edge representation with fuzzy sets in blurred images

E L S E V I E R Fuzzy Sets and Systems 100 (1998) 77-87

IRMI¥ sets and systems

Edge representation with fuzzy sets in blurred images 1 Tae Y o n g K i m * , J o o n H e e H a n

Department of" Computer Science and Engineering, Pohang University of Science & Technology, San 31, Hyoja-Dong, Pohang, 790-784, South Korea

Received August 1996; revised April 1997

Abstract

This paper proposes a representation of edges in blurred images using the concept of fuzzy sets when the images are degraded by various asymmetric and local blurring factors. The proposed representation is expressed by fuzzy membership functions, and it can serve as a relative index of blur. The membership function is derived from the distribution of intensity gradients and the symmetricity of gradient magnitudes, and the function is calculated directly from the blurred image without identifying the point spread fimction or restoring image. In this way, the fuzzy edge representation describes edges with their degradation states by fuzzy memberships instead of the binary description of edges. The index of fuzziness reflects the average amount of ambiguity presented in a fuzzy set, and the moments denote an object in the image quantitatively. These measures are adopted to illustrate the effectiveness of the representation in locally blurred images. (~) 1998 Elsevier Science B.V. All rights reserved

Keywords: Pattern recognition; Edge detection; Edge representation; Fuzzy geometry; Membership function; Blurred image

I. Introduction

Edge detection process has been an essential part in image processing and computer vision. The edge serves to simplify the analysis of images by drasti- cally reducing the amount of data to be processed and by preserving useful structural information about object boundaries. Usually, edge detection processing is performed by smoothing, differentiating and thresholding for finding optimal edges in noisy images [2, 10]. Blur is introduced into the image during the imaging process by such factors as diffrac-

* Corresponding author. Fax: +82 562 279 2299; e-mail: kimty@ hawk.postech.ac.kr, [email protected].

I This research was supported by KOSEF through the Automa- tion Research Center at POSTECH.

tion, lens aberration, motion of the object, wrong focus and atmospheric turbulence. Since intensity variations are degraded in a locally blurred image, it is hard to detect edges and it is also diffi- cult to select proper threshold automatically. Local peaks of intensity gradient magnitudes or zero- crossing points that denote the positions of edges are changed by the effect of asymmetric blurring factors.

The widely accepted standard linear model [5] describes the imaging process by convoluting an unknown original image o(x,y) with point spread function (PSF) h(x, y):

f (x, y )=o(x , y) • h(x ,y) + n(x, y) (1)

where f ( x , y) represents the observed image, n(x, y) denotes the noise factors and the PSF h(x, y) describes

0165-0114/98/$19.00 (g) 1998 Elsevier Science B.V. All rights reserved PII: S0165-0114(97)00190-5

78 T E Kim, J.H. Han/Fuzzy Sets and Systems 100 (1998) 77 87

(a) (b) (c)

Fig. 1. Edges detected by a conventional method and errors: (a)

the imaging system. However, if an image was blurred locally or asymmetrically as in a case that an image obtained from three-dimensional object which is focused on the top portion but defocused on the bottom portion or as in a case of an image having objects moving against the stationary background, the PSF cannot be shift invariant or circular-symmetry. In a case like this, an image is hard to restore by frequency domain or spatial domain methods.

Fig. 1 represents positions of edges changed by the effect of local and asymmetric blurring. The edges are detected by a conventional method when a camera system focuses on the bottom portion of a three-dimensional object, and when focus is on the top portion of the object are shown in Figs. l(a) and (b), respectively. Fig. l(c) is the result of superimposing Fig. l(a) on (b), and it shows that same edges are detected differently depending on where the focus was on, particularly at the circled part. This result demonstrates that the edges change their positions when they are defocused, and errors may be propagated to the high level processing.

In many cases, we do not need to know the original image itself; we only need to know some representation of it. The representation should be immune to

Focusing on bottom; (b) focusing on top; (c) superimposing (a) on (b).

blur or noise and should represent the original image as precisely as possible including its degradation state.

The theory of fuzzy set provides a suitable algo- rithm in analyzing complex systems when the pattern indeterminacy is due to inherent variability or fuzziness rather than randomness. Since a gray tone pic- ture possesses some ambiguity within pixels due to the possible blur or noise, it is justifiable to apply the concept of fuzzy set rather than ordinary set theory to an image processing problem. Fuzzy geometric prop- erties as defined by Rosenfeld [14, 15] seem to pro- vide a helpful tool for analysis. Fuzzy geometry can deal with an image as its natural state that it does not make any prior model and guess any degree of noise. So its applications deal with obtained image directly and get much of information through many kinds of researches [9, 11-13, 16]. Since intensity data have much redundancy and are too sensitive in noise, it is desirable to use more significant feature in image processing.

In this paper, we propose a fuzzy edge representation (FER) in blurred image with the fuzzy membership which is derived from gradient magnitudes of the elements and the symmetricity of magnitudes in the spatial domain. We have also considered an

T.Y. Kim, J.H. HanlFuzzy Sets and Systems 100 (1998) 77~87 79

index for local blurring by the Gaussian normalization of the gradient magnitudes. In this way, we can avoid the PSF identification, image restoration and threshold selection. It can prevent error propagation from wrong edge detection to high level processing such as matching or feature identification. It should be noted that a similar approach has been used in some applications for the purpose of computing [6] or reasoning in decision processes [8], but our work presents a more general approach to describe the edges in the case of local blurring.

The representation is verified by the "index of fuzziness" [13] that reflects an image quality as a quantita- tive measure and "moments" [ 1,7] which are invariant under some deformation (scaling, rotation and translation). "Hough transform" [3] is also considered for verifying the availability of the representation.

In developing FER, the following assumptions and criteria are applied. • It is assumed that step edges are degraded by asym-

metric and/or local blurring factors. • The representation should well cope with the de-

formation of objects, and should be immune to degradation.

• The representation includes the original edges which are not affected by any degradation.

• The representation can be an index for comparison among edges having different kinds or amounts of degradation.

• The representation reflects the state of ambiguity in an image for high level processing. The FER introduced in this paper are based on

the gradients of edges. In Section 2 we define fuzzy sets of an image, and we describe the method for representing fuzzy edges in Section 3. In Section 4 moments, Hough transform and index of fuzziness are introduced for verification measures. FER is compared with binary edge representation by the moments and Hough transform in experiments.

2. Gradient image and fuzzy set

With the concept of fuzzy set, set X in a gradient image F of M × N dimension can be considered as a universal fuzzy set. The gradient image F(x, y) is obtained from intensity image f ( x , y) by convoluting with the Gaussian G(x, y; ag) and differentiat-

ing in two dimension. Filtering is used to smooth the data before differentiating it. Each element of F has a magnitude with direction.

F(x, y) = V[f(x , y) • G(x, y)]. (2)

Edge set A is defined on the universal set X, which takes feasible edge points as its members each of which has its own membership. In a crisp set, char- acteristic function is (~(x)~{O, 1}, depending on edge or not. But in a fuzzy set, the membership function is vA(x)~[0,1], where x E X . Edge set A consists of many subsets Ai whose members are points that are perpendicular to the object boundary in one-dimensional gradient direction. The members of subset Ai are connected and have the same sign of gradient magnitude.

In the notion of fuzzy set, we may there- fore write that the edge set A = U A i so that VAiuAj = max[v&, vAj], where A i = {x E X I VA,(X)>O } and vA, :X ~ [0, 1]. With the fuzzy set definition, a one-dimensional gradient signal following the gradient direction is shown in Fig. 2.

This signal has positive and negative magnitudes. The large magnitude of the signal represents a high possibility of edge. The members of edge subset A i a r e

obtained by dividing gradient curve at the point where the sign of magnitude is different from its neighbor. In Fig. 2, the edge subsets are represented by closed area in gradient curve and a magnitude of Ai, Mi, is determined by Mi = Y'~x~Aj IF(x)l. A threshold of the conventional method is determined by the estimated amount of noise, but this threshold may omit locally blurred edges which still have much information. In FER, edges are thresholded by their subset magnitude represented in Fig. 2. In this case, thresholding can deal with blur as well as noise.

The membership function denotes the degree of edgeness relative to gradient magnitude and variance. This membership function can be viewed as a weight- ing coefficient which reflects the ambiguity in A. We define a possible membership function for the fuzzy subset close to "if ' as follows:

VA,(x) = kl/{1 + k2(x - p)2} (3)

where kl and k2 are constant values to decide the shape of the function, and this function signifies the degree to which that number is close to "p". k~ and k2 will be derived in Section 3.

80 T.Y. Kim, J.H. Han/Fuzzy Sets and Systems 100 (1998) 7747

T.hhmshold A

JI. ,~.¢,J ~,,1 II

(a)

Threshold ~

fl e,,,~l',,, fl

|

(b)

~ U I ~'II "~

~ -Edge set magnitude -'tic

~h

Fig. 2. Threshold in gradient curve: (a) Conventional threshold; (b) edge subsets and subset threshold.

• .............,........""'"""

%/q (a) (b)

I [-ti x

,

x

(c)

Fig. 3. Mapping function curves for fuzzy membership: (a) Initial fuzzy membership with standard deviation; (b) fuzzy membership function with bad symmetricity; (c) fuzzy membership function with good symmetricity.

3. Fuzzy edge representation

Because an image is locally degraded, the fuzzy membership of an edge should be a comparable index among all edges in an image, voi is a maximum value of a fuzzy subset Ai, and it may play a key role in comparing a fuzzy subset with others in an image• v0i is calculated by

Voi = {1 - (1 - ag/ai) 2} (4)

where 0 ~< (7g/(7 i ~ 1.

Eq. (4) is generated by the ratio of standard deviation (ag) of Gaussian convolution which was used for smoothing in Eq. (2) and standard deviation (ai) of subset A i. Gaussian normalization using members in Ai results in parameters, ai and ,//i as standard deviation and mean, respectively. Because ag is constant in an entire image, and also because ai is varied according to blur in subset Ai, voi is a comparable index for whole edges in an image• If an edge is affected only by smoothing and the value of O'g/O'i is close to 1, then the subset is not a fuzzy set any more. In this case, only the Gaussian normalization mean (#i) has

T.Y. Kim, J.H. HanlFuzzy Sets and Systems 100 (1998) 77~7 81

membership value 1 as a crisp set. After finding a maximum value of the fuzzy subset Ai, memberships of the rest are calculated by using the symmetricity and the membership function.

The symmetricity of a subset Ai is defined by

1 S; = - - ~ {[F(~g - t)l - [F(l~i +/ )1} 2 (5)

Mi (#iq-t)EAi

where t-- 1,2,3 . . . . and 54i = ~x~A, ]F(x)]. The value Si denotes the possibility of displace-

ment in the edge position. A large value represents the fuzziness of the edge position caused by asymmetric degradation, and a small value represents correctness of the edge position caused by symmetric degradation. IfS~ is close to 0, only the center point (#~) has fuzzy membership of v0/.

Combining voi with Si and substituting for constants in Eq. (3) gives

k3 2 ]

vA~(x) = if Voi ¢ 1 and S i ¢ O, (6)

Voi if Si = 0 and x = ~ti,

1 if Voi =- 1 and x = #i,

where x E Ai. Fuzzy membership function, defined in Eq. (6), is

inversely proportional to the square of the distance, which reduces the initial membership, voi. Note that kl in Eq. (3) is replaced by Voi, and k2 is replaced by k3/Si, k3 in Eq. (6) is a constant value in an image, and this decides sensibility against the neighboring edges. v~,(x) is little influenced by the distances from center when k3 has a small value.

The relation between the initial value v0~ and the standard deviation is shown in Fig. 3(a). This curve is emphasized for high membership. Figs. 3(b) and (c) show the effect of symmetricity. If the subset members are symmetrical, Si in Eq. (5) has a small value, its inverse (k3/Si) has a large value. This makes the Eq. (6) to have steep descent from the center as in the Fig. 3(c).

Fig. 4 shows the sequencing process of FER. It shows a 1-dimensional blurred intensity signal in Fig. 4(a). The signal is differentiated after it was convoluted by Gaussian for noise removal, which is shown in Fig. 4(b), and the signal has several subsets of edges. It is thresholded by subset magnitudes as in

(a)

~ __ / a t_ _

I I

(b)

(c)

A . " . ~ - . A

(d)

_ _ ~ 2 _ _ ~ _ _ _ A I

(e)

Fig, 4. Process of fuzzy edge representation: (a) Intensity curve; (b) gradient curve; (c) thresholding; (d) Gaussian normalization; (e) fuzzy edge representation.

Fig. 4(c) and Gaussian normalization curves (o'i,12i) of the subsets having sufficient evidence for edge are shown in Fig. 4(d). Finally, FER using Eqs. (5) and (6) is presented in Fig. 4(e). To accomplish edge set A in the two-dimensional image, subsets obtained by possible gradient directions are combined by fuzzy union operation.

4. Verification measures for fuzzy edge representation

To verify the effectiveness of FER, we use the index of fuzziness (IOF) as defined in Eq. (7). The IOF reflects the average amount of ambiguity (fuzziness) presented in a fuzzy subset by measuring the distance between its fuzzy property and the nearest two-level property.

IOF(Ai) = 2 Z min(vA,(x), 1 - vA,(x)). (7) xCAi

N is the number of elements in Ai and IOF(Ai) lies between 0 and 1.

IOF is a value that can justify the value of fuzzy membership function or the quality of image for

82 T Y. Kim, J.H. Han/Fuzzy Sets and Systems 100 (1998) 7 7 ~ 7

(a)

Table 1 Results of FER for each blurring model in Fig. 5

Blurring models Model 1 Model 2 Model 3

Si 0.004926 6 .413953 0.060345 ~ri 2.753435 3 .723615 6.968353 v0i 0.594464 0 .464990 0.266418

Gaussian IOF 0.481046 0 .457081 0.307003 Average fuzzy 0 . 2 6 9 0 2 0 0 .228540 0.153501 FER IOF 0.055363 0 .896359 0.171429 Average f u z z y 0 . 0 3 8 5 2 4 0 .448179 0.085714

(b)

(c)

(d)

(e)

AVA

Fig. 5. Comparison FER with crisp edges by blurting models: (a) Blurting models; (b) blurred objects; (c) cross-sections of blurred step edges; (d) real edges, crisp edges and FER; (e) cross-sections of real edges and FER.

processing. I f a subset has a large value o f IOF, that is, the edges in the subset have erroneous membership values ( = 0.5), the subset is hard to process because the members contain little information about edgeness. Consequently, the IOF can determine a fuzzy subset reasonable or not.

The shape of the boundary can be described quantitatively by moments which are popular in representation. Normalized central moments// jk are defined as follows:

Mjk= ~ Z xJykE(x'Y)' X y

F,j~ = ~ ~ ( ~ - ~ ) q y - ;,)~E(~, y), x V

Yc = Mlo/Moo, fi = Mol/Moo,

r = ( j + k ) / 2 + l , for j + k = 2 , 3 . . . .

//jk = ~j~/~;o (8)

where E(x, y) is a representation for moments, and these moments are invariant to translation and scale change.

From these normalized moments a set of Hu 's seven invariant moments ~b(1), ~b(2), ~b(3) . . . . . q5(7) is derived [7]. This set of moments is invariant to translation, rotation and scale change.

c~(1 ) = q2o +/ /02,

q~(2) = (//20 - / / 0 2 ) 2 -]- 4//2j,

~b(3) = (//30 - 3/ /12) 2 ~- (3//21 - / / 0 3 ) 2,

~b(4) = (//30 -[- / /12) 2 -Jr- (/721 -]- / /03) 2,

~ ( 5 ) = (//30 - 3q12)(/ /30 + / / 1 2 )

[(//30 q- q12) 2 -- 3(//21 q- //03) 2]

-+- (3//21 -- //03)(//21 -J- //03)

× [3(q30 + //12) 2 -- (//21 + //03)2],

~b(6) = (//20 - / / 0 2 ) [ ( / / 3 0 -]- / /12) 2 - (//21 - ] - / /03) 2]

@4//11(/ /30 -F- //12)(//21 -+" //03),

~b(7) = (3//21 -//o3)(//30 +//12)

× [(//30 +//12) 2 - 3(//21 +//03) 2]

-- (//'130 - - 3//12)(/121 q- //03)

× [3(//30 + //12) 2 -- (//21 + / / 0 3 ) 2 ] • (9)

T.Y. Kim, J.H. Han/Fuzzy Sets and Systems 100 (1998) 77~7 83

. . . . . . . . .

(a) (b) (c)

Fig. 6. Experiment of FER: (a) Original scene; (b) crisp edges; (c) fuzzy edges.

A problem in image processing is the detection of straight lines in digitized images. Hough replaced the original problem of finding collinear points by a mathematically equivalent problem of finding concur- rent lines. This method involves transforming each of the figure points into a straight line in a parameter space [3].

The stability of a representation is defined such that the representation does not change much with the effect of degradation or deformation. In other words, if two images are similar in some manner, then their representation should be similar too, and vice versa [4].

5. Experimental results

In the experiments, we use 0.005 for the sensibility constant (k3) in Eq. (6), 8 gradient directions, and 1 for the variance (ag) of the Gaussian smoothing. The conventional edge detection method for crisp edges is used by Canny's edge detector (a --- 1 and a single threshold) with small edge removal (less than 4 pixels) and linking. We assume the same degree of noise in all images.

Fig. 5 shows FER of a circle-shaped object which is blurred by various symmetric or asymmetric blurring models. Fig. 5(a) shows three kinds of blurring

84

Table 2 Hu moments and error

T. E Kim, J.H. Han/Fuzzy Sets and Systems 100 (1998) 77~87

4~(1) ~b(2) q~(3) ~b(4) ~b(5) ~b(6) ~b(7)

Tool 1

C-edge (bot.) 2.502761 0.066269 0.077719 0.094351 0.006202 -0 ,016916 -0.005179 C-edge ( top) 1.713168 0.105744 0.320809 0.373294 0.117352 0.079386 -0.054002 Error (el) 0.789593 -0 .039475 -0 .243090 -0 .278943 -0 .111151 -0.096303 0.048824 ~ e 2 0.785930

F-edge (bot.) 1.060594 0.027966 0.045179 0.002881 -0 .000017 -0.000228 0.000028 F-edge ( top) 0.910672 0.052855 0.009117 0.011451 0.000056 0.001468 0.000103 Error (el) 0.149921 -0.024888 0.036062 -0 .008569 -0 .000073 -0 .001697 -0.000075 Ze~ 0.024473

Tool 2

C-edge (bot.) 2.489556 2.902592 1.791827 0.204405 -0.102752 0.324451 -0.068882 C-edge ( top) 2.085227 2.057463 0.054409 1.754661 -0.407603 2.474763 -0.357480 Error (ei) 0.404329 0.845129 1.737418 -1.550257 0.304851 -2.150312 0.288598 ~e~ 11.099707

F-edge (bot.) 0.981603 0.625864 0.006925 0.026628 0.000358 0.020926 -0.000048 F-edge ( top) 0.976012 0.594582 0.008417 0.000692 0.000001 -0.000307 0.000001 Error (ei) 0.005591 0.031282 -0.001492 0.025936 0.000357 0.021233 -0.000050 ~e/2 0.002136

models which are presented as one-dimensional signals. The second blurring model is asymmetric and arbitrary in form, and the third blurring model is symmetric but has a large variance. When these models convolute with the step edges of a circle-shaped object in one dimension, the results are shown in Fig. 5(b). Fig. 5(c) shows two signals in each box. The lower signal is an intensity signal after the original edge was blurred by the models in Fig. 5(a), and the upper signal is a gradient signal obtained by differentiating the intensity signal. The crisp edges are detected by the conventional edge detection method with a constant threshold. A quarter of an image shown in Fig. 5(d) is obtained by superimposing the fuzzy edge with the real edge and the conventional edge. Real edges of the black circle are compared with the conventional edges denoted in white. FER of gray band includes real edges and has various initial memberships (the darker, the higher value of membership) according to blurring models. The cross-section of FER in Fig. 5(e), which is compared by real edge position (upper line), shows that FER can weight edge posi-

tions reliably by its peak. Since FER describes edges as circular region which includes real edges, we can get edge information selectively by a-cut or detection of local peaks according to applications.

Table 1 shows the IOF of each blurting model in Fig. 5. IOF of each blurring model is compared with Gaussian IOF which is obtained when we use Gaussian as membership function directly. As shown in Table 1, the average value of FER IOF is much lower than Gaussian IOF when the edges are predictable, and the IOF of FER reflects ambiguity when the edges are not predictable. Because blurring model 2 in Fig. 5 is asymmetrically degraded, the IOF must reflect this situation, and the edges which have much fuzziness are handled carefully in the high level processing.

A three-dimensional object can be focused on different points is shown in Fig. 6. When an object is focused on its top/bottom portion and defocused on its bottom/top portion, original images are shown in Fig. 6(a). Fig. 6(b) shows crisp edges detected by the conventional edge detection method with a con-

T.Y. Kim, J.H. HanlFuzzy Sets and Systems 100 (1998) 77-87 85

"0

LU

8

UJ

0.8

0.6

0.4

0.2

-0.2

-0.4

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

TOOL 1 'C-edge' 'F-edge' -4--,

I I I I I

2 3 4 5 6 7 Hu moment

, 0 0 , 2

\ A /

I I . . . . . . . . I

2 3 4 5 6 7 Hu moment

Fig. 7. Errors of both edge representations in Table 2.

stant threshold. FER describes defocused state as in Fig. 6(c). The focused portions are detected crisply, and defocused portions and the shadow are described opaquely. Instead of describing a binary edge, FER represents the uncertain edge as a fuzzy subset, and its maximum membership value describes the degraded state of an edge.

Table 2 illustrates the Hu moments of each representation in the objects of Fig. 6. The moments obtained by FER are not much varied by defocusing, but those of the crisp edges are very sensitive, and it indicates that FER is much more stable than the conventional one. The error distance is calculated by comparing differently focused images in each case. Fig. 7 shows the error distances graphically according to Hu moments in each edge representation and each object.

When we get an image to process an object, we have to focus on the target object. In real image, al- though we focus on the important object, there may be many redundant defocused objects in the image. The conventional edge detection method detects not only edges of the focused object but also edges of defocused objects with no discrimination. So it is very hard to process by these edges in high level processing such as Hough transform for line detection. In FER, we can distinguish edges of focused object from edges of defocused objects.

Fig. 8 shows an application that processes edges differently as their importance. The original image is taken by focusing on near box for detecting boundary of polyhedral object, and the image has local coordinates as shown in Fig. 8(a). The crisp edges detected by the

86 T. E Kim, J.H. Han/Fuzzy Sets and Systems 100 (1998) 77-87

UI X

~, ,~: -~- -~_1 ]1 ,

(a) (b) (c)

~ ~ ~ i : . . . . . . . . . . . . . . . . i~" ~ ; '[ i' ; ~. ~i- ,, . . . . . . . . . . ...... ~ /M, l,r~. 'vrU.~l ~}J ~ '" t{,~, if~{~ } 6"'.. " :~2L~ "" t ''~ ,,. .......... I'~L.~ • ~;~'~ j'~ i ~ i i! ~' II

• . . . . . . . . . . . . . . . . . . h - . - .J" / Ii 1 . . . . .

(d) (e) (f)

Fig. 8. Lines detected by Hough transform: (a) Original image; (b) crisp edges; (c) lines detected from crisp edges; (d) fuzzy edges; (e) local peak points of fuzzy edges; (f) lines detected from local peaks of fuzzy edges•

Table 3 Hough lines and votes

Edge Line Votes Edge Line Votes Distance Angle Distance Angle

Crisp 110 090 93 Fuzzy 110 090 102•5 168 168 81 205 010 64•5 112 003 68 164 167 64.2 080 089 64 132 168 63.8 040 092 62 092 086 54.6 201 008 59 041 135 53.8 130 085 59 137 094 48.5 195 045 54 147 074 47.0

conventional method are shown in Fig. 8(b), and high voted 4 lines detected by Hough transform by using edges in Fig. 8(b) are shown in Fig. 8(c). We can see the lines that are not in the focused object. Fuzzy edges are shown in Fig. 8(d), and local peak points of fuzzy edges are displayed in Fig. 8(e). Local peak points also have their own fuzzy memberships, and they are displayed differently by gray level. Dark pixel denotes high membership (little blurred) and light pixel represents low member-

ship (much blurred). After we have modified the voting mechanism of Hough transform to allow the real value of membership, we find high 4 lines by using edges in Fig. 8(e) and show resultant lines in Fig. 8(f). Since we use edges differently according to their memberships, Hough transform detects lines accurately.

Table 3 shows the results o f voting in Hough transform with high 8 lines' information. The line information is composed of the distance from the origin and angle to the x-axis o f coordinates in Fig. 8(a).

T. E Kim, ,1.14. HanlFuzzy Sets and Systems 100 (1998) 77~7 87

Because the votes of lines from crisp edges are much influenced by background edges, until we reach the 6th high vote of line, we cannot obtain the third fore- ground line. But lines from local peaks of fuzzy edges have very stable votes depending on edges of fore- ground or background.

6. Conclusion

An image obtained from an imperfect imaging system is blurred locally and asymmetrically along with additive Gaussian noise. Since the conventional edge detection algorithms intend to analyze noise, it is hard to overcome blurring degradation.

We proposed a representation of edges using the concept of fuzzy sets. It is derived by describing a gradient image, making fuzzy edge subsets, and finding edge existence. The fuzziness of edge caused by blurring is described by fuzzy membership function which represents the possibility of edge by using standard deviation of Gaussian normalization and the symmetricity. Experiments demonstrate that this FER is much more stable than conventional binary edge representation in dealing with local asymmetric blurring. We also indicate that FER can be a relative index without defining a specific model, and prevent error propagation from edge detection to high level processing.

FER can be used in most of the applications instead of the conventional binary edge representation, and it enables an application to deal with fuzziness without any additional information on such fields as pattern recognition and image matching by selectively using fuzzy membership values in FER.

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edge representation with fuzzy sets in blurred images

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