Summary of Mathematics Doctoral Dissertation: Research image contrast enhancement based on hedge algebra

The thesis aims to study the problems of image processing under the approach of HA theory. Building a gray level transformation function in S form to enhance the contrast for color images to apply HA. New homogeneity with hedge algebra applies to enhance contrast for color images. Building a photo fading transformation that applies to multi-channel images does not lose image detail.

MINISTRY OF EDUCATION

AND TRAINING

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY

NGUYEN VAN QUYEN

RESEARCH IMAGE CONTRAST ENHANCEMENT

BASED ON HEDGE ALGEBRA

MATHEMATICS DOCTORAL DISSERTATION

Major: Math Fundamentals for Informatics

Code: 9.46.01.10

SUMMARY OF MATHEMATICS DOCTORAL

DISSERTATION

Ha Noi, 2018

This work is completed at:

Graduate University of Science and Technology Vietnam Academy of Science and Technology

Supervisor 1: Dr Tran Thai Son

Supervisor 2: Assoc Prof Dr Nguyen Tan An

Reviewer 1: ………

………

Reviewer 2: ………

………

Reviewer 3: ………

………

This Dissertation will be officially presented in front of the Doctoral Dissertation Grading Committee, meeting at: Graduate University of Science and Technology Vietnam Academy of Science and Technology At ………… hrs …… day …… month…… year ……

This Dissertation is available at:

1 Library of Graduate University of Science and Technology

2 National Library of Vietnam

LIST OF PUBLISHED WORKS

[1] Nguyen Van Quyen, Tran Thai Son, Nguyen Tan An, Ngo Hoang

Huy and Dang Duy An, “A new method to enhancement the contrast

of color image based on direct method”, Joural of Research and

Development on Information and Communication technology, Vol 1,

No 17 (37): 59-73, 2017

[2] Nguyen Van Quyen, Ngo Hoang Huy, Nguyen Cat Ho, Tran Thai

Son, “A new homogeneity measure construction for color image direct

contrast enhancement based on Hedge algebra”, Joural of Research

and Development on Information and Communication technology, Vol

2, No 18 (38): 19-32, 2017 [3] Nguyen Van Quyen, Nguyen Tan An, Doan Van Hoa, Hoang Xuan

Trung, Ta Yen Thai, “Contruct a homogeneity measurement for the

color image bassed on T-norm”, Journal of Military science and

Technology, No 49: 117-131, 2017

Trung, Ta Yen Thai, “A method to construct an extent histogram of

multi channel images and applications”, Journal of Military science

and Technology, No 50: 127-137, 2017

S-shaped gray-scale transformation function that enhances images contrast using Hedge Algebra, Proceedings of the 10th National Conference on Fundamental and Applied Information Technology Reseach (Fair’ 10), 884-897, Da Nang, 2017

Introduction

Contrast enhancement is a very important issue in processing and

and (2) direct method of contrast enhancement

a) About indirect method

There are many indirect techniques, which were proposed in references They only modifies the histogram, without using any contrast measure

In recent years, many researchers have applied fuzzy set theory to develope new techniques to enhance the contrast of the image

Fuzzy approach algorithms often lead to the requirement of designing a gray-scale transformation S-shape function (The function is continuous monotonous increase, decreasing the input gray level when the input is below the threshold, and increasing the value gray level input when the input is above the threshold) However, the selection of functions in the fuzzy rule inference to produce the gray-scale transformation S-shape function is not easy With the following simple fuzzy rule

R1: If luminance input is dark then luminance output is darker

R2: If luminance input is bright then luminance output is brighter

R3: If luminance input is gray then luminance output is gray

So, the fuzzy reasoning results using fuzzy sets is not obvious and it is quite difficult to obtain the appropriate gray-scale S-shape function

b) About direct method

For a long time to date, almost only the studies by Cheng and coworkers have followed direct approaches, the authors have been proposed a method which modify the contrast at each pixel of gray-scale image based on the definition of image’s homogeneity measure

In addition, Cheng and coworkers have proposed an algorithm that uses the S-function which have parameters to transform the multi-level gray-scale input image and then enhance the image’s contrast by direct method

Cheng's algorithms are the basis of the contrast enhancement of grayscale images However, these algorithms still have some limitations when applying to color images, multichannel images :

(i) The resulting image after enhancement the contrast may not change the brightness of the color compared to the original image

(ii) Using images that have been modified by Cheng's image modification method as input of contrast enhancement process may lose details of original image

2

For the homogeneity measurement of pixel, Cheng proposed a way to estimate the homogeneity value of the pixel from local values Eij, Hij, Vij, R4,ij When experimenting with color images, we noticed that with this estimate, the resulting image may not be smooth

Actually the pixel's homogeneity is a fuzzy value so that we can apply the fuzzy logic to get this value

If local values Ei j ,Hi j are passed to computing with word then the formula

is formatted T e hEij ,Hij should reflect the fuzzy rule system as follows:

If we add rules with terms like "very", "little", "medium" etc with linguistic variables like "homogeneity", "entropy", "gradient" etc then homogeneity values can be estimated by human inference and thus it will be finer

Because fuzzy set theory has no basis form between the relationships of the linguistic variable with the fuzzy sets and the order of relations between the words, it is important to consider using a fuzzy reasoning method to ensure order

Through surveys, analyses and experiments we have the conclusion:

Firstly, the if-then argument based on the fuzzy set is very difficult to

guarantee the S shape of the gray-scale transformation function The direct contrast enhancement method of Cheng uses a transformation gray-scale function has S-shape but not Symmetric and the gray value may fall outside the gray-area value

Secondly, Cheng's homogeneity measurement has still limited, for

example the resulting image may not be smooth

Thirdly, using Cheng's algorithm directly on the original image channel,

the brightness of the resulting image may be less volatile In order to change the brightness, it is necessary to transform the original image before applying Cheng's contrast enhancement Cheng's image transform method may cause loss

of detail of the original image

The research topic of doctoral dissertation is:

Problem 1: Designing the Gray S-type transformation and symmetry Problem 2: Constructing a local homogeneity measurement of image Problem 3: Constructing fuzzy transformation method for image without

losing details of original image

3

Chapter 1

Overview of contrast enhancement and solving the fuzzy rule system

base on Hedge Algebars

This chapter presents the concepts of hedge algebra and approximate reasoning method based on hedge algebra, overview introduction of contrast enhancement methods such as some indirect and direct methods Analysis proposed use hedge algebra to improve the contrast by direct method

1.1 Hedge algebra: some basic issues

1.1.1 Some basic definition about Hedge algebra

The word-domain X = Dom(X) may be assumed to be a linearly ordered set and can be formalized as a hedge algebra, denoted by AX = (X, G, H, ),

where G is a set of generators, H is a set of the hedges and “”is the semantic

order relation on X Assume that, In G there are constant elements 0, 1, W which

are, respectively, the least, neutral and greatest words of Dom(X) We call each

word value x  X is term in hedge algebra

If X and H are linearly ordered then AX = (X, G, H, ) is linear hedge algebra In addition, if equipped two artificial hedges  và  with the meaning

of which is taking the infimum and supremum of H(x) - the set generated from x then we get the linear complete hedge algebra, denoted by AX = (X, G, H, , ,

) Because in this dissertation, we only care about linear hedge algebra, since

speaking the hedge algebra also means linear hedge algebra

When operand h  H in x  X, will have hx Every x  X, denoted H(x) is

set of every terms u  X from x by apply hedges in H and write u = h n …h1x,

where h n , …, h1 H

The H set include positive hedges H + and negative hedges H - The

positive hedges increase the semantics of a term and the negative hedges decrease the semantics of a term Without loss of generality, we always assume

that H - = {h-1 < h-2 < < h -q } and H + = {h1 < h2 < < h p}

1.1.2 Measurement function in the linear hedge algebra

In this section, we use linear hedge alge AX = (X, C, H, ) with C = {c-,

c+}  {0, 1, W} H = H-  H+, H- = {h-1, h-2, , h-q} satisfy h-1 < h-2 < < h -q and

H+ = {h1, h2, , hp} satisfy h1< h2 < < hp and h0 = I with I is unit operator

Let H(x) be the set of elements of X generated from x by the hedges Thus, the size of H(x) can represent the fuzziness of x The fuzziness measurement of

x, we denote by fm(x) is the diameter of the set (H(x)) = {f(u) : u  H(x)}

Definition 1 Let AX = (X, G, H, , , ) is linear complete hedge algebra An fm: X  [0,1] is said to be a fuzziness measure of terms in X if:

(1) fm(c - ) + fm(c +) =1 và hH fm(hu) = fm(u), uX;

) ( )

(

) (

y fm

hy fm x

fm

hx fm

 , that is it does not depend on specific elements and is called fuzziness measure of h, denoted by(h)

Proposition 1 For each fuzziness measure fm and  which defined in

Definition 1, the following statements hold:

Definition 2 The function Sign : X  {-1, 0, 1} is a mapping defined

recursively as follows, where h, h'  H and c  {c - , c +}:

(1) Sign(c - ) = -1, Sign(c +) = 1;

(2) Sign(hc) = -Sign(c) if h is negative c; Sign(hc) = Sign(c) if h is

positive c;

(3) Sign(h'hx) = -Sign(hx), if h'hx  hx and h' is negative h;

Sign(h'hx) = Sign(hx), if h'hx  hx và h' is positive h;

(4) Sign(h'hx) = 0, if h'hx = hx

(1.2)

Proposition 2 For any h and xX, if sign(hx) =+1 then hx > x and if

sign(hx) = -1 then hx < x

Definition 3 Let fm be a fuzziness measure on X A semantically

quantifying mapping (SQM) v on X (associated with fm) is defined as follows:

( )

j Sign

x j h Sign x x j

x h

j Sign

(1.3)

5

(h j x) = (x) +   1 ( ) ( ) ( ).

2

1 ) ( ) ( )

x h

j Sign

Proposition 3. xX, 0  v(x)  1

1.1.3 Interpolative Reasoning using SQM

Let consider the fuzzy multiple conditional Reasoning (FMCR) has form:

If X1 = A11 and and X m = A 1m then Y = B1

If X1 = A21 and and X m = A 2m then Y = B2

If X1 = A n1 and and X m = A nm then Y = B n

(1.4)

where A ij , B i , j = 1, , m and i = 1, …, n, are not fuzzy sets, but are linguistic

values The Reasoning problem is with given input Xj = A 0j , j = 1, …, m, linguistic model (1.4) will assist us in finding the output Y = B0 Without a general reduction we can suppose that the input is a vectors have semantic value normalized into the interval [0,1]

A0 = (a0,1, …, a 0,m), a0,j  [0, 1] với j = 1, 2, … m, the output is numberic

value is normalized into the interval [0, 1] too

FMCR problem is transposed to surface interpolation and is solved base

on any interpolation method In hedge algebra, this method is done as following:

Step 1: Define hedge algebra for linguistic variable

X j and Y are: AX j = (X j , G j , C j , H j, j ) and AY = (Y, G, C, H, ) correlative

Set of all parameters included, for j = 1, …, m:

*) p + q – 1 fuzzy parameters of AY: (hq), , (h1), (h1), , (h p)

In practice, these parameters can be assigned by experience or determined

by an optimization algorithm such as using genetic algorithms

Construct Xj and Y are SQM of hedge algebras AX j and AY of orrelative

linguistic variables Xj and Y, j = 1, 2, … m Let   j 

1 ,

1 , 1

Vector (X1, …, Xm, Y) of SQMs, Xj , j = 1, …, m, and Y map S L is

transformed into S norm: (X1, …, Xm, Y) : S L S norm

Step 2: Define Interpolative Reasoning on Snorm

Construct SQMs v (A ),v (B )(j  1,m i,  1,n)

 can be defined by m input aggregate oprator

1.2 Cheng’s contrast enhancement

1.2.1 Auto extracting argument (from multi gray-scale) with Cheng

algorithm

a The gray dynamic range is [a,c] is compute based on image’s histogram

b The image transformation use S-function

I  I i j S I a b c  S  fu n c I i j a b c

where [a, c] is gray dynamic range which have parameters are auto estimated when survey the top of histogram and are estimated based on the maximum of fuzzy entropy

c Compute local arguments of original image (or transformed image) and

is normalized into the interval [0,1], gradient E ij , entropy H ij , standard deviation

V ij , the 4th moment of the intensity distribution R 4,ij

d Compute homogeneity measurement of original image’s pixel (or transformed image) by assosiation operator from 4 local values

where H Oij  Eij *V ij*Hij *R4 ,ij  1 Eij * 1 Vij * 1 Hij * 1 R4 ,ij (1.7)

đ Compute non-homogeneity gray value of original image’s pixel (or transformed image)

at that pixel is lower (RCE-rule of contrast enhancement)

Since a image transformation method is monotonically increase, usually preserving the edge intensity of the image and the local entropy value, the RCE rule is generally satisfied with the original image even if the direct contrast enhancement method using a transformed images

1.2 Some Criterias for image quality assessment

1.2.1 Entropy criterion, average for many image channel {I1,I2,…,IK}:

Use common entropy criterion for every gray-scale image:

K k k

1.2.2 Fuzzy Entropy criterion, average for many image channel {I1, I2,…, IK}:

Use common fuzzy entropy for every gray-scale image:

1

1 ,

( ) ( )

K k k

8

If the value of fuzzy Entropy is lower, differentiation the brightness or darkness of an pixel is higher, that is the image having bright -dark contrast height, Gray levels of Ik channel pixels are higher than the "gray" level in the middle

i ,j , i j '

( , ) ( , ) ( , )

Chapter 2

Transform multi-channel image and construct the s-shape transformation function based on hedge algebra and apply to

contrast enhancement for multi-channel image

This chapter presents fuzzy histograms, the method for identifying multiple gray-scale dynamic ranges based on fuzzy histograms - the basis for constructing multichannel transformations and constructing S-shape gray-scale transformation functions based on hedge algebra

2.1 Estimating multiple gray-scale dynamic range based on FCM

Use FCM to estimate the gray-scale dynamic range of each image channel

of a multi-channel image Note that in some color representations such as RGB, image channels are not independent but highly correlated, so the method of estimating the dynamic range of each independent image channel is not entirely appropriate in general case

Using FCM, it is easier to estimate the gray-scale dynamic range of clusters due to the high homogeneity of the grayscale value in a cluster

For a combination of K image channels of image I, for convenience we denote 1, { I ,I , ,I }1 2 K

K

I  , using FCM clustering algorithms cluster I 1 , K to C clusters, C ≥ 2 The loop FCM algorithm minimizes the objective function:

9

2 2

2.2 Fuzzy Histogram with FCM

Definition 2.1 Fuzzy Histogram

Suppose i j c, ,  is the membership degrees value table which satisfies of

the formula (2.2), The fuzzy histogram of each cluster c projected onto the Ik

channel of image I (in a color representation), 1  c  C, 1  k  K, is denoted as

2.3 Estimating multiple gray-scale dynamic range algorithm based on FCM

Estimate each gray area of the fuzzy histogram This is the principle to determine the gray-scale dynamic range of a image channel of multi-channel image

Algorithm 2.1: Estimate C gray-scale dynamic range of a cluster of a

combination of image channel used fuzzy histogram

Input: K channels of image I (in a color representation), I1,K  { I , ,I } 1 K , parameter C N,C  2, threshold fcut (fcut > 0, small enough), M x N is size of image I

2.4 Transforming image channel

Definition 2.2: The image channel transformation of a combination of

image channel in a color representation of input image

Consider K channels of the image I, 1, { I , ,I }1 K

dynamic ranges defined by Algorithm 2.1

For k  1,K , We define a Fk image transformation method for the Ik channel

Comment: Proposition 2.2 expresses the resulting image properties after

transformed, the detail of the input image channel is preserved in gray-scale value domain There is not the case where after transforma, the pixel have small gray-scale value become the pixel have large gray-scale value

For comparison with Cheng's image transformation method, we chose 6 images illustrating the results as follows:

(in data set TID2013 ) channel of remote sensing image, #6: size 633x647 (the color

Lac Thuy district, Vietnam)

Figure 2.1 Original image #1 - #6 for experiment

Figure 2.2 Fuzzy images of image # 1, # 5 use Cheng's image transformations

method (a), (c), and the resulting image uses the proposed algorithm 2.1 (b), (d)

On the fuzzy images of the images # 1, # 5 uses the Cheng’s transformation method, the detail of image at rectangular marked area are lost, while the transformed images by uses many gray-scale dynamic ranges which Estimated from FCM clustering algorithm for a combination of RGB image channels (algorithm 2.1), the detail of images be kept better

Table 2.1 Comparison of Havg values on R, G and B channels of images

as a result of fuzziness transformation image method

Image Cheng's average fuzzy

entropy measurement

The average fuzzy entropy measurement of algorithm 2.2 proposed

2.5 Enhancing image contrast in combination with image transformation

Algorithm 2.2: Contrast Enhancement for color image using HSV color

representation

Input: Color image in the RGB color representation, size M x N

Parameter C, threshold fcut (fcut> 0, small enough), d (d x d is the window size)

Output: Color image RGB Inew, and return: CMR, CMG,CMB, Eavg , Havg

Step 1: Suppose (IH, IS, IV) is the color representation of image I in the HSV color space Quantify the IS and IV channels as grayscale images

Step 2: With input data is a image channel combination (IS, IV), number of cluster argument is C and fcut threshold, call Algorithm 2.1 to estimate the gray-scale dynamic ranges for IS, IV channels (see formula (2.2), (2.3) and (2.4))

Step 3: Determine the FS, FV transform image of channel IS, IV by to Equation (2.5), defining 2.2 with the gray-scale dynamic range estimated from step 2 for each IS and IV channels

Step 4: Compute non-homogeneity gray values {δS,ij}, {δV,ij}, exponential amplifications {S,ij}, {V,ij} at each pixel of FS and FV channels

Step 5: Compute contrast and defined a new gray channel of the FS and FV

I ( , )

1

, 1

t t t t

C g C

C g C

I ( , )

1

, 1

t t t t

C g C

C g C

Note that: S channel is indexed k = 1, V channel is indexed k = 2

Step 6: Transform IH, IS,new, IV,new images in the HSV color representation

to RGB color representation, we get Inew

Step 7: Optional step, Compute criterion CM{R,G,B}, Eavg and Havg

7.1: Calculates the new gray-scale values {δR,ij}, {δG,ij} and {δB,ij} of IR, IG

an IB channels

7.2: Compute CMR, CMG, CMB by formula (1.18):