A new fuzzy image filter controlled by
interval-valued fuzzy sets (IVFS) is
proposed for removing noise from images.
Based on IVFS entropy application. IVFS
entropy is used as a tool to perform
histogram analysis.
The main advantage of the proposed
technique is to restrict the number of
thresholds or parameters which have to
be tuned.
IVFS(Interval-valued fuzzy sets)
This uncertainty of membership function of a precise FS is modelled using the
length of the interval A(x) in an IVFS A(the longer A(x) the more uncertainty), so
choice of functions U(x) and L (x) is crucial. Tizhoosh [26] applied IVFS to gray
scale image thresholding. He used interval-valued fuzzy sets with the following
functions U(x) and L (x):
Upper limit:  ( x ) U :  ( x ) U  [  ( x ; g ,  )] 1 /  ( with   2 )
Lower limit:

 ( x ) L :  ( x ) L  [  ( x ; g ,  )]
 1  x  g 2 
 ( x ; g ,  )  exp   
 
 2    
For each element x ∈ X of the IVFS, the imprecision
of the FS is defined by closed intervals delimited by
the upper membership function U(x) and the lower
membership function L (x). These membership
functions L (x) and U(x) of A are two FS membership
functions, which fulfill the following condition:
IVFS entropy application
If we consider a digital image as a composite of
regions, and each define region as a fuzzy subset of
pixels, where each pixel in the image has a
membership degree to each region.
The entropy of IVFS identifies the indetermination
degree (Tizhoosh named it ultrafuzziness of an IVFS).
An IVFS may represent the uncertainties in the
membership function of an FS. So, if in an FS, the input
data contain noise, transferring this uncertainty to
membership function uncertainty, and rebuilding an
imprecise FS (or IVFS) can be accomplished
IVFS entropy application
For an M × N image subset A ⊆ X with G gray levels g ∈ [0, G −
1], Tizhoosh [26], intuitively proved that it is very easy to extend
the concepts of FS (proposed by Pal [33]) for IVFS, and to define
the(linear)index of ultrafuzziness as follows:
The larger the Γ(x) is , the larger the homogeneity degree of the
region.
Homogram Segmentation
Histogram vs. Homogram
The histogram of an image can be separated into
a number of peaks (modes), each corresponding
to a region.
The advantage of such method is that it does not
need any a priori information about the image.
Histogram-based methods consider only gray
levels and do not take into account the spatial
correlation of the same or similar valued elements
(pixels).
Homogram Segmentation
Homogram
First, fuzzy homogeneity vector, which sums the degree of
homogeneity occurring between the pixel with gray level t and its
neighbors with different angle and neighboring distance d, is
defined as:
Homogram Segmentation
Homogram
Homogram Segmentation
Peak Finding Algorithm
The algorithm includes three steps:
Find all peaks
If a point is larger than its next and previous point, the
point becomes a peak
Find significant peaks
Repeat Step1, the selected points are more significant
than Step1.
Thresholding
This step includes three steps
Homogram Segmentation
Peak Finding Algorithm
Step 3
If a point’s height h(i) is smaller than largest point
h(max), and h(i)/h(max) < 0.05, this point will be
removed.
If two points i, j, are too close, such that |I – j|<16, we
choose the bigger one, and remove smaller one.
If the valley between two peaks is not obvious.
Suppose havg is the average value among the points
between peak p1 and p2.
H
Then if
 0 . 75
avg
(
h ( p 1)  h ( p 2 )
2
)
we will remove the peak with the smaller value from
candidates, since the valley isn’t deep enough.
Noise filtering
After we found significant peak, for each peak k ∈ {0,….k}
find tmin(k) and tmax(k) the gray-level values
corresponding to the boundaries of Rk for which Γ is
minimum.
Rk=[tmin(k), tmax(k)]
Then for each pixel I(n,m),
if its gray level value x ∈ Rk
with k ∈ {0, . . . , K} remain
the pixel, otherwise do
median filter.
The core idea of this paper was to
introduce the application of intervalvalued fuzzy sets, this idea seems to be
very promising.
Proposed method has superior
performance compared to other
existing fuzzy and non-fuzzy filters for
the full range of impulse noise ratio.
References
A Hierarchical Approach to Color Image Segmentation
Using Homogeneity H. D. Cheng and Ying Sun Dept. of
Computer Science Utah State University Logan, UT
84322-4205
Color image segmentation based on homogram
thresholding and region merging H.D. Cheng∗, X.H.
Jiang, Jingli Wang
Fuzzy filter based on interval-valued fuzzy sets for image
filtering - André Biganda, OlivierColot 2009
http://en.wikipedia.org/wiki/Fuzzy_logic
1965年，美國加州柏克萊大學教授扎德(L. A.
Zadeh) 在資訊與控制的專門性學術雜誌上，
發表模糊集合(FUZZY SET)論文。是一門模
仿人類思考，處理存在於所有物理系統中的
不精確本質的數位控制方法學。
*L. A. Zadeh，攝於2002年11月
新加坡國際學術研討會
模糊理論認為，人類的思考邏輯是模糊的，
即使是條件和資料不明確時，仍必須作下判
斷。而現代電腦是兩極邏輯，非0即1，這和
人類思考方式剛好背道而馳，毫無改變空間。
但模糊邏輯理論卻能提供一種方法，將研究
對象以0與1之間的數值來表示模糊概念的程
度 ， 稱 為 「 歸 屬 函 數 」 ( membership
function)將人類的主觀判斷數值化，使得研
究結果更能符合人類思考模式。
一般人類口語上，常常會含有混淆不清或
模棱兩可的意思，尤其是在形容一件事物
或一個人時，這種不確定性往往非常明顯。
什麼是明確集合 (Crisp Sets)？
Ex: 有一個集合{1, 4, 5, 7, 8}有五個正整數，請問，這個集合有
沒有3？這個集合有沒有5？
Ans: 沒有3，但是有5。
每一個人都能夠很明確地分辨 (非0即1)。
一個由明確集合 A 所定義出的一個特性函數
(CharacteristicFunction)
1, 當 x  A ΦA如下所示：
 A ( x)  
0, 當 x  A
什麼是模糊集合 (Fuzzy Sets)？
若有一個集合 A ，它的特性函數ΦA(x)介於0到1之
間。當ΦA(x1)>ΦA(x2)表示x1屬於A的程度比x2屬於
A的程度大。
我們稱這個集合為“模糊集合 (Fuzzy Sets)”，而
它的特性函數被稱為歸屬函數 (Membership
Function)。這個函數的表示法不再是用ΦA(x)來
表示，而是用A(x)或是A(x)來表示，如下所示：
模糊集合的表示法會因為：(1) 對象、環境，
(2) 描述者的主觀意識不同而不同。
對象、環境不同
主觀意識不同
雖然模糊集合的表示法會因為一些條件的
不同而不同，但是基本的特徵還是要把握，
不可有顛倒是非之描述。
合理的描述
不合理的描述
模糊集合的表示可分為兩類：
離散方式 (集合X屬於有限集合的場合)
假設集合 X = {x1, x2, …, xn},
Separator Union
連續方式 (集合X屬於無限集合的場合)
Separator
舉例：
離散方式
假設集合 X = {-2, -1.5, -1, 0, 1, 1.5, 2}，則：
A = 0/-2 + 0.25/-1.5 + 0.5/-1 + 1.0/0 +
0.75/1 + 0.5/1.5 + 0/2
= 0.25/-1.5 + 0.5/-1 + 1.0/0 + 0.75/1 +
0.5/1.5