effective.

2. Image medium similarity measure

Medium principle was established by Zhu and Xiao in 1980s

who devised medium logic tools to build the medium mathe-

matics system [11].

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/neucom

Neurocomputing

http://dx.doi.org/10.1016/j.neucom.2014.03.019

Corresponding author.

E-mail addresses: zhounn@njupt.edu.cn (N. Zhou),

hongl@njupt.edu.cn (L. Hong), adzsb@163.com (S. Zhang).

Neurocomputing 140 (2014) 219–227

In medium mathematics system [11], predicate is represented

by P, any variable is denoted as x, with x completely possessing

quality P being described as P(x). The symbol “╕”stands for inverse

opposite negative and it is termed as “opposite to”. The inverse

opposite of predication is denoted as ╕P. Then the concept of a pair

of inverse opposite is represented by both P and ╕P. Symbol “ ”

denotes fuzzy negative which reﬂects the medium state of “either-

or” or “both this-and that” in opposite transition process. The truth

degree of x related to P or ╕P can be scaled by distance ratio

function h(x) [12,13].

2.1. Medium similarity measure of two pixels

Similarity degree between two pixels can be scaled by the

difference or the radio of the gray level between pixels. This paper

mainly discusses a new similarity measure based on the measure

of medium truth degree [12,13].

Given: There are two pixels x(i,j) and f(i,j) in grey images whose

range of gray level is 0–255. First, correspond the gray level of

pixel to a number axis. Second, predicate S(x(i,j),f(i,j)) represents

that x(i,j) is similar to f

(i,j), ╕S(x(i,j),f(i,j)) represents that x(i,j)is

different from f(i,j), and S(x(i,j),f(i,j)) transition, as shown in

Fig. 1. According to the measure of medium truth degree [12,13],

the similarity degree between x(i,j) and f(i,j) can be calculated by

the distance ratio function hh(f(i,j),x(i,j)) [12].

Deﬁnition 1. For pixels x(i,j) and f(i,j), we deﬁne

hhðf ði; jÞ; xði; jÞÞ ¼

½hðf ði; jÞ; xði; jÞÞ þ hðxði; jÞ; f ði; jÞÞ ð1Þ

where

hðf ði; jÞ; xði; jÞÞ ¼

dðf ði;jÞ;1Þ

dðxði;jÞ;1Þ

f ði; jÞo x ði; jÞ

1 f ði; jÞ¼xði; jÞ

dðf ði;jÞ;256Þ

dðxði;jÞ;256Þ

f ði; jÞ4 x ði; jÞ

ð2Þ

as the medium similarity measure of two pixels, where d(a,b)is

the Euclidean distance between a and b. Under the one-

dimensional circumstance, d(a,b)¼|a b|. The larger the value is,

and the higher the similarity degree between x(i,j) and f(i,j)

becomes. The smaller the value is, and the lower the similarity

degree between x(i,j) and f(i,j) becomes.

2.2. Medium similarity measure of two image sets

Deﬁnition 2. For an image X¼ [x(i,j)]

M  N

of M  N pixels and an

image F¼ [f(i,j)]

M  N

of M  N pixels, de ﬁne

s ¼

M  N

∑

i ¼ 1

∑

j ¼ 1

hhðf ði; jÞ; xði; jÞÞ ð3Þ

as the medium similarity measure of two image sets, where hh(f(i,

j),x(i,j)) is the distance ratio function deﬁned in (1).

The value of the distance ratio function hh(f(i,j),x(i,j)) reﬂects

the similarity degree between the two pixels respectively located

at corresponding position in the images X and F.Sos can be used

to measure the similarity degree of the images X and F. The larger

the value of s is, and the higher similarity degree of the images

X and F becomes.

2.3. Properties of the medium similarity measure

Property 1. hhðf ði; jÞ; xði; jÞÞA ð0; 1

Proof

In gray images, 0r xði; jÞr 255; 0

r f ði; jÞr 255.

According to expression (2), there are three cases as follows:

(1) When f ði; jÞo xði; jÞ;

hðf ði; jÞ; xði; jÞÞ ¼ dðf ði; jÞ; 1Þ=ðdðxði; jÞ ; 1Þ

¼jf ði; jÞþ1j=jxði; jÞþ1j

¼ðf ði; jÞþ1Þ=ðxði; j

Þþ1Þ

Since f ð i; jÞo xði; jÞ ; f

min

ði; jÞ¼0; 0o xði; jÞr 255;

Then 0 o hðf ði; jÞ; xði; jÞÞ o 1

(2) When f ði; jÞ¼xði; jÞ; hðf ð i; jÞ; xði; jÞÞ ¼ 1

(3) When f ði; jÞ4 xði; jÞ;

hðf ði; jÞ; xði; jÞÞ ¼ dðf ð

i; jÞ; 256Þ=ðdðxði; jÞ; 256Þ

¼jf ði; jÞ256j=jxði; jÞ256j

¼ð256f ði; jÞÞ=ð256 xði; jÞÞ

Since f ð i; jÞ4 xði; jÞ; f

max

ði; jÞ¼255; 0r xði; jÞo 255;

Then 0 o hðf ði; jÞ; xð i; jÞÞo 1

Conclusion: According case (1)–(3), we can get. 0o hðf ði; jÞ;

xði; jÞÞr 1

Similarly, we can prove that 0o hðxði; jÞ; f ði; jÞÞr 1

Then 0 o ½hðxði; jÞ; f ði; jÞÞþhðf ði;

jÞ; xði; jÞÞ=2r 1

That is hhðf ði; jÞ; xði; jÞÞ A ð0; 1

Property 2. hhðf ði; jÞ; xði; jÞÞ ¼ 1; only if f ði; jÞ¼xði; jÞ

Property 3. hhðf ði; jÞ; xði; jÞÞ ¼ hhðxði ; jÞ; f ði; jÞÞ

According to Deﬁnition 1

, we can easily get Properties 2 and 3.

3. Applications of the image medium similarity measure

3.1. Image medium edge detection algorithm

Edge is digitally presented as a set of pixels whose neighbors

are located at an orthogonal step or roof transition in gray level.

Paggio stated [14] ‘Edge may correspond to object boundaries or

not, but it owns a great feature; it provides us reduced information

required to process and preserves useful structural information

about the object boundaries as well’.Hedeﬁned edge detection as

the operation of measuring, detecting and locating changes in gray

level of the image.

In this paper, a novel image edge detection algorithm based on

the medium mathematic systems is proposed. It uses the medium

similarity measure to inspect the degree of similarity between a

pixel and its neighbors. Then it makes use of two thresholds and

applies the method of restraining the non-maxim value in domain

for edge determination.

∼

Fig. 1. Relation between the gray level of pixel x(i,j), f(i,j).

Tangent

Edge direction

Fig. 2. Edge direction.

N. Zhou et al. / Neurocomputing 140 (2014) 219–227220

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