FIRE operators for image processing

r

ELSEVIER Fuzzy Sets and Systems 103 (1999) 265-275

FUZZY sets and systems

FIRE operators for image processing Fabrizio Russo*

D.E.E.I., University of Trieste, Via A. Valerio 10, Trieste, 1-34127, Italy

Received May 1998

Abstract

Fuzzy inference ruled by else-action (FIRE) operators are a class of nonlinear operators which process image data by using fuzzy reasoning. The latest developments in the field of FIRE operators are presented in this work focusing on two very important research and application areas: nonlinear filtering of noisy images and edge detection. First, a new family of filters for images corrupted by impulse noise is presented. Due to the adoption of piecewise linear fuzzy sets, the proposed approach is able to combine noise cancellation and detail preservation. A method for automatic generation of the fuzzy rulebase using the Genetic Algorithms is also presented. Then, a new class of noise-protected operators for edge detection is proposed. By suitably choosing fuzzy sets and fuzzy aggregation mechanism, these operators are able to detect edges in images corrupted by different noise distributions. Many experimental results are reported showing that the proposed operators perform significantly better than other techniques in the literature. (~) 1999 Elsevier Science B.V. All rights reserved.

Keywords: Pattern recognition; Image filtering; Edge detection; Fuzzy rules; Genetic algorithms

1. Introduction

In the last few years fuzzy technology has success- fully entered the area of low-level computer vision and it is becoming competitive with classical methods. In particular, focusing on nonlinear filtering and edge detection, many different approaches have been proposed demonstrating that fuzzy reasoning is a very powerful resource when uncertainty affects the process of extracting information from data corrupted by noise [2-4, 8-10, 12, 13, 15-17, 19, 24-28, 30, 31].

In this framework, fuzzy inference ruled by else- action (FIRE) operators are a family of nonlinear operators which adopt fuzzy rules to process image data.

* Tel.: +39-40-6763015; fax: +39-40-6763460; e-mail: rusfab@ univ.trieste.it.

Originally proposed in [16], the special structure of these operators has been progressively improved. A collection of FIRE operators is now available for a va- riety of image processing problems including detail- preserving smoothing of data corrupted by different noise distributions [18, 21,23], image sharpening [22] and edge extraction [20].

The aim of this paper is to present the state-of-the- art of the research work on FIRE operators. Focusing on nonlinear filtering of noisy images, we first present a new class of FIRE operators for the removal of impulse noise. These operators, called PWL-FIRE filters, are based on piecewise linear fuzzy sets whose shapes are dynamically adapted depending upon the local characteristics of the image. By adopting this design, a very effective cancellation of noise pulses can be obtained without degrading the quality of fine

0165-0114/99/$ - see front matter (~) 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0114(98)00226-7

266 F. Russo/Fuzzy Sets and Systems 103 (1999) 265-275

details and textures. A method for the automatic generation of the fuzzy rulebase is also described. Many results of computer simulations are reported demonstrating the effectiveness of the proposed approach and its ability to largely outperform a number of methods in the literature. The second part of the paper is devoted to FIRE operators for edge detection. Indeed, edge detection is a very important step in a complete image understanding system. In this respect it is rel- evant to be able to select object contours which are significant with respect to human perception, without being deceived by the noise which could be present in the data. After reviewing a current technique for edge extraction based on the FIRE approach, a new class of noise-protected operators is presented. The proposed FIRE operators effectively combine in the same structure fuzzy rules for detail-preserving smoothing and edge detection as well. As a result, a strong noise insensitivity can be obtained with respect to different noise distributions. This paper is organized as follows: Section 2 presents the new family of PWL-FIRE filters, Section 3 describes a method for the automatic generation of their rules, Section 4 shows a collection of experimental results, Section 5 focuses on edge detection in presence of noise, Section 6 presents the new class of FIRE operators with embedded filtering capabilities, and, finally, Section 7 reports conclusions.

2. FIRE filters based on piecewise linear fuzzy sets (PWL-FIRE filters)

A FIRE filter is a special fuzzy system based on I F - T H E N - E L S E fuzzy reasoning [18,21]. The op- eration is window-based: for each pixel of the noisy image to be processed, a set of neighboring pixels is considered. The FIRE operator processes this neighborhood information by using fuzzy rules in order to estimate a correction term which aims at cancelling the noise (THEN-action). If no rule is satisfied, the central pixel is basically left unchanged (ELSE-action).

More formally, let x(n) be the pixel luminance at location n = In1, n2] in the noisy image and let W(n) = {xj(n); j = 1 . . . . . 8} be the set of eight neighboring pixels which belong to a 3 × 3 window around x(n), as represented in Fig. 1. The input variables of the

X~ X2 X3

X8 x x,

x7 x 6 x5

Fig. 1. 3 × 3 window.

mLp~

0 a b L-1

Fig. 2. Piecewise linear fuzzy set LP.

operator are the luminance differences defined by

Axj(n) : xj(n) - x(n). (1)

The output variable A y ( n ) is the correction term which, added to x(n), yields the resulting luminance value y ( n ) = x ( n ) + Ay(n) . By using fuzzy rules, the operator nonlinearly maps the set of input variables to the output variable in order to yield a correction term which is able to remove noise pulses. In order to increase this effect, the filter is recursively applied to image data, i.e., the new value y(n) is assigned to the luminance x(n) at the end of the processing:

If we deal with images having L gray levels, input and output variables take values in the interval [ - L + 1 , L - 1]. Fuzzy sets for input variables are two piecewise linear fuzzy sets labeled large positive (LP) and large negative (LN). The membership function mLp of fuzzy set LP is shown in Fig. 2. The membership function mLN of fuzzy set LN is simply defined by

mLN(U) = rnLp(--U). (2)

Fuzzy rules deal with significant patterns of pixels in order to detect noise pulses [23]. As an example, let us consider the pattern formed by three pixel luminances x2,x4 and x6 in Fig. 1, i.e., the pattern briefly denoted by the set of indexes: {2, 4, 6}. Three corresponding luminance differences are evaluated

F. Russol Fuzzy Sets and Systems 103 (1999) 265-275 267

according to relation (1): Ax2 ---x2 - x , Ax4 = x 4 - - X

and Ax6 = x 6 --X. Due to the symmetry of the processing, a pair of fuzzy rules can be defined as follows:

IF (Ax2,LP) AND (Ax4,LP) AND (Ax6,LP)

THEN (AT, PO),

IF (Ax2,LN) AND (Ax4,LN) AND (Ax6,LN)

THEN (Ay, NE),

where PO (positive) and NE (negative) repre- sent singletons centered on L - 1 and - L + 1, respectively. It should be noticed that the above rules are designed to address negative and positive noise pulses, respectively. In general, more than one pattern must be considered in order to take care of many possible combinations of adjacent noisy pixels. For example, we could define a simple FIRE filter dealing with the following patterns: {X2,X4,X6}, {X4,X6,X8} , {X6,Xg,X2} and {xs,x2,x4}. These patterns are briefly identified by the sets of indexes A1 = {2,4,6}, A2 = {4,6, 8}, A3 = { 6 , 8 , 2 } and A 4 = {8,2,4}, respectively. For each pattern, a pair of fuzzy rules is defined as described in the previous example. The output Ay is numerically evaluated by the FIRE inference mechanism. Let us consider the general case of a FIRE filter dealing with N patterns defined by the sets of indexes AI,A2 . . . . . AN. The inference mechanism can be summarized as follows:

A y = ( L - 1)(21 - 22), (3)

where

21 = MAX{MIN{mLp(AX/), j E A i},

i = 1 . . . . . N}, (4)

22 = MAX{MIN{mLN(AXj ) , j E A i},

i = 1 . . . . . N}. (5)

The detail-preserving behavior of the filter mainly de- pends on the choice of fuzzy set parameters a and b (Fig. 2). Indeed, the particular shapes of fuzzy sets LP and LN aim at performing a full correction of noise pulses if their amplitude is large. In the presence of small amplitude pulses, on the contrary, the smoothing action is reduced in order to better preserve fine

a 2 a2

~o

Fig. 3. Fuzzy set MD.

details and textures. In particular, very effective results can be obtained by modulating the smoothing effect according to the local characteristics of the image. A simple method can be implemented as follows:

a = amaxmMD(X), (6)

where MD (medium) is a fuzzy set whose membership functions is shown in Fig. 3 (usually arnax = b). In this case, the detail-preserving mechanism is designed to affect pixels having luminance values in the "medium" range. It will be shown in the next section how fuzzy rules can be automatically generated from a collection of noisy data.

3. Generating the fuzzy rulebase of a PWL-FIRE filter using genetic algorithms

The proposed method for automatic generation of fuzzy rules is based on the genetic algorithms (GAs). As is well known, GAs are techniques which search for the optimal solution of a problem by applying the mechanisms of natural selection and natural genetics to a population of potential solutions [5, 6, 11 ]. Many applications of GAs to the training of artificial neural networks and self-learning fuzzy systems have been proposed in the literature. Indeed, very interesting fea- tures of GAs are robustness, a capability of addressing hill-climbing problems and no requirement for special constraints on the system structure. This feature, in particular, permits us to generate the rulebase of a FIRE filter without any change in its peculiar inference mechanism. An efficient method for encoding the fuzzy rulebase of a PWL-FIRE filter has been adopted. Let us consider the group of eight elements


Substring #1 Substring #M r r

01010100 • • • 1 . . . . • • 0

Fig. 4. Encoding of the fuzzy rulebase.

X1, X2, X3, X4, X5, X6, X7, X8 w h i c h f o r m t h e neighborhood of the central pixel x (Fig. 1). We can encode a pattern by means of a binary string of eight elements dl d2 d3 d4 d5 d6 d7 d8 defined as follows: if xj belongs to the pattern then dj = 1, else dj = 0. As an example, the pattern {x2,x4,x6} is encoded by the string: 01010100. If we exploit the symmetry of the processing, we can further increase the efficiency of the encoding process. In fact, we associate a given pattern with three corresponding patterns obtained by 90 °, 180 ° and 270 ° rotations. Thus, a binary string of eight elements can automatically encode a group of four patterns. Let us consider a binary string composed of M 8-bit substrings, as shown in Fig. 4. According to the above considerations, this string is able to encode 4M patterns. Since each pattern defines a pair of rules, the string can efficiently encode 8M fuzzy rules.

The method starts with a randomly generated population of strings (individuals) and produces the sub- sequent populations by using reproduction, crossover and mutation operators. A test image corrupted by impulse noise is used as input data. The individuals having the best fitness have more chances of being reproduced. A very simple way to evaluate the fitness is to adopt a payoff function F (object function) and we resort to the mean-square error (MSE) of the output data estimated with respect to the uncorrupted image:

1 (N1 - 2)(N2 - 2) F - - MSE X--,N ~ --2 N2 --2

Z-~nI=I ~ n 2 = l ( y ( n l , n z ) -- s ( n l , n 2 ) ) 2

(7)

where y(nl, n2) and s(nl, n2) denote the luminance values at location [nl, n2] in the filtered and in the uncorrupted image, respectively (n l - -0 . . . . . N1 - 1; n2 =0 , . . . ,N2 - 1).

More sophisticated choices may take care of aspects related to human visual perception [10].

4. Exper imenta l results

In order to show the performance of the proposed approach, we have performed some computer simulations. In the first experiment we have adopted as a training image a 256 × 256 version of the "Pentagon" picture having 256 gray levels (Fig. 5a) and we have corrupted this image by superimposing impulse noise with probability 0.18 (Fig. 5b). We have set the parameters of membership functions as follows: a0 = 128, al = 120, a2 = 40, b-- 127. We have chosen M = 3 (i.e. a string of 24 elements) and a small population of 20 individuals. The result yielded by the genetic algorithm after 14 genera- tions is shown in Fig. 5c. The corresponding binary string representing the fuzzy rulebase is: 11000101 11010000 10100111.

We have performed a second experiment in order to verify the ability of this rulebase to process different image data. For this purpose we have chosen the well-known 256 × 256 "Lena" picture (Fig. 6a) and we have produced a noisy version of it by using the same amount of impulse noise (Fig. 6b). The result yielded by the PWL-FIRE filter is shown in Fig. 6c (MSE ---- 30). It can be favourably compared to the re- suits yielded by some well-known nonlinear operators [14,29]: a 3 x3 median filter (Fig. 6d, MSE= 110), a 3 x 3 recursive median filter (Fig. 6e, MSE-- 109), and the 3 × 3 recursive weighted median filter (Fig. 6f, MSE ---- 126) defined by the following mask: ml 1 = m13 : - m31 ----: m33 = l , m12 -= m2l : - m23 = m32 =

2, m22 = 5.

The better performance of the fuzzy operator is clearly perceivable: unlike median operators, the new PWL-FIRE filter is able to yield a very effective noise cancellation without degrading the image structure.

Finally, the results yielded by the new fuzzy technique have been compared to those yielded by the very powerful, recently introduced SD-ROM operator [1]. For this purpose, we have considered a 512 x 512 version of the "Pentagon" picture and we have cre- ated five test images by superimposing impulse (salt and pepper) noise with increasing probabilities from 0.1 to 0.4. The resulting MSE values are reported in

F Russo/Fuzzy Sets and Systems 103 (1999) 265-275 269

Ca) (b) (c)

Fig. 5. Original test image (a); test image corrupted by impulse noise (b); result of the processing (c).

(d) (e) (t3

Fig. 6. Original test image (a); test image corrupted by impulse noise (b); results yielded by: the PWL-FIRE filter (c); the 3 × 3 median filter (d); the 3 × 3 recursive median filter (e); the 3 × 3 recursive weighted median filter (f).

270 F Russo/Fuzzy Sets and Systems 103 (1999) 265-275

Table 1 MSE values

Noise probability SD-ROM Filter PWL-FIRE Filter

0.10 26 9 0.18 41 21 0.26 60 37 0.33 87 61 0.40 130 93

Table 1. It can be observed that the PWL-FIRE operator is able to largely outperform the competing filter.

5. Designing a FIRE operator for edge detection

As shown in [20], FIRE operators can be designed to detect edges in noisy images. Let us consider a simple 3 × 3 operator whose input variables are defined by relation (1). Let medium positive (MP) be the fuzzy set whose membership function is represented in Fig. 7 and let medium negative (MN) be the fuzzy set defined by mMN(U) = mMp(-- U). In the proposed design we choose: c -- L/2. Let us suppose that bright pixels in the resulting image denote possible uniform regions and dark pixels denote possible object contours. Thus, a FIRE operator can be designed according to the following fuzzy reasoning: ifa pixel belongs to a border region then make it black, else make it white. A group of fuzzy rules for the detection of border regions could be expressed as follows:

IF (Axl,MP) AND (Ax2,MP) AND (Ax3,MP)

AND (Axs, MN) AND (Ax6, MN) AND (Ax7, MN)

THEN (y, BL),

IF (Ax2,MP) AND (Ax3,MP) AND (Ax4,MP)

AND (Ax6,MN) AND (Ax7,MN) AND (Ax8,MN)

THEN (y, BE),

IF (Ax3,MP) AND (Ax4,MP) AND (Ax5,MP) AND (Ax7,MN) AND (Axs,MN) AND (Ax1,MN)

THEN (y, BE),

IF (Ax4, MP) AND (Axs, MP) AND (Ax6, MP) AND (Axs,MN) AND (AxI,MN) AND (Ax2,MN) THEN (y, BE),

IF (Axs, MP) AND (Ax6, MP) AND (AXT, MP)

AND (AxI,MN) AND (Ax2,MN) AND (Ax3,MN)

THEN (y, BL),

IF (Ax6,MP) AND (AxT,MP) AND (Ax8,MP)

AND (Ax2,MN) AND (Ax3,MN) AND (Z~X4, MN ) THEN ( y, BL)

IF (Ax7, MP) AND (Axs, MP) AND (Axl, MP)

AND (Ax3,MN) AND (Ax4,MN) AND (Axs,MN)

THEN (y, BL),

IF (Axs,MP)AND (AxI,MP) AND (Ax2,MP)

AND (Ax4,MN) AND (Axs,MN) AND (Ax6,MN)

THEN (y, BL),

where AND denotes a fuzzy aggregator, y is the output luminance value and black (BL) is a singleton centered on zero. If no border region is de- tected, the following action is performed: ELSE (y, WH), where white (WH) is a singleton centered on L - 1 (the maximum luminance value). It can be observed that detection of edges is ac- tually performed by considering the following patterns: {Xl,X2,X3 }, {x2,x3,x4}, {x3,xa,xs}, {xa,xs,x6}, {xs,x6,x7}, {x6,xT,xs}, {XT,X8,Xl} and {x8,xl,x2}. As done in Section 2, we briefly identify such patterns by means of the corresponding sets of indexes: B1 = {1,2,3}, B2 = {2, 3,4}, B3 ~--- {3,4,5},

m ~

-c 0 3c

Fig. 7. Fuzzy set MP.

B1 B2 B3 B4

B5 B6 B 7 B s

Fig. 8. Patterns adopted for edge detection.

F Russo/Fuzzy Sets and Systems 103 (1999)265~75 271

B4={4,5,6}, B5={5,6,7}, B6={6,7,8}, B7 = {7, 8, 1}, B8 = {8, 1,2}. The patterns are graphically represented in Fig. 8.

If we choose the arithmetic mean as the fuzzy aggregator, the output y of the operator is yielded by the following relationship:

y = q ( L - 1)(1 - MAX{2i, i = 1 . . . . . 8}), (8)

where

2i = ~ mMp(Ax/) + ~_~ mMN(AX/) ,

jcB +4

i = 1,...,4, (9)

)~i+4 : ~ mMp(Axj) + Z mMN(AXj) .jCB,,,~ jCB,

i = 1 .. . . . 4 (10)

and q represents a scaling factor (typically q = 2). Ex- perimental results show that the fuzzy technique performs better than the well-known Sobel operator [7]. It should be observed that the arithmetic mean has been chosen as the fuzzy aggregator in relations (9) and (10) in order to make the operator less sensitive to noise. This is the only mechanism which aims at re- ducing the effect of the noise. A new approach to edge detection based on the integration of smoothing rules in the same operator will be shown in the next section.

6. A new class of noise-protected FIRE operators for edge detection

The new operator combines in the same structure rules for edge detection and noise cancellation. By suitably choosing fuzzy sets and fuzzy aggregation mechanism, the proposed operator (here called the L-FIRE operator) is able to detect edges in images corrupted by very different noise distributions. Let us consider a 3 x 3 window (Fig. 1) and'let us define the input variables by adopting relation (1)again. Rules deal with the pattems defined by the following set of indexes: Cl = {8, 1,2,3}, C2 = {2,3,4,5}, C3={4,5,6,7}, C4={6,7,8,1} and Cs={8,2}. These patterns are graphically represented in Fig. 9.

C l C 2 C 3 C4 C 5

Fig. 9. Patterns adopted by the L-FIRE operator.

For each pixel to be processed, the proposed method operates as follows. First, smoothing rules are acti- vated. These rules deal with the patterns identified by C1, C2, C3 and C4. If the image is corrupted by impulse noise we adopt fuzzy sets LP and LN (Section 2) for input variables, singletons PO and NE for the output variable and the minimum operator as the fuzzy aggregator. The group of rules is formally expressed as follows:

IF (Axs, LP) AND (Axl, LP) AND (Ax2, LP)

AND (Ax3, LP) THEN (Ay', PO),

IF (Ax2, LP) AND (Ax3, LP) AND (Ax4, LP)

AND (Axs,LP) THEN ( A y ' PO),

IF (Ax4, LP) AND (Axs, LP) AND (Ax6, LP)

AND (AxT, LP) THEN (A J , PO),

IF (Ax6, LP) AND (Ax7,LP) AND (Ax8,LP)

AND (Axe, LP) THEN (A J , PO),

IF (Axs, LN) AND (Axl, LN) AND (Ax2, LN)

AND (Ax3, LN ) THEN ( A y , NE ),

IF (Ax2, LN) AND (Ax3, LN) AND (Ax4, LN)

AND (Ax5, LN) THEN (AJ,NE) ,

IF (Ax4, LN) AND (Ax5, LN) AND (Ax6, LN)

AND (AxT, LN) THEN (AJ,NE) ,

IF (Ax6, LN) AND (Ax7, LN) AND (Axs, LN)

AND (Ax~,LN) THEN ( A J , NE).

The correction term Ay(n) is evaluated according to the above described PWL-FIRE inference mechanism:

~ x y ' = ( L - 1)(,V~ - , ~ ) ,

where

(11)

)/1 = MAX{ MIN{mLp( Axi), J E C,},

i = 1 . . . . . 4}, (12)

272 F. Russo /Fuzzy Sets and Systems 103 (1999) 265-275

b2 b2

0

Fig. I0. Fuzzy set ZE.

212 = MAX{ MIN{mL~q( Axj), j E Ci},

i = 1 . . . . ,4}. (13)

Then, the new value x(n) + Ay'(n) is assigned to the luminance x(n) in the input image. Edge detection is thus performed according to the following fuzzy reasoning: if a pixel belongs to a uniform region then make it white, else make it black. Such a reasoning is applied to the pattern identified by C5:

IF ((x - x2), ZE) AND ((x - x8), ZE)

THEN (y", WH) ELSE (y", BL),

where zero (ZE) is a trapezoid-shaped fuzzy set defined in the interval [ - L + 1, L - 1 ] and centered on zero (Fig. 10). The luminance y'(n) in the output image (representing the edge map) is evaluated by means of the following relationship:

y" = (L - 1)MIN{mzE(x - x2), mZE(X -- x8)},

(14)

where mzE is the membership function of fuzzy set ZE. It should be observed that the 3 × 3 window scans the input image from the upper-left comer to the bottom- right comer: thus, due to the recursive nature of the processing, the luminance values x2 and x8 in relation (14) are the results of the filtering action performed in the previous steps.

The proposed design is very simple and permits the easy integration of smoothing rules. If the input image is highly corrupted by noise, more rules for noise cancellation can be embedded in the same operator.

As mentioned above, the operator can be designed to deal with different noise statistics by changing fuzzy sets and aggregation mechanism in the group of eight

rules devoted to noise cancellation. Let the input image be corrupted by uniformly distributed noise in the interval [-An,A,]. In this case, we adopt fuzzy sets MP and MN (Section 5) for input variables by setting the parameter c as follows: c=An. We also choose the arithmetic mean instead of the minimum operator. Fuzzy singletons for the output variable are now centered on c and - c , respectively. The processing is very similar to the previous case, because the correction term Ay'(n) is yielded by the following relationships:

Ay ' = c(2,1, _ ,~,t-~2 J, (15)

where

2':=MAX( (~ -~mMP(Ax j ) I ' i=I \ jeci / . . . . . 4 / '

(16)

i: l ..... 4 }

(17)

In this case too, the luminance y"(n) in the output image is yielded by relation (14).

In order to show the performance of the new operator, some experimental results are reported in Figs. 11 and 12. In the first experiment, the "Lena" image has been corrupted by superimposing impulse noise with probability 0.1 (Fig. l la) . The result of the application of the L-FIRE operator is shown in Fig. 1 lb (a0=128, a1=120, a2=100, b=160 , b1=48, b2 = 8). For comparison, the result yielded by the Sobel operator is reported in Fig. 1 lc. It can be seen that the L-FIRE operator performs significantly better than the other non-fuzzy technique. In the second experiment, the "Lena" picture has been corrupted by using uniformly distributed noise in the interval [ -24, 24] (Fig. 12a). The result yielded by the fuzzy operator is shown in Fig. 12b (bl =52, b2= 12, c = 24). It can be favorably compared to the result yielded by the Sobel technique (Fig. 12c).

F. Russo/Fuzzy Sets and Systems 103 (1999) 265-275 273

(a) (b) (c)

Fig. 11. Test image corrupted by impulse noise (a), result yielded by the L-FIRE operator (b), result of the application of the Sobel operator (c).

(a) (b) (c)

Fig. 12. Test image corrupted by uniform noise (a), result yielded by the L-FIRE operator (b), result of the application of the Sobel operator (c).

7. Conclusions

The latest results of the research work in the field of FIRE operators have been presented. Focusing on nonlinear filtering of noisy images, a new class of rule-based operators called PWL-FIRE filters have been proposed. These operators adopt piecewise linear fuzzy sets whose shapes are dynamically adapted depending upon the local characteristics of the image. As a result, a very effective cancellation of noise pulses can be obtained while preserving the image details very well. A method for the automatic gener-

ation of the fuzzy rulebase has also been described. Focusing on edge detection, a new class of FIRE operators with embedded filtering capabilities has been presented. A key aspect of the proposed approach is the combination of fuzzy rules for noise cancellation and edge detection in the same structure. This design choice improves the noise insensitivity of a FIRE operator and makes it able to deal with different noise distributions too. Experimental results reported in the paper have shown that new FIRE operators are able to outperfom a number of widely adopted methods in the literature. This opens up new vistas to the application


of soft computing techniques in low-level computer vision.

Acknowledgements

This work has been partially supported by M.U.R.S.T.

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FIRE operators for image processing