Introduction

2. Introduction • 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.

3. 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): • Upperlimit: • Lowerlimit: Foreachelement x ∈ X oftheIVFS,theimprecisionoftheFSisdefinedbyclosedintervalsdelimitedbytheuppermembership function U(x) andthelowermembershipfunctionL (x). Thesemembership functionsL (x) and U(x)of A are twoFSmembershipfunctions,whichfulfillthefollowingcondition:

4. 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

5. 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.

6. 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).

7. 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:

8. Homogram Segmentation • Homogram

9. 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

10. 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 havgis the average value among the points between peak p1 and p2. • Then if • we will remove the peak with the smaller value from candidates, since the valley isn’t deep enough.

12. 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.

16. Conclusion • The core idea of this paper was to introduce the application of interval-valued 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.

17. 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

18. 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。 • 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。 *L. A. Zadeh，攝於2002年11月 新加坡國際學術研討會

19. 簡 介 • 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。

20. 明確集合之複習 • 什麼是明確集合 (Crisp Sets)？ • 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)ΦA如下所示：

21. 模糊集合之基本型態 • 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：

23. 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

24. 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。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

25. Union Separator Separator • 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}, • 連續方式 (集合X屬於無限集合的場合)

26. 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